J . fi. Ijto tLcL IN MEMORIAM FLORIAN CAJORI THR THEORY OF SUBSTITUTIONS AND ITS APPLICATIONS TO ALGEBRA. By dr. EUGEN NETTO, Professor of Mathematics in the Univeksity of Giessen. Revised by the Author and Translated with his Permission By F. N. cole. Ph. D., Assistant Professor of Mathematics in the f University of Michigan. ANN ARBOR, MICH,: the register publishing company. Ube f nlanb prees. \t PREFACE. The presentation of the Theory of Substitutions here given differs in several essential features from that which has heretofore been custom- ary. It will accordingly be proper in this place to state in brief the guiding principles adopted in the present work. It is unquestionable that the sphere of application of an Algorithm is extended by eliminating from its fundamental principles and its general structure all matters and suppositions not absolutely essential to its nature, and that through the general character of the objects with which it deals, the possibility of its employment in the most varied directions is secured. That the theory of the construction of groups admits of such a treatment is a guarantee for its far-reaching impor- tance and for its future. If, on the other hand, it is a question of the application of an aux- iliary method to a definitely prescribed and limited problem, the elab- oration of the method will also have to take into account only this one purpose. The exclusion of all superfluous elements and the increased usefulness of the method is a sufficient compensation for the lacking, but not defective, generality. A greater efficiency is attained in a smaller sphere of action. The following treatment is calculated solely to introduce in an elementary manner an important auxiliary method for algebraic inves- tigations. By the employment of integral functions from the outset, it is not only possible to give to the Theory of Substitutions, this operat- ing with operations, a concrete and readily comprehended foundation, but also in many cases to simplify the demonstrations, to give the various conceptions which arise a precise form, to define sharply the principal question, and— what does not appear to be least important — to limit the extent of the work. The two comprehensive treatises on the Theory of Substitutions which have thus far appeared are those of J. A. Serret and of C. Jordan. The fourth section of the " Alg^bre Supdrieure " of Serret is devoted to this subject. T^^e radical difference of the methods involved here and there hardly permitted an employment of this highly deserving work for our purposes. Otherwise Avith the more extensive work of Jordan, the "Traits des substitutions et des Equations alg^briques." Not only the new fundamental ideas were taken from this book, but it js proper to mention expressly here that many of its proofs and pro- IV PKEFACE. cesses of thought also permitted of being satisfactorily employed in the present work in spite of the essential difference of the general treat- ment. The investigations of Jordan not contained in the " Traits " which have been consulted are cited in the appropriate places. But while many single particulars are traceable to this "Traits" and to these investigations, nevertheless, the author is indebted to his honored teacher, L. Kronecker, for the ideas which lie at the foundation of his entire work. He has striven to employ to best advantage the benefit which he has derived from the lectures and from the study of the works of this scholarly man, and from the inspiring personal inter- course with him; and he hopes that traces of this influence may appear in many places in his work. One thing he regrets: that the recent im- portant publication of Kronecker, "Grundztige einer arithmetischen Theorie der algebraischen Grossen," appeared too late for him to derive from it the benefit which he would have wished for himself and his readers. The plan of the present book is as follows: In the first part the leading principles of the theory of substitutions are deduced with constant regard to the theory of the integral func- tions; the analytical treatment retires almost wholly to the background, being employed only at a late stage in reference to the groups of solvable equations. In the second part, after the establishment of a few fundamental principles, the equations of the second, third and fourth degrees, the Abelian and the Galois equations are discussed as examples. After this follows a chapter devoted to an arithmetical discussion the necessity of which is there explained. Finally the more general, but still elementary questions with regard to solvable equations are examined. Stkassburg, 1880. To the preceding I have now to add that the present translation differs from the German edition in many important particulars. Many new investigations have been added. Others, formerly included, which have shown themselves to be of inferior importance, have been omitted. Entire chapters have been rearranged and demonstrations simplified. In short, the whole material which has accumulated in the course of time since the first appearance of the book is now turned to account. In conclusion the author desires to express his warmest thanks to Mr. Y. N. Cole who has disinterestedly assumed th#task of translation and performed it with care and skijl, EUGEN KETTO. CllESSEN, 1892, TRANSLATOE'S NOTE. The translator has confined himself almost exclusively to the function of rendering the German into respectable English. My thanks are especially due to The Register Publishing Company for their gener- ous assumption of the expense of publication and to Mr. C. N. Jones, of Milwaukee, for valuable assistance while the book was passing through the press. F. N. COLE. Ann Arbor, February 27, 1892. TABLE OF CONTENTS. PART I. Theory of Substitutions and of Integral Functions. CHAPTER I. SYMMETRIC OR SINGLE-VALUED FUNCTIONS ALTERNATING AND TWO-VAL- UED FUNCTIONS. 1-3. Symmetric and single- valued functions. 4 . Elementary symmetric functions. 5-10. Treatment of the symmetric functions. 11. Discriminants. 12 . Euler's formula. 13. Two- valued functions; substitutions. 14 . Decomposition of substitutions into transpositions, 15. Alternating functions. 16-20. Treatment and group of the two- valued functions. CHAPTER II. MULTIPLE-VALUED FUNCTIONS AND GROUPS OF SUBSTITUTIONS. 22. Notation for substitutions. 24. Their number. 25. Their applications to functions. 26-27 . Products of substitutions. 28 . Groups of substitutions. 29-32 . Correlation of function and group. 34. Symmetric group. 35. Alternating group. 36-38 . Construction of simple groups. 39-40. Group of order p/. CHAPTER III. THE DIFFERENT VALUES OF A MULTIPLE-VALUED FUNCTION AND THEIR ALGEBRAIC RELATION TO ONE ANOTHER. 41-44 . Relation of the order of a group to the number of values of the corresponding function. Till CONTENTS. 45. Groups belonging to the different values of a function. 46-47. Transformation. 48-50. The Cauchy-Sylow Theorem. 51 . Distribution of the elements in the cycles of a group. 52. Substitutions which belong to all values of a function. 53. Equation for a p-valued function. 55. Discriminants of the functions of a group. 56-59. Multiple- valued functions, powers of which are single- valued. CHAPTEE ly. TRANSITIVITY AND PRIMITIVITY. SIMPLE AND COMPOUND GROUPS. ISOMORPHISM. 60-61 . Simple transitivity. 62-63. Multiple transitivity. 64 . Primitivity and non-primitivity. 65-67 . Kon-primitive groups. 68. Transitive properties of groups. 69-71. Commutative substitutions; self -con jugate subgroups. 72-73. Isomorphism. 74-76. Substitutions which affect all the elements. 77-80. Limits of transitivity. 81-85. Transitivity of primitive groups. 86. Quotient groups. 87. Series of composition. 88-89. Constant character of the factors of composition. 91 . Construction of compound groups. 92. The alternating group is simple. 93 . Groups of order p'^. 94. Principal series of composition. 95. The factors of composition equal prime numbers. 96 . Isomorphism . 97-98. The degree and order equal. 99-101 . Construction of isomorphic groups. CHAPTER V. ALGEBRAIC RELATIONS BETWEEN FUNCTIONS BELONGING TO THE SAME GROUP. FAMILIES OF MULTIPLE-VALUED FUNCTIONS. 103-105. Eunctions belonging to the same group can be rationally expressed one in terms of another. 106, Eamilies; conjugate families. 107. Subordinate families. CONTENTS. IX 108-109. Expression of the principal functions in terms of the subordinate. 110. The resulting equation binomial. 111 . Functions of the family with non- vanishing discriminant. CHAPTER VI. THE NUMBER OF THE VALUES OF INTEGRAL FUNCTIONS. 112. Special cases. 113. Change in the form of the question. 114-115 . Functions whose number of values is less than their degree. 116. Intransitive and non-primitive groups. 117-121. Groups with substitutions of four elements. 122-127 . General theorem of C. Jordan. CHAPTER VII. CERTAIN SPECIAL CLASSES OF GROUPS. 128. Preliminary theorem. 129. Groups i2 with r = w — p. Cyclical groups. 130. Groups i2 with r = n=p, q. 131. Groups Q with r = n=p'^. 132-135. Groups which leave, at the most, one element unchanged.— Metacyclic and semi-metacyclic groups. 136. Linear fractional substitutions. Group of the modular equations. 137-139. Groups of commutative substitutions. CHAPTER VIII. ANALYTICAL REPRESENTATION OF SUBSTITUTIONS. THE LINEAR GROUP. 140. The analytical representation. 141 . Condition for the detining function. 143. Arithmetic substitutions. 144. Geometric substitutions. 145. Condition among the constants of a geometric substitution. 146-147. Order of the linear group. PART n. Application of the Theory of Substitutions to the Algebraic Equations. CHAPTER IX. the equations of the second, third and fourth degrees. — group of an equation. — resolvents. 148. The equations of the second degree. X CONTENTS. 149 . The equations of the third degree. 150. The equations of the fourth degree. 152 . The general problem formulated. Galois resolvents. 153-154 . Affect equations. Group of an equation. 156. Fundamental theorems on the group of an equation. 157 . Group of the Galois resolvent equation. 158-159. General resolvents. CHAPTER X. THE CYCLOTOMIC EQUATIONS. 161. Definition and irreducibility. 162 . Solution of cyclic equations. 163. Investigation of the operations involved. 164-165. Special resolvents. 166 . Construction of regular polygons by ruler and compass. 167. The regular pentagon. 168 . The regular heptadecagon. 169-170 , Decomposition of the cyclic polynomial. , CHAPTER XI. THE ABELIAN EQUATIONS. 171-172. One root of an equation a rational function of another. 173. Construction of a resolvent. 174-175. Solution of the simplest Abelian equations. ' 176 . Employment of special resolvents for the solution. 177 . Second method of solution. 178-180. Examples. 181. Abelian equations. Their solvability. 182. Their group. 183. Solution of the Abelian equations; first method. 184-186. Second method. 187 . Analytical representation of the groups of primitive Abelian equations. 188-189. Examples. CHAPTER XII. EQUATIONS WITH RATIONAL RELATIONS BETWEEN THREE ROOTS. 190-193 . Groups analogous to the Abelian groups. 194. Equations all the roots of which are rational functions of two among them. 196. Their group in the case n=p. 197 . The binomial equations. CONTENTS. XI 199. Triad equations. 200-201 . Constructions of compound triad equations. 202 . Group of the triad equation for n = l. 203-205. Group of the triad equation for 7i = 9 206 . Hessian equation of the ninth degree. CHAPTER XIII. THE ALGEBRAIC SOLUTION OF EQUATIONS. 207-209. Rational domain. Algebraic functions. 210-211. Preliminary theorem. 212-216. Roots of solvable equations. 217. Impossibility of the solution of general equations of higher degrees. 218. Representation of the roots of a solvable equation. 219. The equation which is satisfied by any algebraic expression. 220-221 . Changes of the roots of unity which occur in the expres- sions for the roots. 222-224 . Solvable equations of prime degree. CHAPTER XIV. THE GEOUP OF AN ALGEBRAIC EQUATION. 226. Definition of the group. 227 . Its transitivity. 228. Its primitivity. 229. Galois resolvents of general and special equations. 230. Composition of the group. 231 . Resolvents. 232-234. Reduction of the solution of a compound equation. 235 . Decomposition of the equation into rational factors. 236-238. Adjunction of the roots of a second equation. CHAPTER XY. ALGEBRAICALLY SOLVABLE EQUATIONS. 239-241 . Criteria for solvability. 242 . Applications. 243 . Abel's theorem on the decomposition of solvable equations. 244. Equations of degree p^; their group. 245. Solvable equations of degree _p. 246. Solvable equations of degree _p^ 248-249 . Expression of all the roots in terms of a certain number of them. EKKATA. p. 7, footnote, for transformatione read transmutatione. p. 15, line 10, read (p = S s/~^ . p. 16, line 5, read s^i — tp^ — ^S^s/ A. p. 28, line 9, for cp read ^. p. 29, line 12, for ^read ^. p. 29, line 9, from bottom, for 4^ read ^. p. 31, line 8, read G — [1, {x^xi) {x^x^, {xxX^ {x^x^, ix^x^ {^i^^^ p. 41, line 7, for p read p-^. p. 52, line 13, read aip^-\-\)il,\. p. 52, line 5, from bottom, for -— read — . da. Cia p. 89, line 2, for not more read less. p. 93, line 9, for a group H read a primitive group H. p. 94, line 2, for n — q-\-2 read n — q-\-k. p. 98, line 19, for §J',§, read §«§^(t,. p. 101, line 3, for ^2, read Q\. p. 103, line 14, read [1, (2:12:2)]. p. 125, Theorem XI, read: In order that there may be a pp-valued function / a prime power /f of which shall have p values, etc. p. 159, lines 10, 11 from bottom, read: Since ^3^2 » • • • p. 174, line 2, for Cj read 2ci . p. 210, foot note, for No. XI, etc., read 478-507, edition of Sylow and Lie. p. 219, line 10, for ^{cos a) read ffi(cos a). p. 224, line 2 from bottom, read: which leaves two elements with successive indices unchanged. p. 248, line 3 from bottom, read V^%V = ^a + ^(Va+2, • . •)• PART I. THEOKY OF SUBSTITUTIONS AND OF THE INTEGKAL FUNCTIONS. CHAPTER I. SYMMETRIC OR SINQLE - VALUED FUNCTIONS. ALTERNA- TING AND TWO-VALUED FUNCTIONS. § 1. In the present investigations we have to deal with n ele- ments ^1, X., . . . x„, which are to be regarded throughout as entirely- independent quantities, unless the contrary is expressly stated. It is easy to construct integral functions of these elements which are unchanged in form when the x^^s are permuted or interchanged in. anyway. For example the following functions are of this kind: ^i" + 0-2* + a?/ -f . . . + x^", x^'^ x./ -\- Xi'^ x./ -\- . . . -{- x.^ 0?!^ -1- a?/ ^^3^ -f- ... + X,'' Xi^ ... + x,," x^,,_i, [x^ — x,)^ (xi — x^y {x.2 — x^Y . . . (x„_i — x„y, etc. Such functions are called symmetric functions. We conjQne ourselves, unless otherwise noted, to the case of integral functions. If the Xx's be put equal to any arbitrary quantities, a^ , a2) • • -(^m ?o that Xj = ttj, Xo = a.y, ... x„ = a„, it is clear that the symmetric functions of the x^'s will be unchanged not only in form, but also in value by any change in the order of assignment of the values a^, to the X\S. Such a reassignment may be denoted by X = tti , Xo = cii , ... Xn = a, where the a,^ , a,-^ , . . . denote the same quantities a^, a2, ... in any one of the possible n ! orders. Conversely, it can be shown that every integral function, ^(Xj, X2 . . . x„) , of n independent quantities x^, x^, ... x,,, which THEORY OF SUBSTITUTIONS. is unchanged in value by all the possible permutations of arbitrary- values of the X\S, is also unchanged in form by these permutations. Theorem I. Every single-valued integral function of n independent elements x^, a?2, ... x,^ is symmetric in these elements. § 2. The reasoning on which the proof of this theorem is based will be of frequent application in the following treatment. It seems, therefore, desirable to present it here in full detail. (A). If in the integral function A=n <1) / (.t) =2 ttA a;' /. = o all the coefficients a^ are equal to zero, then / (x) vanishes identi- cally, i. e., f(x) is equal to zero for every value of x. Conversely, if f{x) vanishes for every value of x, then all the coefficients a\ are «qual to zero. For if f{x) is not identically zero, then there is a value C^ guch that for every real x of which the absolute value j .t | is greater than Co , the value of the function f{x) is different from zero. For Co we may take the highest of the absolute values of the several roots of the equation f{x) = 0. Without assuming the existence of roots of algebraic equations, we may also obtain a value of Co as follows:* Let % be the numerically greatest of the n coefficients Go, cii, . . a„_i in (1), and denote a„\ by r. We have then Ctn- X" + . . . + f/o < (h < ^+i.^r^+...-M) (r-l) Hence, for any value of x not lying between —1 r and + r, < i^r SO that the sign of f{x) is the same as that of a„ x". Consequently, we may take Co = ^*. a,, _,x"- ^ + a„_2a:"-^+ . ..+a. a,, *L. Kronecker. Crelle 101, p. 347. SYMMETRIC AND TWO-VALUED FUNCTIONS. 6 (B). If no two of the integral functions (2) ./, (4 /.(*),... /.,(*) are identically equal to each other, then there is always a quantity Xo such that for every x the absolute value of which is greater than Co, the values of the functions (2) are different from each other. For, if we denote by i\p the value determined for the function fa {^) — //3 {x)i as r was determined in (A), we may take for Co the greatest of the quantities Va^. (C). If in the integral function /(.rj, X,, . . . X,) = S «Al X2. . . \>n Xi^^ x^2 . . . xj>» all the coefficients a are equal to zero, then the function / vanishes identically, i. e., the value of / is equal to zero for every system of values of x^ , x., , . . . x„. To prove the converse proposition we put (3) x^, = g, a:, = gv, x,=:gi>\ , , . x„ = gi-"'-' a^i^^ .T> . . . .r,/'" then becomes a power of g, the exponent of which is ^Ai A2 . . . A.- = ^1 + ^-^ + ^;)>' + . . . Ky-' From (B), we can find a value for v such that for all greater val- ues of V, the various rxi X2 • • • Am are all different from one another. We have then / (^1, OC2, ... X,,) = 2 aAlA2 . . . \,n fy'"^^ ^i'" ^n But, from (A), if all the coefficients a do not disappear, we can take g so large that / is different from 0. The converse proposition is then proved. (D). If a product of integral functions (4) /i (^1 , ^2 ) • • • '^n) J 2 (^1 ) ^2 ) • • • X„) . . . f^i {Xi , X2, . . . X,^) is equal to zero for all systems of values of the x^^s, then one of the factors is identically zero. For, if we employ again the substitution (3), we can, from (C), select such values ga and v/^for any factor /« {x^, x^, ... ir„) which does not vanish identically, that for every system of values which arises from (3 ) when g > ga and > > v^ the value of /« is different from zero. If then we take g greater than gx, gii - - - Q,n and at the same time > greater than v,^ , ^2 , ... ^,n we obtain systems of values 4 THEORY OF SUBSTITUTIONS. of the Xxs for which (4) does not vanish, unless one of the factors vanishes identically. The proof of Theorem I follows now directly from (C). For if

^y*. 1 ? tAyi 1 t4^9 *^ 3 9 • • • tA^I «A-9 *^3 • • • '^'A 5 • • • and the function Ci* Ca^ c^"^ - . . has for its highest term ^^a + ^ + V + . . . ^2^ + V + . . . a^gV + • . ._ In order, therefore, that the highest terms of the two expressions, Ci* Co^ Cg'y . . . and c^^-' c^^' c-^' . . . may be equal, we must have /5 + r+ ... = /?'+/+ ... r + . . . = r' + . . . that is, a = a', /? = y9', ^ = ^', . . . It follows that two different systems of exponents in c^ c.,^ c-i' . . , give two different highest terms in the Xv's. Again it is clear that x^^X2^x;i .,. ( « i /5 > r = ^ . . .) is the highest term of the expression c^°-~^ c<^~ '^ c^~^ . . . and that * Deinonstratio nova altera etc. Gesammelte Werke III, § 5, pp. 37-38. Cf. Krou- ecker, Monatsberichte der Berliner Akademie, 1889, p. 943 seq. 6 THEORY OF SUBSTITUTIONS. all the terms in the expansion of this expression in terms of the e^^'s are of the same degree. § 6. If now a symmetric function S be given of which the highest term is A a-i*^ x/x^y x^^ . . . ( « = /5 > ^ > o\ . . ) the difPerence S— A c,--^ c/-v c,y-^ ... =S, will again be a symmetric function ; and if, in the subtrahend on the left, the values of the Ca.'s given in (7) be substituted, the highest term of S will be removed, and accordingly a reduction will have been effected. If the highest term of S^ is now A^ x{^' x.f' x^y' x^^' . . . , then s. — A.c.'^'-^'cf-y'c/-^'... = s, is again a symmetric function with a still lower highest term. The degrees of S.2 and S^ are clearly not greater than that of S, and since there is only a finite number of expressions Xi^ x^}^ Cfg^^ . . . of a given degree which are lower than x^°- x.f x^y . . . , we shall finally arrive by repetition of the same process at the symmetric function ; that is S, - A, c.«<^> - ?« c/''->'- vW . . . = ; and accordingly we have S = A^ Ci--^c/-Y ... + A2Ci«'-^'c/'-v'... + ... § 7. It is also readily shown that the expression of a symmetric function of the cca's as a rational function of the Cx's can be effected in only one way. For, if an integral symmetric function of x-^, x^-, ... x^ could be reduced to two essentially different functions of c-^, c^, ... c„, 9? (Cj, C2, . . . c„) and v'' (ci, C2,. . . c„), then we should have, for all values of the a-^'s, the equation

^ ) B) s, — CiS,._i -{-c,s,._2 — . . . + ( — 1)^ r c,.— {r ^n) These two formulas can be proved in a variety of ways. The formula A) is obtained by multiplying the right member of (6) by x^^^, replacing a?by a?x, and taking the sum over / = 1, 2, . . . w . The formula B ) may be verified with equal ease as follows. If we represent the elementary symmetric functions oi X2, x^, . . . x^ by c/, Co', . . . c'„_i, we have ^1 = »^1 -f" Cj J C2 ^ Xi Ci ~t~ C2 5 C3 := CC] C2 -p C3 , . . . and accordingly, if r ^ n, we have x/ — c, x/^' + C2 o^/-^— . . . (— l)^c, . = ^1^— (^1 + c/) ^1^-^ + (a^c/ +C2')^/- — . . . + (-1/ {x,c\._,-^ cj) = (-l/c/ and hence, replacing x^ successively hj x^, x^, . . . x^, and, corres- pondingly, c,/ by c/', c/", . . . c/"\ and taking the sum of the n resulting equations s,. — CiS,._i + C2S,._2 — . . . ( — lyc,. n = ( — 1 )r (c/ + cj' 4- c,!" + . : . + c/" ). * Newton: Arith. Univ., De Transformatione Aequationum. O THEORY OF SUBSTITUTIONS. The right member is symmetric in cci , X2,. . . x„, and contains all the terms of c,. and no others. Moreover, the term x^ x., . . . x,. , and consequently every term, occurs 71 — 7' times. Accordingly we have s, — CiS,_i + CoS,_o— . . . -i- {—ly c,n = {—ly {n—r)c,, .'. s,. — Cj s,. _ 1 + C2 s,. _2 — ... ( — 1 )'' c,. r = 0, and formula B) is proved. * The formula A) can obviously be veri- fied in the same way. § 9. The solution of the equations A) and B) ior the successive values of the s^s gives the expressions for these quantities in terms of the Cx's. The solution is readily accomplished by the aid of determinants. We add here a few of the results. f C) So = n 02 Cj ^^2 S3 = Ci^ ' dCiC2 + SCg 54 = Cj^ — 4ci^C2 + 4C1C3 -\- 2c2^ — 4iC^ ■ 55 = Ci^ — 5ci^C2 + 5ci% + ^CiCo^ — ^c^c^ — 5C2C3 + 5C5 It is to be observed here that all the c^'s of which the indices are greater than n are to be taken equal to 0. This is obvious if we add to the n elements x^, x^, ... x,^ any number of others with the value 0; for the c^'s up to c„ will not be affected by this addition, while c„+i, c„+2, ... will be 0. § 10, The observation of § 5 that Cj* Cg^ c^y . . . gives for its highest term a^i* + ^ + >' + ••• a?2^ + "^ + ••• x^y,-^,--; can be employed to facili- tate the calculation of a symmetric function in terms of the C\S. We may suppose that the several terms of the given function are of the same type, that is that they arise from a single term among them by interchanges of the X\S . The function is then homogeneous; suppose it to be of degree j>. We can then obtain its literal part at once. For, if the function contains one element, and consequently all elements, in the m^^ and no higher power, then every term of the corresponding expression in the c^'s will be of degree m at the high- est. For, in the first place, two different terms Ci<^ c/ c^y . . . and ♦Another, purely arithmetical, proof is given by Euler; Opuscula Varii Argumenti. Demonstrat. genuina theor. N«ewtoniani, II, p. 108. + C/. FaA di Bruno: Formes Binalres. SYMMETRIC AND TWO-VALUED FUNCTIONS. 9 Cj*' c.f c/ . . . give different highest terms in the a^x's, so that two such terms cannot cancel each other; and, in the second place, €j;^€.f c^y ... gives a power X}^'^ + ^ + y +••-, so that « + A^ + r + . . . i m. Again the degree of x^"" -^ ^ + y + ••• x.,^^ ^ -^ ••• x^y -^ •.••. . is a + 2 ,? + 3 r + . . . = ^ and since the given expresssion is homogeneous, the sum u-\-2l^-\-dy-\- ... must be equal to v for every term c^^ c./ e^y . . . These two limitations imposed on the exponents of the c^'s that « + /5 + r + . . . £ ^n, a + 2 ,5 + 3 r . . . = ^ exclude a large number of possible terms. The coefficients of those that remain are then calculated from numerical examples. The quantity a-^2l3-\-Sy-\-... is called the ivelght of the term ^1** ^2^ CgV . . . and a function of the c^^s whose several terms are all of the same weight is called isobarlc. For example S (xi^ X2^ xi a?4 ) = go^T + gi Cg c^ + g., % c, + q^ c^ c^ (m = 2, v = 7) S [Soc, — x,)\x, — x,)'(x^ — Xiy] ^^oCfi + giCsC, -\-q2C^c, + ^3 G, (^x + ^4 C3' + gs C3 C2 Ci + q^ C3 Ci' 4- q, c/ + gs c/ c{ (m = 4, V = 6) where the g's are as yet undetermined numerical coefficients. In the second example we will calculate the g's for the case n = 3, for which therefore c^ = %=Gi^ — 0. It is obvious that for different values of n the coefficient g's will be different. Taking I. a?i =: 1, X2 — — 1, x.^ =: 0, we have c^ = 0, Cg = — 1, c^ — O .••S = 4 = -3,; • 3: = -4. II. Xi = X2 = 1, a?3 = 0, . Cj = 2, C2 = 1, Cg = .-. S = = — 4 + 4^3; gs^l. III. Xi = X2= 1, x^ = — 1, Ci = 0, 0.2 — — 3, C3 = — 2 .-. S ==0 = 4g, +4 -27; g, = — 27. lY. Xi = Xo = 2, x^ = — 1, Ci = 3, C2 = 0, Cg = — 4 .♦. S = = — 2716— 4-27g6; gfi = — 4. V. a?i =^ a?2 = a?3 = 1, Cj = 3, C2 = 3, Cg = 1 .-. S = = — 27 + 9 gg — 135; q, ==18. + 18 Cg C2 Gi 4C3 Ci^ 4:C.2^ + 62^ Cjl 10 THEOKY OF SUBSTITUTIONS. This expression, (x^ — XoY { X2 — x-^y^x^ — x^Y = A, is called the discriminant of the quantities x^, x^, x^. The characteristic property of this discriminant is that it is symmetric and that its vanishing is the sufficient and necessary condition that at least two of the Xx^ are equal. § 11. In general, we give the name "discriminant of n quanti- ties Xi, a?2, ... a?„"to the symmetric function of the x^^ the van- ishing of which is the sufficient and necessary condition for the equality of at least two of the Xx^ . If a symmetric function S of the XxS is to vanish for x^ = x.2, it must be divisible by x^ — a?2, and consequently by every difPerence Xa — x^. Suppose S = {Xi — X2) Si. Now S, and consequently (x^ — x.2) S^ , is unchanged if x^ and X2 be interchanged. But this changes the sign of x^ — x., and there- fore of S^ . Consequently S^ vanishes if x^ = X2, and accordingly S^ contains x^ — a?2 as a factor. The symmetric function S is therefore divisible by (x^ — Xo)^ and consequently by every (Xa — ^^Y', that is it is divisible by J = \\ (xx — x^Y {^^ < /^-; ^ =1,2, ... n — 1; ,a = 2, 3, . . . 71) AU. ~~ V^^l '^2) \'^1 ^^3) \'^1 "^ij • • • V*^! '^nj (8) (a?2 — Xsf (X2 — x.f. .. (X2 — x„f This quantity A already satisfies the condition as to the equality of the XxQ , and, being the simplest function with this property, is itself the discriminant. It contains ^n{n — 1) factors of the form (xx—Xf^)"^', its degree is n{n — 1), and the highest power to which any Xx occurs is the (ii — 1)^^. It is the square of an integral, but, as we shall presently show, unsymmetric function, with which we shall hereafter frequently have to deal. § 12. Finally we will consider another symmetric function in which the discriminant occurs as a factor. Let the equation of which the roots are x^, x^, . . .x^ be, as before, f(x) = 0. Then if we write SYMMETRIC AND TWO- VALUED FUNCTIONS. 11 we have, for all values >^. = 1, 2, . . . n, the equation f'{x^) = {xx — a?j) (xx —X2}... {xx — ^A_i) (a?A — a^x + 1) . . . {x;, — x,,y We attempt now to express the integral symmetric function in terms of the coefficients Cj , Co , . . . c„ of / ( x), Every one of the 71 terms of S is divisible by Xi — a'2, since either /'(xj) or /'(;r2) occurs in every term. Consequently, by the same reasoning as in § 11, S is divisible by {x^ — x._,y, and therefore being a symmetric function, by every {Xa — XpY, that is by ^ =TT(«^A — ^,^y (^ < /^-; A = 1, 2, ... 71 — 1; /. = 2, 3, . . . ny S is therefore divisible by the discriminant off{x), i. e., by the dis- criminant of the n roots of f (x). Now f {x\) is of degree of n — 1 in Xa and of degree 1 in every other Xfj,', and therefore ^Y • f'i^-z)' /'(a) • . . /'(^«) is of degree a + n — 1 in x^ x.2^ . /'(^i) . f'{x^) . . . f'{x„) is of degree 2n — 3 in a?i . Consequently, if a S is of degree 2 n — 3 in Xj , while A is of degree 2n — 2 in x^. But since J is a divisor of S, it fol- lows that S is in this case identically 0. (9) S [.r,». /'(X3) . fix,) . . . f'(x,.)} = 0, (« < n - 1.) Again, if a = n — 1, then S and J can only differ by a constant factor. To determine this factor we note that the first term of S is of degree 2 n — 2 in a^j , while all the other terms are of lower degree in x^. The coefficient of x^^"^^ is therefore y J-) {X2 x^) , . . {X2 Xn) \x^ X2) . . . [x^ x,t) . . . n(n-l) {X, — Xo) . . . {X„ — X„ _ 1) = (— 1) 2 {X2 — X^f (X.2 — X,y . . . In J the coefficient of a"i^"~^ is n(n — l) The desired numerical factor is therefore ( — 1) 2 and wa have 12 THEORY OF SUBSTITUTIONS. n(n-l) (10) S[x-\f{x,),r{x,)...f'{x,^)-] = {-l)^- J. Formulas (9) and (10) evidently still hold if we replace x^'^ or ^i''~^ by any integral function c? (x) of degree a ^ n respectively. Moreover since n(n—\) {-'^)~^~ ■^=nx,).f'{x,)...f'{x.,) ■we have p) 2^ ^ = or 1*, ^a) according as the degree of c is less than or equal to n — 1. § 13. If an integral function of the elements a*! , a:, , . . . a^„ is not symmetric, it will be changed in form, and consequently, if the Xxs are entirely independent, also in value, by some of the possi- ble interchanges of the a-^'s. The process of effecting such an inter- change we shall call a suhstitution. Any order of arrangement of the Xx8 we call a permutation. The substitutions are therefore operations; the permutations the result. Any substitution whatever leaves a symmetric function unchanged in form ; but there are other functions the form of which can be changed by substitutions. For example, the functions I Xi^ X2^ -}- X^^ X^^, Xi X2 Xz -\- X^Xr^-\- Xf,, X^ -f" X.2 + '^'i take new values if certain substitutions be applied to them; thus if iPj and X.J be interchanged, these functions become JLJ. tt/j ~J~ X2 ~j~ X^ X^ , iCj t4?2 «^3 '|~ '^Sj^o ~\ "^6 ) 2 1" 1 ~r" 3 ' The first two functions are unchanged if x^ and x^ be inter- changed, the second also if x^ and x^ be interchanged, etc. Functions are designated as one- two-, three-, ??i-valued according to the number of different values they take under the operation of all the n! possible substitutions. The existence of one- valued functions was apparent at the outset. We enquire now as to the possibility of the existence of two-valued functions. In § 11 we have met with the symmetric function J, the dis- criminant of the n quantities cCi, a'2, ... x„. The square root of J is also a rational integral function of these n quantities : *Tlie formula D is due to Euler; Calc Int. II § 116i?. SYMMETEIC AND TWO-VALUED FUNCTIONS. IS /y/J = (a?i — X2) {Xi — 0:3) {xi — x^) . . .{xi — x„) (x^ -- x^) (a?2 — Xi). . . {x, — Xn) \Xn — i ^n)' Every difference of two elements x^ — x^ occurs once and only once on the right side of this equation. Accordingly if we inter- change the aVs in any way, every such difference still occurs once and only once, and the only possible change is that in one or more cases an Xa — ^^ may become x^ — Xa. The result of any substitution is therefore either + V-J or — \^ J,i. e., the function V-^ is either one-valued or two-valued. But if, in particular, we interchange a?i and X.2 , the first factor of the first row above changes its sign, while the other factors of the first row are converted into the correspond- ing factors of the second row, and vice versa. No change occurs in the other rows, since these do not contain either ^1 or x., . Since then, for this substitution, \/ J becomes — \/ -^ > it appears that we have in V'-' 3- two-valued function. This function is specially characterized by the fact that its two values only differ in algebraic sign. Such two- valued functions we shall call alternating functions. Tlieoreiii III. The square root of the discriminant of the n quantities x^, x.j, ... Xn is an alternating function of these quanti- ties. § 14. Before we can determine all the alternating functions, a short digression will be necessary. xln interchange of two elements we shall call a transposition. The transposition of Xa and x^^ we will denote by the symbol {Xa. x^). We shall now prove the following Tlieoreni IV. Every substitution ca7i be replaced by a series of transpositions. Thus, if we have to transform the order x^, X2, x^, . . . x,^ into the order X[^, Xi„, x,:^, . . . xi,^, we ^apply fi rs t the tran sposition (a'l Xi^). The order of the x^s then becomes aj^^, X2, x^^ . . . xi^ — i iTj , Xi^ + I, ... x„ , and we have now only to convert the order a'2 . . . Xi^ _ 1, a-i , Xi^ + I, ... x„ into the order x^ y xi^, , . . xi^. By 14 THEORY OF SUBSTITUTIONS. repeating the same process as before, this can be gradually efPected, and the theorem is proved. Since a symmetric function is unaltered by any substitution, we obtain as a direct result Theorem V. A function which is unchanged by every transposition is symmetric. § 15. There is therefore at least one transposition which •changes the value of any alternating function into the opposite Talue. We will denote this transposition by (Xa oc^) , and the alter- nating function by c'-, and accordingly we have (,.' yXi, X^^ . . . Xa ; • • • X^ ) • • • ^n) ^' V^^l ) '^21 • • • ^P^ • • • '^aj • • • ♦^'«) Accordingly, if Xa = Xp, we must have v'' = 0. Consequently the -equation c'' (xi, X2, . . . z, . . . xp , . . . X,) = regarded as an equation in z has a root z = x^ and the polynomial (/' is therefore divisible by z — x^. The function therefore contains Xa — cc^s as a factor, and, consequently, c''- con- tains {Xa — Xp)- as a factor. But since, for all substitutions, c'' either remains unchanged or only changes its sign, c'-^ must be a symmetric function; and, accordingly, since is determined by aid of the following Theorem VI. Every alternating integral function is of the form S. \/ J , where \/ J is the square root of the discriminant and S is^an integral symmetric function. That >S. V'J is an alternating function is obvious. Conversely, if (}> is an alternating function, it is, as we have just seen, divisible by V-J- I^et (V-j)'" be the highest power of V-J which occurs as a factor in c''. Then the quotient is either a one- or a two-valued function, since every substitution SYMMETRIC AND TWO-VALUED FUNCTIONS. 15 either leaves both numerator and denominator unchanged or changes the sign of one or both of them. But this quotient cannot be two- valued, for then it would be again divisible hj\/J, which is con- trary to hypothesis. It must therefore be symmetric, and we have accordingly = s, . (V J)'" Now if m were an even number, the right member of this equa- tion, and consequently the left, would be symmetric. We must therefore have m = 2n -\- 1. And if we write Si. J" = S, we have ^^ = ^ . V J Corollary. From the form of an alternating function it follows that such a function remains uyichanged or is changed in sign simultaneously with \/ J for all substitutions. § 16. Having now shown how to form all the alternating func- tions, we proceed to the examination of the two-valued functions in general. Let ). If in this expression we interchange the position of the elements x^ in such a way that for x^, X2, ... x,, we put X;^ , X;„, . . . o?,,^ respectively, where the system of numbers i^ , i, , . . . ^„ denotes any arbitrary per- mutation of the numbers 1, 2, ... n, we obtain from the original function

is symmetric this complex will comprise all the n\ substitutions; if ^ is a two- valued function, the complex will contain all substitutions which are composed of an even number of transpositions, and only these. Again, for example, consider the case of four elements iCj, 0^2? ^3> ^4i and suppose

4 ele- ments, the order of the corresponding group becomes 8 • (n — 4) ! , the group being obtained by multiplying the 8 substitutions of § 25 by all the substitutions of the elements Xr^, x^, . . . x„: CORRELATION OF FUNCTIONS AND GROUPS. 27 § 29. The following theorem is obviously true : Theorem III, For every single- or multiple -valued func- tion there is a group of substitutions which, applied to the function^ leave it unchanged. To show the perfect correlation of the theory of multiple-valued functions and that of groups of substitutions we will demonstrate the converse theorem : Theorem IV. For every group of substitutions there are functions ivhich are unchanged by all the substitutions of the group and by no others. We begin by constructing a function ^ of the 7i independent elements x^, X2, . . . x,^ which shall take the greatest possible num- ber of values, viz : n ! ; ^ is therefore to be changed in value by the application of every substitution difPerent from unity. Taking n -\-l arbitrary and difPerent constants «o, a^, . . . a.,, we form the linear expression ^ = «0 + «i 0^1 + «2 + • • • + «« ^« If now two substitutions Sa=(x^,0Ci^ ...)... and s^={XsX^,^ ...)..., on being applied to . The «^'s are to be regarded here, as before, as arbitrary quanti- ties, and, as the ^a's are also arbitrary, it follows at once that 9'' is an w!-valued function. For, if then we must have identically x^^^x/'^x/^ . . . x^^n = x{y^ x-^y^ x^y^ . . . x,yn, and, from § 2, C, this is only possible if every ,3; is equal to the cor- responding )^f^ that is, if the substitutions (t and r are identical. We denote the functions which proceed from v'' under the opera tion of the substitutions of G by d> — d' d' (!> d> and form now the sum ^■=^'s, + «1, «3 > «1 + «2, «4 > «1 + «2 + «3 , , in particular if a^—1, as = 2, a3 = 4, a, z=z 8, «5 = 16, E. g. If a,j + a._^ -|- _ . -^ a.^ = 13^ ^e must have i^ = 1, u = 3, 4 = 4, / = 3. Example. — We will apply the two methods given above to the familiar group G = [l, (x.QCo), (x,x,), (x.x,) (_x,x,), {x,x,X^23c,), (x,Xi) (x,x,), (XjX.,X2Xi), {x^x^x^Xo)'], {n = 4:, r = 8) taking as fundamental functions

/y> •^^y» ° I /y» 'y« •^,-yi " I /y» ^y» "^y, o I /y» ^y« 2^y, 3 I /y. /y. 2-, O | ^y, /y, 2^y, 3 ^ eAy2«^3 «^^ I eA^JtA^g «A^^ j «A-2«^'4 «^3 | eA^^jM^'^ M>g | eA^^eX/j «A^2 r eA-qeA^2 *^1 ~p" «//^U?2 «^i ~)~ t3-'3U?j y>2 = (^1 + ^2) (^3 V + ^4^') + fe + 3C,) {Xlx./ + ^-2^') ^^^^ {^Xi "T~ ^2} \'^3 "! ^i) \p^9 '^i "T~ •^l ^^2 /• Neither of the two methods furnishes simple results directly. But from ^ >ve may pass at once to the function [{Xi — x^y + (a?3 — a?,)-] [{x, — x,f + (x, — x,y] , and from ^F to the two functions (a?i + X2) (x^ -\- Xi) and XiXo-j- x^x^ , the latter being already known to us. It is clear also that by alter- ing the exponents which occur in '/^ we can obtain a series of func- tions all of which belong to G. Among these are included all functions of the form (/y» O I ry* o\ ( r^ O. I /y» a\ /y« O /yi O I /y» tt ^y» tt X\ 1^ •^2 / \'^3 I '*'4 / ? Xj ^2 n^ t^3 t/.4 . CORRELATION OF FUNCTIONS AND GROUPS. 31 In general we perceive that to every group of substitutions there belong an infinite number of functions. It may be observed however that we cannot obtain all functions belonging to a given group by the present methods. Thus the function belongs to the group G = [1, (xix,), (x^x,), (xiX,) (xo, a?,)], but cannot be obtained by these methods. More generally, if the functions v^'', V'", v''"'? • • • belong respectively to the groups H', H", H'", . . . , and if the substitutions common to these groups (Cf. § 44, Theorem VII) form the given group G, then the function where the a's are arbitrary, belongs to the group G. § 32. We now proceed to consider the case where the elements x^,X2, . . . x„ are no longer independent quantities. Theorem V. Even where any system of relations exists among the elements Xi,X2, ... x„ , excluding only the case of the equality of two or more elements, we can still construct n\-valued functions of x^, x^, . . . Xn.^ Using the notation of the preceding Section, we start from the same linear function cp —a^-\- a^x^ -f a^X. + . . . + a„a?„ and form the product of the differences of the ^'s n\{n\ 1) this product being taken over the \ c, — possible combinations of the ^'s in pairs. Expanding we have In no one of these factors can all the parentheses vanish, since otherwise either the substitutions fff and r,: must be identical, or else the x'q are not all different. The product, regarded as a function of the a's , therefore cannot vanish identically (§ 2, D). Consequently, *C/. G. Cantor: Math. Annalen V, 133; Acta Math. I. 372-3. 32 THEORY OF SUBSTITUTIONS. (§ 2), there are an infinite number of systems of values of the a's for which all the n\ values of

4. A series of other anal- ogous results will also be obtained. For the present we shall concern ourselves only with the con- struction and the properties of some of the simplest, and for our purpose, most important groups. * § 34. First of all we have the group of order 7i ! , composed of all the substitutions. This group belongs to the symmetric func- tions, and is called the symmetric group. In Chapter I we have seen that every substitution is reducible to a series of transpositions. Accordingly, if a group contains all • Cf. Serret: Cours d'alg^bre sup6rieure. II, §§ 416-429. Cauchy : loc. cit. COREELATION OF FUNCTIONS AND GROUPS. 33 the transpositions, it contains all the possible substitutions and is identical with the symmetric group. To secure this result it is how- ever sufficient that the group should contain all those transpositions which affect any one element, for example x^ , that is the transposi- tions .{OCiXo), [XiX^), {XlX^), . . . [XiX,i). For every other transposition can be expressed as a combination of these n^l-, in fact every (XaXp) is equivalent to a series of three of the system above, (XaXp) = (x^Xa) (XiXp) (x^Xa), (where it is again to be noted that the order of the factors is not indifferent). We have then Theorem VI, A group of n elements Xi, x^, . . . x„ which contains the n — 1 transpositions is identical with the symmetric group. Corollary. A group ivhich contains the transpositions (XaXp), (XaXy), . . . {Xa,X^) contains all the substitutions of the symmetric group of the elements Xd , «A/|3 J Xy , . . . X^. § 35. We know further a group composed of all those substitu- tions which are equivalent to an even number of transpositions. For all these substitutions, and only these, leave every two-valued func- tion unchanged, and they therefore form a group. We will call this group the alternating group. Its order r is as yet unknown, and we proceed to determine it. Let L) Sj = 1, §2 , S3, ... s,. be all the substitutions of the alternating group, and let ii) Si , S.f , S^ , . . . Sf be all the substitutions which are not contained in I), and which are therefore composed of an odd number of transpositions. We select now any transposition 1. The number of resulting cycles is then equal to d. For example, [XiX2X^X^X^X^) =: {XiX^Xt^j {X2X^X(^)j yXlX2X^X^Xr,XQ) ^= [XiX^j [XoX^j yX^X^j^ {XiX2X^X^Xr^XfJ ^ \XiXr,X-^) {X2X(^X^), (/y r^ /y r^ /y r^ \'' / /y /y .'yt ry r^ q^ \ If the number of elements of a cycle be m, then the m^^\ (2w)*^, (3m)^'\ . . . powers of the cycle, and no others, will be equal to 1. E. g. {x^X2X'^X^Xr,X^'^ — {XyX2X-;.X.^Xr,X^^'^ = ... =1. If a substitution contains several cycles -with m^ , m, , mg , . . . elements respectively, the lowest power of the substitution which is equal to 1 is that of which the exponent r is the least common multiple of m, , mj , w^ , . . . Thus [{x,X2X.i) {x.Xr) {XfflCT)J = 1. This same exponent r is also the order of the group formed by the powers of the given substitution. For if we calculate CORRELATION OF FUNCTIONS AND GROUPS. 37 SfSj,,,S jO — J. J a further continuation of the series gives merely a repetition of the same terms in the same order: s'-+' = s, s^-+' = s\ s^-+' = sV . . s''-' = s>-\ s'^- = s'=l, ... Moreover the powers of s from s^ to s' are different from one another, for if then we should have contrary to hypothesis s'* = 1 (// < r). The extension of the definition of a power to include the case of negative exponents is now easily accomplished. We write SO that we have gfc g-fc _ 2. The substitution s^ therefore cancels the effect of the substitu- tion s~% and vice versa. The negative powers of a substitution are formed in the same way as the positive powers, only that in forming ( — 1)^^ ( — 2)^,( — 3)^, . . . powers, we pass backward in each cycle 1, 2, 3, . . . elements, the last element being regarded as next pre- ceding the first. It maybe noted that (sQ~^ = ^-^s~\ For (st)~^ {st)=l, and by multiplying the members of this equation first into t^^ and then into s~\ we obtain the result stated. The simplest function belonging to the cycle {xi x^ . . . x„) is ^ »X| eX/2 I •^2 3 i • • ' I *^)n 1 ^m I *^ w *^1 • § 37. Given two substitutions Sa and Sp , if we wish to deter- mine the group of lowest order which contains Sa and s^ , we have not only to form all the powers Sa^, s^>^ and to multiply these together, but we must form all the combinations Of the substitutions thus formed we retain those which are dif- ferent from one another, and proceed with the construction until all substitutions which arise from a product of m factors are contained among the preceding ones. For then every product of m-\-l fac- 38 THEORY OF SUBSTITUTIONS. tors is obviously reducible to one of m factors,; and is consequently also contained among those already found. The group is then complete. In case s^Sa = SaSpf^, the con^esponding group is exhausted by all the substitutions of the form Sa^s^K For we have in this case S/s' Sa = Sp- Sa Spf' = S|3 Sa " S^'^ = §« S^"", Sb Sa= Sb- SaSB^'^ = SaSfl^'^, S^'"Sa— Sp"Sa, • Sa = SaSp"'^ Sa = Sa^Sp''^^'^, Sb"'sJ = Sa'sB"""^ Sa = Sa%'"''\ Consequently any product of three factors is reducible to a product of two. Thus S^PSa''S^-' = Sa''S^(-' + P^>''', and the theoreiji is proved. For example, let Si — \XlX2X^^X^X^) , §2 -— V'^2'^3'^5^4/ ) then >1 — yXiX2X^X^) — Sj §2. The group of lowest order which contains s^ and So contains therefore at the most 5 • 4 = 20 substitutions. To determine whether the number is less than this, we examine whether it is possible that If this were the case, it would follow that But in the series of powers of S2 there is only one which is also a pow- er of .9i , and this is the zero power. Consequently we must have (i = yandft = d. The group therefore actually contains 20 substi- tutions. These are the following, where for the sake of simplicity we write only the indices: CORRELATION OF FUNCTIONS AND GROUPS. 39 s,' = i, s, = (2354), s/ = (25) (34), s,' = (2453), s^' = (1 2345), s,s, = (1325), s,s,' = (15) (24), s,s,' = (1435), si^= (13524), s,% = {l^S4:), s,V=(14)(23), s;%' = {12bi), s,' = (14253), sX = (1243), s,W = (13) (45), s,\' = (1523), si* = (15432), si% = (1452), s, V = (12) (35), s^ V = (1342). Analogous results may be obtained, for example, for the case /»! case every Sp^^Sa (a^ = 1, 2, 3, . . . ) can be reduced to the form Sa^sp^, the group of lowest order which contains Sa and s^ is exhausted by the substitutions of the form Sa''Sp\ For by processes similar to those above we can bring every sub- stitution Sjs'^Sa*' to theforms/Sj3^. The proof is then reduced to the preceding. Furthermore if s^"^ is the lowest power in the series sp, s^^^ . . . which occurs among the power of Sa? then the group contains q times as many substitutions as the order k of s^. For in the first place, if the exponent ^ in s/s/ is greater than q — 1, we can replace s^g^ by a Sa'^sp" , where v^q — 1. There are therefore at the most q • k differ- ent substitutions s/S/g^. Again if then we must have, if we suppose A > v, V~'' = «a'*^'' (/^ — v;/? = 1, 2, .. . r') form a group of order rr' which contains G and H as subgroups. For since Sat^ ' Syts— Sa.(tpSy)ts = Sa(Se ' t^)ts = S^ty, the substitutions §« tp form a group of, at the most, rr' substitutions. And we will show also that all of these are different from one an- other. For if, for example, ^a f /3 -— ^y ts then if we multiply both sides of this equation by Sy~^ at the left and t^"^ at the right we obtain Sy~ Sa= t^t^~ . But s^^^^a is a substitution of G and t^t^^^ a substitution of H, and consequently they can only be equal if both are equal to 1. Hence Sa ^ ^7, t^^^t^. The substitutions of the new group are then all different, and the order of the group is therefore 7t'. We denote the group by K=\G,H\. We add without proof, the following generalization of the last theorem. Under the same assumption Sat^ = tySs, if the two groups G and rr^ H have A substitutions in common there is a group of order -^ which contains G and A as subgroups.* § 39. In later developments a group will frequently be required *F. Giudice. Palermo Eeiid. I, pp. 222-223. CORRELATION OF FUNCTIONS AND GROUPS. 41 the order of which is a power of a prime number p. The exist- ence of such a group will be demonstrated by the proof of the fol- lowing proposition, from which the nature of the group will also be apparent. Theorem XII. If p^ be the highest power of the prime number p ivhich is a divisor of the product n! — 1 • 2 • 3 . . . n, then there is a group of degree n and of order p. In the first place suppose n < p\ so that n = ap -\-b {a,b + 1 = ^p + 1 *1 > *1 ^P + 1 ^^ ^P + 1*2 ) *a *iJ + l =S^> + 1 Sa+i , S^, *iJ + 1 ^^ ^p + 1 *1 J *i *jp + r ^^ *i> + 1 *3 ) „A,„ 2 — 2„ A *jj *iJ + 1 ^= S^J + 1 *2 > ^1 Sp + i'* = Sp + i'*Sjx + i J *a Sp + i'* =^ Sp + 1^ Sa ^ ^ , *iJ *ZJ + l'*=S^> + l'^*i> • Accordingly every combination of the substitutions Sj , Sa , . . . s^, + j can be brought, as in § 37, to the form Si^^V . . . V^i^+i" («j /^j r, . . . ^ ^ = 0, 1, 2, . . . p — 1). But we must also show in the present case that we need only take the powers of s^ + ^ as far as the (p — l)**^. We find that ^2>+l ~^ K'^ii'^io'^is • • • "^i^)) Xp^ki^k^ • • • "^k^,) • • • \p^ti'^t» • • • '^<„) = Si §2 . . . Sj, , i) + 1 — *1 *2 • • • ^p • Consequently, if A;>p, we can replace the highest power of s^,^i^' which occurs in Sj^,j^^ by powers of Sj, Sg, . . .s^,, and these can then be written in the order above. The question then remains whether the p^' + ^ =p^ substitutions thus obtained are all distinct. If two of them were equal n an B O <■<} " — o "■' o /3' o t'o f' bl 6.2 ' ' • ^p^p + l — oi <^2 ' ' ' f>p Op + 1 > we should have ^ „ k — k' — Qa' — aoBf—B q J — t ^P + i — ^1 ^-^ '^ . . . Sj, But the substitution on the right does not affect the first subscripts i, k, . . . , while that on the left does, unless /. = /.'. The proof then proceeds as before. If n>p'^ but • • • ^r'^2; G ■ <72; ^2' We show then, as in Chapter II, § 35, 1) that all substitutions of this line convert ^j into ^2? for since ^^^ = ^1, it follows that • • • S^p of the function

, of (p. For we have _n\ _n\ P _ fi ^ ~ r r^^ ' ' p^~ r' Corollary. If a function it follows that We may note however that the substitutions of the second line are not necessarily different from those of the first. In fact the identical substitution is of course always common to both and other substitutions may also occur in common. {Cf. § 50). From the three properties of the second line obtained above, it follows that the r substitutions of this line form the group of (p^. We will denote this group by G^,. That these substitutions from a group can also be shown formally; for MULTIPLE-VALUED FUNCTIONS ALGEBRAIC RELATIONS. 49 {(TrT ^80.(^2) {^2~ ^8^ (7.2) = (7-2" ^Sa (^2'^2~ ') ^/S ^2 = <^2'\SaS^) ^2 , SO that, if Sa? s^» . . • form a group, as was assumed, the same is true of the new substitutions. Similar results hold for all the other values, [Xi) (a?2^3^4)? '•^2) (^1^3^4)5 (^3) (^1^2^4)5 (^4) V^1^2^3)j [Xi) [X^X^X^), (^2) \'^l'^4'^3)j ('^^3) C^l^4'^2)) V^'i) (^i^3'^2J' Here the number of cycles with one element is 12, which is equal to the order of the group. The number of elements which occur in cycles of the second order is also 12. But, for A; = 3, the number of elements is 24 = 2 • 12. Correspondingly it is readily shown that the group permits of replacing any element by any other one; that the cycles of order 2 are all conjugate; but that the cycles of order 3 divide into two sets of four each: \«^i«^2'^3/j yp^i^'i'^i/i \XiX^X2)j \X2X^Xr^^f \"^l"^3'^2/5 V'^1'^4'^3J) \'^l«^2'^4/? \'^2'^3'^4J) the second set being non- conjugate with the first. § 52. We return now to the table constructed in § 45. This table did not possess the last of the four properties noted in § 41 ; the substitutions of one line were not necessarily all different from those of the other lines. For every group certainly contains the identical substitution 1, which therefore occurs p times; and again in the example of § 46 three other substitutions ' 'A . {^XiX2) [X^X^Jj {X1X2) (^£ - • - ^p any arbitrary substitution t, we obtain These values must coincide, apart from the order of succession, with the former set, for ^j , ^^2? - - • ^p are all the possible values of (f, and the p values just obtained are all different from one another. The groups which belong to the latter (j~^G^ 4, the substitution of H which affects the least number of elements cannot contain more than one cycle. For otherwise H would contain substitutions of the form and therefore the corresponding conjugate substitutions with respect to 'T = (x^iCg) Sfi =■ {XiX2) yX^X^) . . . , Sp =^ (^XiX.2X^) [X^X^Xq) . . . Consequently the corresponding products Sa''Sa'= (X,) {X2) (a?3 ...)... , Sp-'Sp' = (X,) (^2) (0:^3) (x^Xr^X^) . . . , which are not 1, but affect fewer elements than s, must also occur in H, which would again be contrary to hypothesis. If then n > 4, either H consists of the identical substitution 1, or H contains a substitution (Xf^x^), or a substitution {x^x^Xy). In the second case H must contain all ' the transpositions, that is H is the symmetric group. In the third case H must contain all the cir- cular substitutions of the third order, that is H is the alternating group. (Of. §§ 34-35). Returning from the group H to the group G, it appears that if Gi, 0-2, . . . Gp have any substitution, except 1, common to all, then either the second or the third case occurs. H, which is contained in G, includes in either case the alternating group; G is therefore either the alternating or the symmetric group, and p = 2, or p — 1. If, however, n = 4 we might have, beside s^ = 1, another substi- tution in the group. With this its conjugates, of which there are only two, §2 := [XiX'j) {X2Xi), Si = {XiX^) {X2X^)j must also occur. The group H cannot contain any further substitu- tion without becoming either the alternating or the symmetric group. .We have then the exceptional group A 60 THEORY OF SUBSTITUTIONS. and this actually does transform into itself with respect to every sub- stitution. Returning to the group G it follows from § 43, Theorem II, that the order of (r is a multiple of that of H, that is, a multiple of 4; again from Theorem II the order of G must be a divisor of u 4! = 24. The choice is therefore restricted to the numbers 4, 8, 12, and 24. The last two numbers lead to the general case already dis- cussed where p = 2, or 1. The first gives G = H, /> = 6, and for example, (p2 — \X-iX^ -\- Xt^X^J \XiX^ -\- i3?2«^3/? ^4 "~" V'^1'^3 \ '^2'^^) [XiXo, ~\~ »^3'^4/ ^5 ~~ \XiX^ -\- X-iX-^J \X1X2 ~\~ '^3'^4 j? ^6 ~~ \X]X^ -\~ X2X^j yXiX^ "T" «^2'^4/ • In the second case, ?^ = 8, G contains H as a subgroup. To obtain G we must add other substitutions to those of H. None of these can be cyclical of the third order, for in this case we should have r = 12 or 24. If we select any other substitution, we obtain the group of § 46, which is included among those treated in § 39. For this group p = 3, and, for example, f /ft /y» /'v* I ry* rv* /ft /v» rv* i ry* /y* /ft ^V» rg* I y* /y* V 1 tA/-^tA^2 1^ tAy^tAy^ , V 2 tt/JeA/3 \^ tA/O'^i J V 3 tAyji4y^ |^ t4y2«^3 • Theorem XI. If n ^4: there is no function, except the al- ternating and symmetric functions, of ivhich all the p values are unchanged by the same substitution (excluding the case of the identi- cal substitution). 7fw = 4, all the values of any function belong ing to the same group with

- valued function from the point of view of the theory of substitutions. We turn now to the consideration of the algebraic relations of these values. We saw at the beginning of the preceding Section that the sys- tem of values ^1, ^'2? • • • ^p belonging to a function

) = rifefa . . • S^p = ^-^pC^i, C., . . .C,), the -R's are the coefficients of an algebraic equation of which S^i 5 S^2 ? • • • ^P a^^6 the roots. Theorem XII. The /> values c^i, t^^, . . . (pp of a p-valued integral rational function (f are the roots of a7i equation of degree p the coefficients of ivhich are rational integral functions of the ele- mentary symmetric functions Cj , Co, . . . c„ of the elements Xi,X2, ... x,,. § 54-. As an example we determine the equation of which the three roots are O-j^ — Ju^JLo I t^gt-C'^ ^ y- 2 — 13 I «^2*^4 5 * 3 — Ju-^JL^ | tX'2*^3 5 where Xi,X2, x^, x^ are themselves the roots of the equation fix) = a?^ — CjX*^ -|-- <^2'^^ - ^3^^ + ^4 = 0. We find at once fx -V f 2 + f^3 = S {X^X^ = C2; and again, by § 10, Chapter I, ^iS^2 + 9'2^3 + f 3f 1 = S (a^i^^a^a) = ^^^4 + ^-CiCg + ^-Cgl The numerical coefficients a, /?, y are readily found from special ex- amples. They are « = — 4, ,5 = 1, ^ = 0. Hence ^1^2 + ^2?'3 + f%{\ — C1C3 — 4 C4 . Finally ^'X^l'fz — ^Vm?! X2 X^ I ~j~ XiX2X^X^ OliCj ) "~~ ^1 ^4 4C2C4 -J- C3 , Accordingly the required equation is 62 THEORY OF SUBSTITUTIONS. /(^) = ^' — C2^' + (C1C3 — 4c,)^ — (ci'c^ — 4c,C4 + Cg'O = 0. ^ We examine the discriminant of this equation, i. e., of its three roots. To determine this function it is not necessary to employ the the general formula obtained in § 10, Chapter I. We have at once ft the highest power of J which is contained as a factor in J^, then, as A contains n{n — 1) factors Xa — x^, and consequently zl' contains n{n — 1)^ such factors, we must have „(„_,),>,[^fcl)-g], t > '' '^'' 2 n(re — 1) nin - 1^ The number t can be only when q = , that is, when all the transpositions occur in. G^. c? is then symmetric and /> = 1. Again q can be only when G contains no transposition. One of the cases in which this occurs is that where G is the alternating group or one of its subgroups. Theorem XII. //

-valued functions whose />*^^ powers are symmetric. For (> = 2, we already know that (p= \f d satisfies this condition. In treating the general case we will assume that, if the required function (p contains any factors of the form V J , these factors are all removed at the outset. If the resulting quotient is ^'', so that then, since cp-p and (V-j)"^ ai'^ both symmetric, v''"^ is symmetric also. We write then If d\ be any root of this equation, and if m be a primitive (2/>)^^ root of unity, then all the roots are and consequently From Theorem XIII this discriminant must be divisible by J, unless 4> is itself symmetric. But the factors containing w are constant and therefore not divisible by J, and by supposition (l\ does not con- tain V -^ as a factor. Consequently v''i is symmetric, and we have, according as a is odd or even,

= 1 or 2. In the last case p = n\. All the values of a function are of the same type, and consequently there are substitu- tions which transform one into another. Suppose, in the case p = n\, that T converts the value we obtain Multiplying the^e three equations together and removing the func- tional values, wG have a.«=rl, g = 3. MULTIPLE-VALUED FUNCTIONS ALGEBRAIC RELATIONS. 67 If now WG assume n > 4, then the group of (J' cannot contain all the circular substitutions of the fifth order, (Theorem X, Chapter II). If r is one of those not occurring in the group of (J', then V''T+V''n but and consequently, if ^^ be a q^^ root of unity different from 1, It follows from this, precisely as above, that, since r" = 1, and consequently o>^ = 1, g = 5. But this is inconsistent with the first result. It follows there- fore that n is not greater than 4. Theorem XVII. If n> A^ there is no multiple-valued function a power of ivhich is two-valued, if the elements x are independent quantities. § 59. We conclude these investigations by examining for n^4 the possibility of the existence of functions having the property dis- cussed above. The case n = 2 requires no consideration. In the case n = 3 we undertake a systematic determination of the possible functions of the required kind. We begin with the type and attempt to determine «, /5, y so as to satisfy the required condi- tions. For this purpose we make use of the circumstance that some where y^T is equal to the product of / is two- valued is also readily shown, if we write X]X2 ~\~ oc^x^ — 2/1 ) XiXq -J- x.2X^ ^=: 2/2 ) x^x^ -J- x^x^ ^ 2/3 • For then (p^ coincides with the expression obtained above for the. case 91 = 3 ; and since 2/1 ? 2/2 5 2/3 > ^^^ the roots of the equation y' — C2 2/' + (C1C3 — 4cJ 2/ — {c;\ — 4C2C4 + C3'') = 0, where the c's are the coefficients of the equation of which the roots a?i, a?2, a?3, a?4, (§ 54), we can translate the expression obtained for w = 3 directly into a two- valued function of the four elements a?i, a?2, a?3, CC4, since we have (§ 54) A,, = J^. CHAPTER IV. TKANSITIVITY AND PRIMITIVITY. SIMPLE AND COMPOUND GROUPS. ISOMORPHISM. § 60. The two familiar functions ry^ /y» L. /y* ^y* -'V O^ O^ O^ differ from each other in the important particular that the group belonging to the former G = [1, (XYJC^), (x^Xi), (X^X,) (x^Xi), (X^X^) {X,X,), (XiX,) (x^x-^), {XiX^X2X^)j {XlX^X2XQ)^ contains substitutions which replace x^ by X2 , Xr^ , or x^, while in the group belonging to the latter G, = [1, (x,X2), {x,Xi), (x.x^) (x,x^)'] there is no substitution present which replaces x^ by x^ or x^. The general principle of which this is a particular instance is the basis of an important classification. We designate a group as transitive, if its substitutions permit us to replace any selected element x^ by every other element. Otherwise the group is intransitive. Thus 6r, above, is transitive, while Gi is intransitive. It follows directly from this definition that the substitutions of a transitive group permit of replacing every element Xi by every element x,,. For a transitive group contains some substitution s = (xiXi ...)... which replaces x^ by Xi, and also some substitu- tion t= {x^Xk ...)... which replaces x^ by x,,. Consequently the product s~^t, which also occurs in the group, replaces x,- by a?^. The same designations, transitive and intransitive, are applied to functions as to their corresponding groups. It appears at once that the elements of an intransitive group are distributed in systems of transitively connected elements. For example, in the group G^ above a?, and 0^35 and again, x^ and 0^4, are transitively connected. Suppose that in a given intransitive group there are contained substitutions which connect x^ X2, . , . Xa transi- GENERAL CLASSIFICATION OF GROUPS. il tively, others which connect x^+i, x„^2i • • - ^a + bj and so on, but none which, for instance, replace x^ hj x„^\ (^ = 1)5 and so on. The maximum possible number of substitutions within the several sys- tems is a ! , 6 ! , . . . , and consequently the maximum number in the given group, if a, b, . . . are known, is a! 5! . . . If only the sum a-\-b-\- ... =n is known, the maximum number of substitutions in an intransitive group of degree n is determined by the following equations : {n—l)l 1! = ^^^(n— 2)! 2! > {n — 2)l 2!, {n > 3) f, 9 {n-2)l2l=^^{n — ^)\^\>{n — ^)\^\, (n>5). Theorem I. The maximum orders of intransitive groups of degree n are (n—1)!, 4(^—1)!, {n—2y. 2l,{n — 2)\, (n — 3)!3!, (w — 3).!2!, . . . The first two orders here given correspond to the symmetric and the alternating groups of (n — 1) elements, so that in these cases one element is unaffected. The third corresponds to the combination of the symmetric group of (n — 2) elements with that of the re- maining two elements. The fourth belongs either to the combina- tion of the alternating group of (n — 2) with the symmetric group of the remaining two, or to the symmetric group of {n — 2) ele- ments alone, the other two elements remaining unchanged; and so on. The construction of intransitive from transitive groups will be treated later, (§ 99). § 61. We proceed now to arrange the substitutions of a transi- tive group in a table. The first line of the table is to contain all those substitutions 7 Sj = 1 , §2 , S3 , . . . S„j which leave the element x^ unchanged, each substitution occurring only once. From the definition of transitivity, there is in the given group a substitution t., which replaces x^ by 'X2. For the second line of the table we take 72 THEORY OF SUBSTITUTIONS. We show then, 1) that all the substitutions of this line replace x^ by X2; for every s^ leaves x^ unchanged and (t^ converts a?i into X2', 2) that all the substitutions which produce this effect are contained in this line; for if r replaces x^ by x.2, "^^2"^ leaves x^ unchanged and is therefore contained among the s^'s; but from t(T2~^ — s^ it follows that r = s^^.2 ; 3) that all the substitutions of the line are distinct ; for if Saf^2 = s^ ''■2 ? we obtain by right hand multiplication by 't," \ Sa = Sp', 4) that the substitutions of the second line are all different from those of the first; for the latter leave Xi unchanged, while the former do not. We select now any substitution t.^ which converts x^ into x^ and form for the third linq of the table ^3 , S2'T3 , S3 of the latter, and again to every § of @ corresponds either one substitution, or a certain constant number of substitutions s of G. And this correspondence is moreover of such a nature that to the product of any two s's corresponds the product of the two corres- ponding §'s. If to every § there corresponds only one s, then there is only one substitution, identity, in (S which leaves the order of the systems A unchanged. The two following groups may serve as an example of this type. Suppose that (j- = |_1, {X1X2) yX^X^j [X^Xq)^ (XiX^) {X.2Xq) \X^X^)J [XiX^Xf,) (X2'^b'^3)j i^XiX^) (^X2X^) (XqXq)^ yXiX^x^) (^i^2'^3'^5JJ' Here the systems A^, Ao, and A^ are composed respectively of Xi and X2, x^ and Xq, and Xi and Xr^. The corresponding substitutions of the A^s form the group (S = [1, (A3A3), (A,A2), (A1A3A,), (A1A3), (AA2A3)]. § 67. We examine more closely the subgroup G-j = [_Si , §2 , S3 , . . . s„ J of the group G oi % 65. Since Gi cannot replace any element of one system by an element of another system, it follows that Gi is intransitive. Any arbitrary substitution t ot G transforms G^ into t~^G^t= G\. The latter is also a subgroup of G\ it is similar to G-^\ and it evidently does not interchange the systems of G. It follows that G\ = G,. Suppose that any system of a non-primitive group consists of the elements x\, x'2, x'^,. . . The subgroup Gi therefore permutes the elements x' among themselves. We proceed to examine whether these elements are transitively connected with one another by the group Gi , or whether this is the case only when substitutions of G are added which interchange the systems of elements. Suppose that x\,x\j.. . . x^a, and again a:^'a + ij . . . ^'s» etc., are transitively connected by (xj. Then G contains a substitution of the form t={x\x'a + i ...)... f and since t~^Git= G^, it follows that t re- places all the elements a?'i , cc'2 > • • • ^'a by x'a+i . . . x'p. Further consideration then shows that x\ , x'2, . . . x'a form a system of non- primitivity. 78 THEORY OF SUBSTITUTIONS. Accordingly if the systems of non-primitivity are chosen at the outset as small as possible, then the group G^ connects all the ele- ments of every system transitively. Assuming the systems to be thus chosen, we direct our attention to those cycles of the several substitutions of G^ which interchange the elements x'^^x'^, . . . of any one system of non-primitivity. These form a transitive group H'. Similarly the components of the several substitutions of Gj which interchange the elements a?/'*^ x.2^°-\ . . of any second system form a group H^'^X The groups H', H", . . . are similar, for if t = {x\x^^'^K ..)... is a substitution of G, then the transformation t-^Git= G^ will convert H' into IK'^X The order of H' is a multiple of — and a divisor of — ^ ! , where /f. is the number // /J. of systems of non-primitivity. § 68. The following easily demonstrated theorems in regard to to primitive and non-primitive groups may be added here: Theorem VI. If from the elements x^^ x^., . . . Xn of a tran- sitive group G any system x\^ x'2, . . . can be selected such that every substitution of G ivhich replaces any x'a by an x'^ permutes the x'^s only among themselves, then G is a non-primitive group. Theorem VII. If from the elements x^, Xo, . . . x„ of a transitive group G two systems x\, x'2, . . . and x'\, x"^, . . • can be selected such that any substitution which replaces any element x'a by an x" ^ replaces all the x'^s by a?"'s, then G is a non-primitive group. Theorem VIII. Every primitive group G contains substi- tutions iL'hich replace an element x'a of any giveyi system x'^, x'2, . . . by an element of the same system, and ivhich at the same time replace any second element of the system by some element not belongincj to the system. * § 69. The preceding discussion has led us to two general prop- erties of groups which, together with transitivity and primitivity, are of fundamental importance. In§ 67 the subgroup Gj of G possessed the property of being reproduced by transformation with respect to every substitution t of *Eudio: Ueber primitive Grunpen. Crelle CI. p. l. GENEEAL CLASSIFICATION OF GROUPS. 79 G, so that for every t we have t^^G^t—G^. We may conveniently indicate this property of G^ by the equation G-'G,G=^G, or G,G=GG,. It is to be observed however that this notation must be cautiously employed. For example, if G^ is any subgroup of G, we have always Gi~'^GGi = G, and from this equation would apparently fol- low GGi — GiG, and consequently Gi — G~^ G^ G. But this last equation holds only for a special type of subgroup G^ . The reason for this apparent inconsistency lies in the fact, that in the equation G~^ GiG= G^ the two G's represent the same substitution and the two G^i's in general different substitutions, while in the equation Gj~^ GGi = G the reverse is the case. We introduce here the following definitions. 1) Two substitutions s^ and Sg ct''^ comnnitatwe * if 12 — ^2 1 • 2) A substitution Sj and a group H are commutative if s,H = Hs,. 3) Two groups H and G are commutative if HG=GH. The last equation is to be understood as indicating that the product of any substitution of H into any substitution of G is equal to the product of some substitution of G into some substitution of H, so that, if the substitutions of G are denoted by s and those of H by t, then for every « and fi. Under 2) s^ may be a substitution of H-, for s^ and H are then always commutative. Under 3) a case of special importance is that, an instance of which we have just considered, for which if is a sub- group of G. In this case Sa and ss of the equation sj^ = tySs are always to be taken equal. A subgroup H of any group G, for which G^^HG = i?, is called a self -conjugate subgroup of G. ♦German: " vertauschbar"; French: "exchangeable", retained as "interchangeable " by Dolza: Anier. Jour. Math. XIII, p. 11. 80 THEORY OF SUBSTITUTIONS. The following may serve as examples: 1) The substitutions s^ = (ccia^g^a) (^^^s^e)? ^2 = (xiXi) {x2X^) (x^x^) are commutative; for their product is (x^Xr^x^x^x^x^), independently of the order of the factors. Every power s« of any substitution s is commutative with every other power s^ of the same substitution. Two substitutions which have no common element are commuta- tive. 2) The group H = [1, {x,x,) (x^x^), (x^x^) (x^x^), (x,Xi) (x^x^)'] is commutative with every substitution of the four elements x^, X2, x^j x^. The alternating group of n elements is commutative with every substitution of the same elements. 3) The group H of 2), being commutative with the symmetric group of the four elements x^, X2, x ^^35 • • • ^% contains all the substitutions of @ which cor- respond to s, and the number q is constant for every s. Similarly to every § correspond the same number p of the substitutions s. It is evident at once that the substitutions of G ((5)) which correspond to the identical substi- tution of @ (G) fonn a group H (§) which is a self- conjugate sub- group of G ((S). The correspondence of two groups as just defined is called iso- morphism. If to every substitution of G correspond q substitutions of ®, and to every substitution of @ p substitutions of (7, then G » and (3 are said to be (p-q)-fold isomorphic, or if p and q are not specified, manifold isomorphic. If p = q = l, the groups are said to be simply isomorphic. * Examples. I. The groups Cr — \_1, \X1X2) {X^Xq) [X^Xr,), yXiXcij yXoXr,) (a^^iCgj, {XiX^j [X^Xf^j {X-^Xr,), yXiX^x^) yX2X^x^), yXiX^Xr^) {X2X^Xq)j, ^ == L-'-J (^1^2)5 (m^s)) (^2^3)) (^1^2^3)? (^1^3^2)J are simply isomorphic, the substitutions corresponding in the order as written. For if any two substitutions of G, and the corres- ponding substitutions of T, are multiplied together, the resulting products again occupy the same positions in their respective groups. II. The groups G = [1, (x,x2)l r = [1, {-A) {^A\ i^A) (4^2?.), (^1^.) (^2^3)] are (1-2) -fold isomorphic. Corresponding to 1 of (r-we may take, beside 1, any other arbitrary substitution of /'. It follows that F is simply isomorphic with itself in different ways. * C/. Camille Jordan : Traite etc., § 67-74, where the names " holoedric " and *nierl- edric" isomorphism are employed. These have been retained by Dolza: Amer.Jour, Vol. XIII. The "simply, manifold, (p-Q)-fold isomorphic " above rei)resent the "ein- stufig, mehrstuflg, (p-g)-stufig isomorph " of the (Jerman edition. 84 THEOBY OF SUBSTITUTIONS. III. The groups Cr = l^i, {X1X2) \X^X^), {X1X2) \X2X^)J \^iXi) \X2X^)_\, r= [1, (?.f,f,), (f,?,?,), (?,?,), (f,e,), (^A)} are (2-3)-fold isomorphic. To the substitution 1 of 6r correspond 1, (ci^2^3)) (^i^3^2) of ^5 and conversely to 1 of F correspond 1, (x^x^) (x^x^) of G. § 73. If G and F are (m-n)-fold isomorphic, then their orders are in the ratio of m:n. If L is a self-conjugate subgroup of G, and if A is the corres- ponding subgroup of T, then A is a self- conjugate subgroup of F. For from G'^LG — L follows at once F-^AF= A. In the case of (p-l)-fold isomorphism, it may however happen that the group A consists of the identical substitution alone. § 74. Having now discussed the more elementary properties of groups in reference to transitivity, primitivity, commutativity, and isomorphism, we turn next to certain more elaborate investigations devoted to the same subjects. The m substitutions of a transitive group G which do not affect the element Xi form a subgroup G^ of G. Similarly the substitutions of G which do not affect X2 from a second subgroup G2, and so on to the subgroup G,, which does not affect x„. All these subgroups are similar; for if ^{n — 1). Theorem IX. Every transitive gromp contains at least (n — 1) substitutions which affect all the n elements. If there are more than {n — 1) of these, then the group also contains substitu- tions which affect less than (n — 1) elements. * Corollary. A k-fold transitive group contains substitutions which affect exactly n elements, and others which affect exactly {n — 1), {n — 2), . . . {n — k-\-l) elements. Those substitutions which affect exactly k elements we shall call substitutions of the fc*'^ class. We have just demonstrated the existence of substitutions of the n^^, or highest class. If we consider a non- primitive group G, there is (§ 66) a second group @ isomorphic with G, the substitutions of which interchange ♦C. Jordan: Liouville Jour. (2), XVII, p 351. 86 THEORY OF SUBSTITUTIONS. the elements Ai, A^, . . , A^ exactly as the corresponding substitu- tions of G interchange the several systems of non-primitivity. Since G is transitive, © is also transitive. From Theorem IX fol- lows therefore Theorem X. Every non-primitive group G contains substi- tutions which interchange all the systems of non-primitivity. § 75. We construct within the transitive group G the subgroup H of lowest order, which contains all the substitutions of the high- est class in G, and prove that this group H is also transitive. H is evidently a self- conjugate subgroup of G. If H were intransitive, G must then be non-primitive (Theorem VI). If this is the case, let @ be the group of § 66 which affects the systems Ai , Ag , . . . Afj, regarded as elements. @ is transitive. To substitu- tions of the highest class in @ correspond substitutions of the high- est class in G. (The converse is not necessarily true). Suppose that ^ is the subgroup of the lowest order which contains all the substi- tutions of the highest class in @.' To § then corresponds either Hot a subgroup of H. If $ is transitive in the A's, H is transitive in the x's. The question therefore reduces to the consideration of the groups @ and §. § can be intransitive only if @ is non-primitive and G^ accordingly contains more comprehensive systems of non- primitivity. If this were the case, we should again start out in the same way from @ and ^, and continue until we arrive at a primitive group. The proof is then complete. Theorem XI. In every transitive group the substitutions of the highest class form by themselves a transitive system. § 76. Suppose a second transitive group G' to have all its sub- stitutions of the highest class in common with G of the preceding Section. If then we construct the subgroup H' for G', correspond- ing to the subgroup H of G, we have H' = H. Moreover the number iVj of the substitutions of the highest class in H is <|[n-2], + |[«-3J, + ..+^^^M.+ ..+^[0]).> where [g]i has the same relation to H as [q] to G. But the number iV"i is, as we have just seen equal to the iV of § 74. Consequently GENERAL CLASSIFICATION OF GROUPS. 87 «H([»-2]-[»-2],) + S([»-3]-[«-3]0+ . . . •••+^^^=^(M-Mi)+---f=o. But, since H is entirely contained in G, it follows that [gJ^Mi, and therefore the left hand member of the equation above can only- vanish if each parenthesis is 0. Consequently G and G' can only difiPer in respect to substitutions of the (n — 1)*^ class. Theorem XII. If two transitive groups have all their sub- stitutioiis of the highest class in common^ they can only differ in those substitutions which leave only one element unchanged. § 77. Let G be any transitive group and Gi^Gi, . . . Gn those subgroups of G which do not affect ccj, a?2, . . . a?„ respectively. These groups are, as we have seen, all similar. If now G^ , and con- sequently (t2, . . . (r„, are A;-fold transitive, then G is at least (k -{-!)- fold transitive. For if it be required that the {k -f- 1) elements Xi,X2i . . .Xj,^^ shall be replaced by Xi^, .t.^? • • • ^^k+i respectively, we can find in G some substitution s which replaces Xi,X2,x^, . . .Xf,^i by a?,i,a?;,„,a?;,3,. . .a?;,^^j, where Xj,^,Xh^,... may be any elements whatever. Again Gi^ contains some substitution t which replaces ^hoi ^/igj ... by a?,2)^»3j • • • Consequently the substitution st ot G satisfies the requirement. From this follows the more general Theorem XIII. // a group G is at least k-fold transi- tive, and if the subgroup of G which leaves k given elements un- changed is still h-fold transitive, then G is at least {k-\-h)-fold tran- sitive.^ § 78. Suppose that those substitutions of a A;- fold transitive group (t, which, excluding the identical substitution, affect the smallest number of elements, are of the q^^ class, i. e., that they affect exactly q elements. The question arises whether there is any connection between the numbers k and q. In the first place suppose k^q, and let one of the substitu- tions of the q^^ class contained in (7 be s = (o^jCCg ...)...(... a^g_ i x.^). Then, on account of its fc-fold transitivity, G also contains a substitu- *G. Frobenius: Ueber die Congruenz nach einem aus zwei endlichen Gruppen geb- ildeten Doppelmodul. Crelle CI. p. 290. 88 THEORY OF SUBSTITUTIONS. tion (T, which replaces a?i , a?2 » • • • ^2-1 ? x^byx^, X2,... Xq_i, x^ (x > q), and which is therefore of the form (T — (xi) (x^) ...{Xg_,) (x^x^ ...). We have then (fT-'s(T)s-' = l{x,x, ...)...(... x^_,x^)']s-' = (x^^.x^Xq), and since this substitution affects only 3 elements, it follows that ^<3. Secondly, suppose k g. It is evident that both are possible, if in the latter case it is remembered that n> q. We obtain then (Ti Siffi = {X^X2 ...)...(... Xk_iXif . . . ), (T^-'^S^ff^ = (X^X2 ...)•••(••• ^A—l) (a^A ••• )» and if we form now {frr^Si(T^)Si~\ the first {k — 2) elements are removed, and there remain, at the most, q-\-(q — k) — (A; — 2) = 2q — 2A; + 2. Similarly, if we form {(Tf ^82(72)82" \ the first (A: + 1) elements are removed, and there remain, at the most, q-\-{q — k-{-l) — (A; — l) = 2g — 2A; + 2. By hypothesis, this number cannot be less than q. Consequently g>2A; — 2. Theorem XIV. If a k-fold transitive group contains any substitution, except the identical substitution, which affects less than (2k — 2) elements, it contains also substitutions which affect at the most only three elements. This theorem gives a positive result only if A; > 2. In this case, by anticipating the conclusions of the next Section, we can add the following GENERAL CLASSIFICATION OF GROUPS. 89 Corollary. // a k-f old transitive group ky2 contains sub- stitutio7is, different from identity, which affect not more than (2k — 2) elements, it is either the alternating or the symmetric group. We may now combine this result with the corollary of Theorem IX. If G is A;- fold transitive, it contains substitutions of the class {n — k-\rV). Accordino^ly q^{n — k-\-l). If G is neither the alternating nor the symmetric group, q>{2k — 2). Consequently (n — /j+1) > (2/^— 2) and A;£^^i^. o Theorem XV. If a group of degree n is neither the alter- nating nor the symmetric group, it is, at the most, I — -f- 1 \- fold tran- sitive. That the upper limit of transitivity here assigned may actually occur is demonstrated by the five-fold transitive group of twelve elements discovered by Matthieu, (y.x) {x,x,) (a-gic^) (XiX^), (2/22/1) (xiX;) (x,x^) (x^x,), (y^Vi) (^1^5) (-^s-^t) (^'4^6), (2/42/3) (^1^3) (^4^-5) {^o^i)\- § 79. Theorem XVI. // a k-f old transitive group {k>l) contains a circular substitidion of three elements, it contains the alternating group. Suppose that s = (^x-iX^x.^) occurs in the given group G. Then, since G is at least two- fold transitive, it must contain a substitution (T = (x-i) {xiX^X\ .,.)... and consequently also r — fT - 's^ = (x-^XiXfj), r~^sr =■ [XiXiX^). In the same way it appears that G contains {XiX2X^), {X^X^X^), . . . Consequently (§ 35) G contains the alternating group. Theorem XVII. If a kfold transitive group (k > 1) con- tains a transposition, the group is symmetric. The proof is exactly analogous to the preceding. For simply transitive groups the l^st two theorems bold only 6a 90 THEORY OF SUBSTITUTIONS. under certain limitations, as appear from the following instances (xi = [1, (a?ia?2), {x^x^)y {X1X2) {x^x^}, (^1^3) (^2^4)? (^1^4) (^2^3)? \XiXsX2X^), [XiX^X2X^)^j 1X2 ::^= ^ Ij [XiX2X^jf [X^XqXqJj yX-jX^Xg)j yXiX^XgX^XQX'jX^X^X^f ^ . Both of these are transitive. But the former contains a substi- tution of two elements, without being symmetric, and the latter a substitution of three elements without being the alternating group. § 80. An explanation of this exception in the case of simply transitive groups is obtained from the following considerations. If we arbitrarily select two or more substitutions of n elements, it is to be regarded as extremely probable that the group of lowest order which contains these is the symmetric group, or at least the alternating group. In the case of two substitutions the probability in favor of the symmetric group may be taken as about |, and in favor of the alternating, but not symmetric, group as about J. In order that any given substitutions may generate a group which is only a part of the n\ possible substitutions, very special relations are necessary, and it is highly improbable that arbitrarily chosen substitutions s,: = \ J J ' ' \S I should satisfy these conditions. The exception most likely to occur would be that all the given substitu- tions were severally equivalent to an even number of transposi- tions and would consequently generate the alternating group. In general, therefore, we must regard every transitive group which is neither symmetric nor alternating, and every intransitive group which is not made up of symmetric or alternating parts, as de- cidedly exceptional. And we shall expect to find in such cases special relations among the substitutions of the group, of such a nature as to limit the number of their distinct combinations. Such relations occur in the case of the two groups cited above. Both of them belong to the groups which we have designated as non-primitive. In G^ the elements x^ , X2 form one system, and a?3, a?4 another; it is therefore impossible that G^ should include, for example, the transposition {x^x^. In G2 there are three systems of non-primitivity x^,X2,x^^ Xi,Xr^,x^, and x-i,x^,x^, G^^ therefore cannot contain the substitution (xiX^x-!). GENERAL CLASSIFICATION OF GROUPS. . 91 It is, then, evidently of importance to examine the influence of primitivity on the character of a transitive group, and we turn our attention now in this direction. § 81. With the last two theorems belongs naturally Theorem XVIII. If a primitive group contains either of the two substitutions it contains in the former case the alternating, in the latter the sym- metric group. The proofs in the two cases are of the same character. We give only that for the latter case. From Theorem VIII, the given group must contain a substitu- tion which leaves x^ unchanged and replaces X2 by a new element X3, or which leaves X2 unchanged and replaces x^ by a new element x^ , or which replaces x^ by X2 or X2 by x^ and the latter element in either case by a new element x-^. If then we transform r with respect to this substitution, we obtain a transposition r' connecting either x^ or X2 with x^ , for example r' = (cCjCCg). The presence of r and r' in the group shows that the latter must contain the symmet- ric group of the three elements x^, X2,x^. From Theorem VIII there must also be in the given group a second substitution which replaces one of these three elements by either itself or a second one among them, and which also replaces one of them by a new element x^. Suppose this substitution to be, for instance, S — I . . . t4v2t*^i . . • X^Xi^ ..,).,, We obtain then t o 1*^2 3/ \ 1 4/5 \ and it follows that the given group contains the symmetric group of the four elements x^^x^^x^, CC4; and so on. • § 82. We can generalize the last theorem as follows: Theorem XIX. If a primitive group G with the elements Xi, X2 . . . Xn contains a primitive subgroup H of degree k i < w affects the ele- n ments x^,X2, . ... X},. In the first pl^ce if A < -^then the group G, on account of its primitivity, contains a substitution s, which replaces one element of x^,x<^, . . . x^hj another element of the same system and at the same time replaces a second element of a?i , a?2 5 • • • ^a by some new element. Then H' = s{~ ^Hs^ contains beside some of the old elements, also certain new ones, so that iiTi = \H,H'\ affects more than X elements, but less than n, since H and H' together n affect at the most (2 A — 1) J ?2. Accordingly if 2 = i ^1 , H'l \ con- tains more elements than H^ but less than G. Proceeding in this way, we must finally arrive at a group K which contains exactly (n — 1) elements. *C. Jordan: Liouville Jour. (2) XVI. B. Marggraf: Ueber primitive Gruppen mit transitiven Unlergruppen geriugeren Grades; Giessen Dissertation, 1890. GENERAL CLASSIFICATION OF GROUPS. 95 If H is transitive, then H', and consequently Hi=\H', H\, and 80 on to Kj are also transitive. From Theorem XIII, G must therefore in this case be at least two -fold transitive. We have then the following Corollary. If a primitive group G contains a transitive subgroup of lower degree, then G is at least two-fold transitive. § 86. We turn now to a series of properties based on the the- ory of self- conjugate subgroups. Let jff = [1, S2,S3, . . . sj be a self- conjugate subgroup of a group G of order n = km. The substitutions of G can be arranged (§ 41) in a table, the first line of which contains the substitutions of H. *1 -— Ij ^2 J ^3 , . . . Stnj *^21 ^2^2i ^3^2) • • • ^m^2i ^3) ^2^Sj ^3*^3) • • • ^m'^Sj From the definition of a self- conjugate subgroup we have then that is, the line of the table in which the product {s^^^a) (^/x'^'/s) occurs depends only on ^a and ^ = 1, 2, . . . m) of the a*^ line of the table. The fs therefore form a group T, which is (1-m)- 96 THEORY OF SUBSTITUTIONS. fold isomorphic with the given group G. The degree and order of T are equal, and both are equal to k (Cf. § 97). To the identical substitution of T corresponds the self-conjugate subgroup H of G. We shall designate T as the quotient of G and H , and write accordingly T—G\ H. § 87. A group G which contains a self-conjugate subgroup H, different from identity, is called a compound group; otherwise G is a simple group. If G contains no other self-conjugate subgroup K which includes H, then i?is a maximal self -conjugate subgroup. If G^ is a compound group, and if the series of groups G, Gj, G^2j • . • G^-, 1 is so taken that every G\ is a maximal self- conjugate subgroup of the preceding one, then this series is called the series belonging to the compound group G, or the series of composition of G, or, still more briefly, the series of G. If the numbers r, rj = r : e^ , ^2 — ^ ^i • ^2 ? • • • ^/u. ^^ ''ha — 1 • ^/u, 5 ^/u. + 1 ^^ '"/* • ^/u. + 1 =^ 1 are the orders of the successive groups of the series of composition of G, then ^i, 63? • •'• ^/oi + i are called the factors of composition of G] and we have r = e^ 63 63 . . . e^ + 1 . If, in accordance with the notation of § 86, we write, G:Gi=^ r^, Gj : G^2 = A J • • • ^m - 1 • ^m = ^m > Gfi • 1 ^= A the order and the degree of every 1\ is equal to Ca (« = 1, 2, . . . /x + 1). All the groups Fa are simple. For F^ is (l-ra)-fold isomorphic with Ga-i, and to the identical substitution in -/« corresponds Ga in Ga-\. Consequently, if !'„, contains a self-conjugate subgroup dif- ferent from identity, then the corresponding self- conjugate sub- group of Ga-i (§ 73) contains and is greater than Ga- The latter would therefore not be a maximal self- conjugate subgroup of G^a-i. The groups /', which define the transition from every Ga to the following one in the series of composition, are called the factor groups of (r. * * O. Holder; Math. Ann. XXXIV, p. 30 ff. GENERAL CLASSIFICATION OF GROUPS. 97 § 88. Given a compound group G, it is quite possible that the corresponding series of composition is nob fully determinate. It is conceivable that, if a series of composition G, Gi, G2, . . . Gfj,,! has been found to exist, there may also be a second series G, G\, G'2, . . . G'v,l in which every G' is contained as a maximal self- conjugate subgroup in the preceding one. We shall find however that, in whatever way the series of composition may be chosen, the number of groups G is constant, and moreover the factors of composition are always the same, apart from their order of succession. Suppose the substitutions of Gi and G\ to be denoted by Sa and s'a respectively. Let r^ = r : e^ be the order of Gi , and r\—r: e\ that of G\ . The substitutions common to Gi and G\ form a group r (§ 44), the order x of which is a factor of both r^ and r\ . We write r^ = xy, r\ = xy\ The substitutions of F we denote by §2^3J • . . ^2^x1 §2^^J where the § belonging to any line is any substitution of G^ not con- tained in the preceding lines. The group G\ can be treated in the same way. We will suppose that in this case, in place of §i , §2 » • • • r we have s'l, ^\^ . . . Every substitution of G' or G\ can, then, be written in the form Again, the product belongs to G^. For, since G'^G^G—G^, it follows that s'^-'s^^, which occurs in the second form of the product, is equal to s^? and the product itself is equal to s'^s^. But, from the third form, this 98 THEORY OF SUBSTITUTIONS. same product belongs to G', since G~^G'G= G' , and therefore Sa~^s'p'~^Sa = s'y, SO that the product is equal to s'ys'^. Conse- quently the product belongs to the group F which is common to Gi and G\ . Hence In particular, since the o-'s belong to both the s's and the s"s, we obtain From this it follows that the substitutions of the form ^a^'^^y form a group @ . For, by repeated applications of the equation B), we obtain The group @ is commutative with G; for we have The group (S is more extensive than Gi or G'; it is contained in G; consequently, from the assumption as to G^ and G\ & must be identical with G. The order of (S is equal to xy ■ y'. For, if §a§'^<^v = ^c^'i^c it is easily seen that a — a, b = /S, c = y. Consequently the order of G is also xyy\ and since we have r = ViBi = xye^ , r — r\e\ = xy'e\ it follows that y'=ei, y = e\. r 7\ r\ This last result gives us for the order of T, x = 1 1 1 ^1 We can show, further, that T is a maximal self- conjugate subgroup of (tj and of G\ , and consequently occurs in one of the series of composition of either of these groups. For in the first place I\ as a part of G\ , is commutative with G^ , and, as a part of G^ , is com- mutative with (r'l , so that we have Gr TG, = G\ , G'r 'rG\ = g, . But since the left member of the first equation belongs entirely to Gi , the same is true for the right member, and a similar result holds for the second equation. Consequently GENEEAL CLASSIFICATION OF GEOUPS. 99 Gr'vG, = r, G'r'rG\ = r. Again there is no self-conjugate subgroup of G^ intermediate be- tween Gi and i' which contains the latter. For if there were such a group H with substitutions ta , then it would follow from A) that that is, iJ is also commutative with G'l. And since Gy and G'l together generate G, it appears that H must be commutative with G. If now we add to the fa's the §'3? S'3, . . . , then the substitu- tions §'atp form a group. For since /' is contained in if and in G^ we have from A) This group is commutative with G, since this is true of its compo- nent groups H and G\. It contains G\^ which consists of the sub- stitutions S'a'^/s. It is contained in G, which consists of the substitu- tions §'a§i3S- ^^^ this is contrary to the assumption that G\ is a maximal self-conjugate subgroup of G. We have therefore the fol- lowing preliminary result: If in two series of composition of the group G, the groups next succeeding G are respeetively G^ and G\ , then in both series we may take for the group next succeeding G^ or G\ one and the same max- imal self-conjugate subgroup l\ which is composed of all the substi- tutions common to G^ and G\. If e^ and e\ are the factors of composition belonging to Gi and G\ respectively, then F has for its factors of composition, in the first series e\, in the second e^. § 89, We can now easily obtain the final result. Let one series of composition for G be 1) G, Gi,G2, G3, . . . , r, r, = r:ei, r. — r^-.e^, r-^^r^-.e^, . . . , and let a second series be 2) G, G\, G'2, G's, . . , ,' r '.e I, r 2 — '^^1*^25 ^'3 — ^'2*^3 Then from the result just obtained, we can construct two more series belonging to G: 100 THEOEY OF SUBSTITUTIOXS. 3) G, G„I\ J,H,... 4) G, G\, F, ^ H, . , . , r,r, = r:e,, r^ — r.-.e'^,..,, r,r\ = r:e\, r'^ = r\\e, ..., and apply the same proof for the constancy of the factors of com- position to the series 1) and 3), and again 2) and 4), as was employed above in the case of the series 1) and 2). The series 3) and 4) have obviously the same factors of composition. The problem is now reduced, for while the series 1) and 2) agree only in their first terms, the series 1) and 3), and again 2) and 4), agree to two terms each. The proof can then be carried another step by constructing from 1) and 2) as before two new series, both of which now begin with G, G^ : 1') G, G,,G^„@,$,3,..., J 'n '2 — 'l'*^2 5 ' 3 — /2-t>2>'-' 3') G,G„ r,®,o,3,..., ^> ^] 5 ^ 2 =^ '^1 • ^ 2 ? ^ 3 ^ '^ 2 • ^2 ) • • • These series have again the same factors of composition, and 1') and 1) and again 3') and 3) agree to three terms, and so on. We have then finally Theorem XXIII. // a compound group G admits of two different series of composition, the factors of composition in the two cases are identical, apart from their order, and the number of groups in the two series is therefore the same. § 90. From § 88 we deduce another result. Since G~^rG belongs to Gi, because G~'^GiG= G^, and also to G\ because G~^G'iG= G\, it appears that G~^rG, as a common subgroup of Gi and G\, must be identical with T, so that F is a self- conjugate subgroup of G. From § 86 it follows that it is possible to con- r struct a group ii of order e^e'^ which is (1 ^)-fold isomorphic with G, in such a way that the same substitution of i2 corresponds to all the substitutions of G which only difPer in a factor (t. We will take now, to correspond to the substitutions 1, §0? ^35 • • • ^/i ot Gi, the substitutions 1, 10^,0)^, . . . w/^ of i2, and, to correspond to the 1, §'2, §'3, . . . S'^j of G'l, the substitutions 1, (o'^, w'3, . . . M, and, since Sa^'is = ^'^^a^^-y, it follows that ^^ ^\ = ^\ P^^ . Moreover every s in G is equal to Sa^'^S, that is ^ = P]P\. We obtain P therefore, by multiplying every substitution of P^ by every one of P\. § 91. We consider now two successive groups of a series of composition, or, what is the same thing, a group G and one of its maximal self -conjugate subgroups H. Suppose that s\ is a substi- tution of G which does not occur in H, and let s\"' be the lowest power of s\ which does occur in H {m is either the order of s\ or a factor of the order). If m is a composite number and equal to p g, we put s V = ^1 ) and obtain thus a substitution Sj which does not occur in IT, and of which a prime power s^^ is the first to occur in H. We then transform Sj with respect to all the substitutions of G, and obtain in this way a series of substitutions Si, S2, . . . s^. No one of these can occur in H. For if this were the case with Sa = c>~^Si(t, then <7Sa^~^ = s^, being the transformed of a substitution Sa of H with respect to a substitution (r~^ oi G, would also occur in H. We consider then the group r={H,s„S2,...s^\. This group contains H and is contained in G. If t is any arbitrary substitution of G, we have t-Tt = t-'{Hsi''s.f. ..\t = t-'Ht-t-'s'^t . t-'s/t . . . = Hs,^%.^... =r. r is therefore commutative with G. These three properties of F are inconsistent with the assumption that H is a maximal self-con- jugate subgroup of G, unless F and H are identical. If we remember further that all substitutions, as Si,S2, . . . s^, which are obtained from one another by transformation, are similar, we have Theorem XXIV. Every group of the series of composition of any group G, is obtainable from the next following {or, every group is obtainable from any one of its maximal self- conjugate 102 THEORY OF SUBSTITUTIONH. subgroups) by the addition of a series of substitutions^ 1) ivhich are similar to one another, and 2) a prime poicer of ivhich belongs to the smaller group. The last actual group of a series of composi- tion consists entirely of similar substitutions of prime order. § 92. The following theorem is of great importance for the theory of equations: Iriieorem XXV, The series of composition of the symmet- ric group of n elements, consists, if n^ 4,, of the alternating group and the identical substitution. The corresponding factors of com- position are therefore 2 and ^n\ The alternating group of more than four elements is simple. We have already seen that the alternating group is a maximal self- conjugate subgroup of the symmetric group. It only remains to be shown that, for n > 4, the alternating group is simple. The proof is perfectly analogous to that of § 52, and the theorem there obtained, when expressed in the nomenclature of the present Chap- ter, becomes: a group which is commutative with the symmetric group is, for n > 4, either the alternating group or the identical sub- stitution. It will be necessary therefore to give only a brief sketch of the proof. Suppose that H^ is a maximal self -con jugate subgroup of the alternating group H, and consider the substitutions of H^ which afPect the smallest number of elements. All the cycles of any one of these substitutions must contain the same number of elements (§ 52). The substitutions cannot contain more than three elements in any cycle. For if H contains the substitution s — ypc-YJC^uC^iC ^ ...).,., and if we transform s with respect to ^r = {x2X.iX^, which of course occurs in H, then s~ V~^srt- contains fewer elements than s. Again the substitutions of H^ with the least number of the ele- ments cannot contain more than one cycle. For if either Sa ^= (^^1^2) y^a'^*) . . . J Sj3 = {X^X2Xj) {X^X^X^j . . . , occurs in H, and if we transform with respect to groups Hy_^, H'v-i . . . which are all similar and all belong to the factor e, and which give Theorem XXIX, If a series of composition of G does not coincide with a principal series, hut if, betiveen tivo groups H and J of the latter, v — 1 groups Hi^H^, . . . H^_i of the former are inserted, theyi to H^, Hoy ... J belong the same factoids of composition e, and the order r' of G is therefore equal to the order r" of J mul- tiplied by £*'. H can be obtained from J by combining with J a series of v groups Hy_^, H\_^, . . . , ivhich are all similar, and of the order r"s. Corollary I. If the factors of composition of a group are not all equal, the group has a principal series. Corollary II, Every non-primitive group is compound if it contains any substitution except identity which leaves the several systems of non-primitivity unchanged as units. If the group con- tains greater (including) and lesser (included) systems of non-prim- itivity, it has a principal series. The instance of the group G = [1, (X^X^) (x^Xi) (x^Xq), (x^X^) (X2X^) (XiXe), (xiX^) (x^x^) (x^x^), [XiX^X^) [X2X^Xq), yXiXr^X^) {^X2XqX2) J shows that non-primitivity may occur in a simple group. In this case the only substitution which leaves the systems Xi,X2i X3,x^, and x^, a?6 unchanged is the identical substitution. Corollary III, The groups H^_^,H\_^,H'\_^, . . . are commutative, i. e., the equations hold ^,_/«) H,_,^^) = H^-i^^^ H, (*). For in the series preceding J we may assume the sequence iZ"v_iH ji2'^_iH-H'v-/^^h • • • to occur. Accordingly we must have GENEEAL CLASSIFICATION OF GROUPS. 107 or Corollary IV. The last actual group M of the principal series of G is composed of one or more groups similar to one another, which have no substitutions except identity in common, and which are commutative with one another. § 95. We have now to consider the important special case where £ is a prime number p. Instead of if'v_i, -ff"v_i, ... we employ now the more conven- ient notation H\ H'\ H"\ . . . ifW. Then H' is obtained from J by adding to the latter a substitu- tion ^1 , the p^^ power of which is the first to occur in J. We may write (§ 91) H' = t,^J, H^'^t^^'J, ir'"=f3«J,... (a = 0, 1, . ..p— 1). Since J is a self- conjugate subgroup of every one of the groups H',H'\ ...,wehave t,-''jti'' = j, t^-'^jt.'^^j, tr''Jtz- = J, '■ ' and, if we denote the substitutions of J by i^ , % , e's , . . . , t J 2.J tj 2'2, ^2 ^1 ^2 ^ ^3 5 ^3 *1 ^3 ~~ ^4 ) • • • ? t\ h ^ Hh (*2%)i ^2 H ^-^ ^1^2'* (%*!/? *3 H ^- ^1^3 V*4*lj) • • • ) that is, the substitutions of H\ of H'\ of H"\ and so on, are com- mutative among themselves, apart from a factor belonging to J. Since we can return from J" to iJ by combining the substitutions of H' and H'\ for example, into a single group (§ 88), we have from § 94, Corollary III ^2 ^1^2 """ *^1 ^15 ^1 ^2 ^1 ^2 ^2) and consequently, by combination of these two results, t\ ^2 ^1^2 ~~ t^ t'^2 ~~ ^2 % ) = t^ ti°-l 1 = f 1* *i 5 t\^ ^1 ^^ *2 ^3 • The left member of the last equation is a substitution of H\ the 108 THEORY 0¥ SUBSTITUTIONS.- right member a substitution of H". Since these two groups have only the substitutions of J in common, the powers of ^i and /, must disappear. Consequently a = 1, /?= — 1, and {tih)(i2h)={¥2){tih)is, (t,%)(t/i,) = {t/H)(t,%)i,. The substitutions of the group formed from J, t^, and 1.2 are therefore commutative among themselves, apart from a factor be- longing to J. The same is true of the group formed from J, t^, and fg, or from J, t.y, and ^3, and consequently of the group \J,ti,t2,tr^\, and so on, to the group H itself. (It is to be noted that Corollary III of § 94 involves much less than this. There it -was a question of the commutativity of groups, here of the single substitutions.) Every two substitutions of H are, then, commutative apart from a factor belonging to J. We will prove now the converse proposi- tion: If two substitutions of H are commutative apart from a fac- tor belonging to J, then e is a prime number. In fact this will be the case, if the substitutions of H' have this property. For, this being assumed, if £ were a composite number, suppose its prime fac- tors to be g, q\ q'\ . . . We select from H\ , in accordance with Theorem XXIV, § 91, a substitution t which is not contained in J. The lowest power of t which occurs in J will then be, for example, P. Transforming, we have and, since by assumption, t'^H' = H'f^J, H'-\t''J)H' = t''J. The group \t, J\ is therefore a self-conjugate subgroup of H\ which contains J and is larger than J. Moreover, it is contained in H', and is smaller than H'. For, if t is commutative with J, then from §§ 37-8 the order of \t,J\ i8r'q•••?, > and therefore the substitutions of F replace $1 by any element ^n ^2 J •••'?»•. Again every substitution of G alters the order of ^1, ^2) • • . ^r) for I is a n! -valued function. Consequently every sub- stitution of /'also rearranges the Ci,?2)--*^r- The order of F is therefore equal to its degree, and both are equal to r. G and F are simply isomorphic. For to every substitution of G corresponds one substitution of F, and conversely to every substitu- tion of F at least one substitution of G. And in the latter case it can be only one substitution of G, since G and F are of the same order. Theorem XXXI. To any transitive group of order r cor- responds a simply isomorphic transitive group, the degree and order of which are both equal to r. Such groups are called regular. § 98. Theorem XXXII. Every substitution of a regular group, except the identical substitution, affects all the elements. A regular group contains only one substitution which replaces a given element by a prescribed element. Every one of its substitutions consists of cycles of the same order. If two regular groups of the same degree are {necessarily simply) isomorphic, they are similar i. e., they differ only in respect to the designation of the elements. Every regular group is non-primitive. * The greater part oE the the theorem is already proved in the preceding Section, and the remainder presents no difficulty. We need consider in particular only the last two statements. Suppose that /', with elements $1, ?2) • • • '^n and substitutions J A 5 ^k"^ A J . • • Similarly r^ may occur in other combinations '^a'a? ''■'a'^Aj ''■"a^a • • • We coordinate now with <>\ all the r^^ t\, r\, . . . , and with t;^ all the (Tx, ^'a, ^"a, . . . , and proceed in the same way with all the sub- stitutions sx of G. The (7a's form a group - and the r/s a group T. Suppose that t;^, ^r^ are coordinated with "a,"^^- Then there are substitutions Sa , s^ , s^ , such that Sk = pairs of values of (f and v's together with the p corresponding groups. For every integral value of /, the function is therefore, like 9^1 + f^- -h • • • H- 9^p or v''i + 4'-2 + . . . + v'v? ^^ iii^®- gral symmetric function of the elements Xy^x.,, . . . x„. For this function is merely the sum of all the values which (fx^4'\ can assume and is accordingly unchanged by any substitution, only the order of the several terms being affected. Accordingly, if (f^ and 0, are integral rational functions of the elements x^, then A^ is an integral rational function of Ci , c.2 , . . . c„ . Taking successively / = 0, 1, 2, . . . /> — 1, we write the corres- ponding equations: . . ^''p, every v''a is obtained ^I'^-'-rS + f^-V^ + ^a^^V'sH- If these equations are solved for c'-, , 6 as a rational function of ^1, ^2? • • • S^p- § 104. We multiply the first i> — 2 equations of the system B) successively by the undetermined quantities ^„, 2/1, 2/2 • • • 2/p-2) and the last equation by ?/p_i = 1, and add the resulting products, wri- ting for brevity 2/p-i9'-^""'-r2/p-29'"'"' + 2/p-3^''~'+ . . . H-^iC^ + 2/o=;^(^). We obtain then (1) <\ '/, (S^i) -h ^''2 / (^2) + . • . V^ Z (s^p) = Ao2/o + AxVx + ^22/2 + . . . . . • ~r -^p -22/p - 2 T~ -^p - i2/p - 1 • From this equation we can eliminate ^''2? ^''3, • • • v'v and obtain ^''1. For this purpose we need only select the t/'s so that we have simul- taneously ■/Xfi) = ^, ;?(fO = 0, ...;f(?p) = 0; z(n) + 0. 116 THEORY OF SUBSTITUTIONS. In Chapter III, § 53, we have shown that ^i, ^2, . . . cTp satisfy an equation of degree p the coefficients of which are rational in c, , C2 , . , . c,, . Again, the quotient ^(^) :==(c._c.,)(^-c.3)...(c.-cg — ip, vanishes if s^ = ctj, (p.>, . . . ^p. But, if o = cTj, we have fe — ^2)^ — ^3) . • • {^Pi — fp) = -^'fe)- The derivative X\(p^ is not zero, for if a?,, it'2, . . . ^„ are independ- ent, the values ^1, ^^2) • • • S^p are all different. We can therefore satisfy the requirements above by taking that is, yp-2 = — A, yp-S=i%, 2/p-4 = — ^4, • . . 2/0= ±i^"p- Or, if we write we have and consequently ^p - 2 == ^1 — «n Up - 3 = fi'— «i '7^1 + '^2 , .?/p - 4 = V^'— ^^-1 t^i'"+ 'h'Pi — «3 , • • . By substitution in (1) we obtain then (2) ,',Xfe)=«fe), ^'■■ = I^J- The value of ^''i thus obtained can be reduced to a simpler form as follows. The product X'{v,)X'{ — 1 with respect to ), and therefore Similar considerations hold for the values 4'2i4'i, - - - 4'^- We have therefore Theorem II. If two p -valued functions (fx and 4'k belong to the same group G\ , theii (/'\ can be expressed as a rational f miction of which the denominator is the discriminant J^ and is therefore rational and integral in Cj, Cg, . . . c„, ivhile the numerator is an integral rational function of ^x, of a degree not exceeding p — 1, with coefficients which are integral and rational in Cj, C2, . . . c„* § 105. The converse of Theorem I is proved at once : Theorem III. If two functions can be rationally ex- pressed one in terms of the other ^ they belong to the same group. *Cf. Kronecker: Cre]le9l, p. 307. 118 THEORY OF SUBSTITUTIONS. In fact, given the two equations it appears from the former that (p is unchanged by all substitutions which leave i!< unchanged, so that the group of cp contains that of 9'', while from the latter equation it appears in the same way that the group of v'' contains that of ^^ X-y OCyOC^ ~\ 3C2 J '\' OCy LiOCy)C2 \~ X"2 , '\' X-y ~j~ ciX-ytlC2 ~t~ 3C2 the expressions under the square root sign are all unchanged by the transposition (7 — {x-^x^. But it remains entirely uncertain whether the algebraic signs of the irrationalities are affected by this substitution. Considerations from the theory of substitutions alone cannot determine this question, and accordingly the sphere of appli- cation of this theory is restricted to the case of rational functions. If, in the last two irrationalities above, the roots are actually extracted and written in rational form 31 yXi XiJ^ IE \X\ |~ «^2/? it appears at once that the transposition <7 changes the sign of the former expression but leaves that of the latter unchanged, while in the case of the first irrationality this matter is entirely undecided. § 106. Theorems I and III furnish the basis for an algebraic classification of functions resting on the theory of groups. All rational integral functions which can be rationally expressed one in terms of another, that is, which belong to the same group, are regarded as forming a family of algebraic functions. The number f) of the values of the individual functions of a .family is called the order of the family. The several families to which the different values of any one of the functions belong are called conjugate families.''^ ♦L. Kronecker: Monatsber. d. Berl. Akad., 1879, p. 212. FUNCTIONS BELONGING TO THE SAME GROUP. 119 The product of the order of a family by the order of the cor- responding group is equal to 7il, ivhere n is the degree of the group. Every function of a family of order ft is a root of an equation of degree p, the coefficients of which are rational in Cj, C2, . . . c,^. The remaining /> — 1 roots of this equation are the conjugate func- tions. The groups which belong to conjugate fdmilies have, if i> > 2, n> 4:, no common substitution except the identical substitution. For p = 2 the tivo conjugate families are identical. For p = Q, n — 4t there is a family which is identical with its five conjugate families. § 107. In the demonstration of § 104 the condition that

. An including group corresponds to an included fam- ily and vice versa. The larger the group the smaller the family, the same inverse relation holding here as between the orders r and p. From the preceding considerations we further deduce the fol- lowing theorems: 120 THEORY OF SUBSTITUTIONS. Theorem V. It is always possible to find a function in terms of which any number of given functions can be rationally expressed. This function can be constructed as a linear combina- tion of the given functions. Its family includes all the families of the given functions. Thus any given functions 9?, 0, Xi • • • can be rationally expressed in terms of where «, /5, ^, . . . are arbitrary parameters. For the group of m is composed of those substitutions which leave -valued function was'^l expressed in terms of a symmetric function by the aid of an equa- tion with symmetric coefficients of which the former was a root. \ From the analogy of the two cases we can state at once the present result: FUNCTIONS BELONGING TO THE SAME GROUP. 121 Theorem VII. If the group of a mp-valued function

• ♦ • 9m 5 and in general the equation (A,) 9'"— A, ( is a symmetric function, there is no longer a discriminant, and the denominator is removed, as we have seen in Chapter III, § 53. § 109. One special case deserves particular notice. If the included group H of the function

• • • ^», therefore belong to one and the same group if, and can consequently all be rationally expressed in terms of any one among them, in accordance with Theorem I. The family of c''i is included in that of n cyclically. Moreover, since r"* corresponds to t"\ it fol- lows that r'"' leaves all the functions 9^1 , ^^2 > • • • 9>n unchanged. Ac- cordingly r'", and no lower power of r, is contained among the sub- stitutions of H^ . Furthermore we readily show that For the substitutions ifi, iJjr, . . . H^t"'~^ are all different and, since H, is of order — '- , there are m — - = — ^ of them. They are all nec- mp mp f) Til essarily contained m Gi, which, being itself of order — , cannot contain any other substitutions. From this it appears again that r is commutative with H^ . Theorem IX. If the equation {A^ is of prime degree my and if the group H^ of (p^ is a self-conjugate subgroup of the group Gi of v''ij then G^ contains a substitution r ivhich permutes fi 5 S^2? ' ■ ' 9^2 + ^Vs + • . • +«>'"" Vm- If we apply to this expression the successive powers of t or r, we obtain 124 THEORY OF SUBSTITUTIONS. /2 = 9''2 + "^S^3 + ^V4+ . . . +^"' Vl = <^~V y.2 — "•" V2 = ^'^~ 7 15 H ■consequently /I — /2 — . . — /«, . We have now to prove 1) that /^ belongs to the group if 1 , and 2) that 7i"' belongs to the group G^ . In the first place, since ^i , 9^2 ? • • • ^^ are unchanged by all the substitutions of H^ , the same is true of /i . Moreover if there were any other substitutions which left /, unchanged we should have, for example, and therefore The latter equation would then have one of its roots, and conse- quently all its roots, in common with the irreducible equation and we must therefore have But we may assume the function (f-^ to have been constructed by the n\ method of § 31 as a sum of — '- terms of the form x.'^x.j^ . . . with ^ mp undetermined exponents. The systems of exponents in 9^1 , f 2 ? • • • 9>n will then all be different, and therefore, since the a^'s are independ- ent variables, the equation •can hold only if (f^ ^ (p/^ and 9?^ = ^,., identically. The function /, therefore belongs to H^. It follows at once that /i'" belongs to G^ • For this function is unchanged by H^ and r, and consequently by G,= \H,,r\. No other substitutions can leave x^" unchanged. For otherwise X"* would take less than /> values, and its m^^ root Xi l^ss than inp Talues, which would be contrary to the result just obtained. FUNCTIONS BELONGING TO THE SAME GROUP. 125 Tlieoreill X. In order that the family belonging to a group H may contain functions the m^^^ power of which belongs to the family of a group G, it is necessary and sufficient that H should be a self- conjugate subgroup of G, or, in other words, that the famil'y of G should be a self- conjugate subfamily of that of H. From Theorems IX and X the following special case of the lat- ter is readily deduced : Theorem XI. In order that the prime power (f^' of a pp- valued function cp may have /> values, it is necessary and sufficient that there should be a substitidion r, commutative with the group H of ip, of which the p^^ power is the first to occur in H. Finally an extension of the last theorem furnishes the following important result: Theorem XII. If the series of groups G, Gi, G2, G3, . . . Gv is so connected that every Ga-i can be obtained from the following Ga by the addition to the latter of a substitution r^ commutative with Ga, of ivhich a prime i^ower, the pj^^, is the first to occur in Gay then and only then it is possible to obtain a i> - Pi ■ Pi . . . Pv-valued function belonging to G^ from a [> -valued function belonging to G by the solution of a series of binomial equations. The latter are then of degree Pi,P2,2h, • • -Pvj respectively. § 111. In the expression of a given function in terms of another belonging to the same family, we have met with rational fractional forms the denominators of which were factors of the dis- criminant of the given function. If we regard the elements x^, X.2, . . . X,, as independent quantities, as we have thus far done, the discriminant of any function ^ is different from zero, for the various conjugate values of c^ have different forms. But if any relations exist among ihe elements x, it is no longer true that a dif- ference in form necessarily involves a difference in value. It is therefore quite possible that if the coefficients in the equations f[x) = X" C-iX"~'^ -\- CoX""'^ — ... ± c„ = are assigned special values, the discriminant 126 THEORY OF SUBSTITUTIONS. may become zero. If this were the case, c> could not be employed in the expression of other functions of the same family. And it is conceivable that the discriminant of every function of a family might vanish. It is therefore necessary, in order to remove this uncertainty, to prove Theorem XIII. If only no Uvo x^s are equal, then ivhatever other relations may exist among the x^s, there are in every family functions the discriminants of which do not vanish. A proof might be given similar to that of § 30. It is however more convenient to make use of the result there obtained, that under the given conditions there are still ti!- valued functions of the form if = «(, -\- a^x^ + a,^x.2^ + . . . + «„^„ . We suppose the a's and the .r's to be free to assume imagin- ary (complex) as well as real values. This being the case, if the n\ values of mod. v''a + 1) • We take then the integer e so great that vv > (v''A+r+^''A+/+ . . . + v''.r) {^^ = 1, 2, . . . (^i!-i)). THE NUMBER OF VALUES OF INTEGRAL FUNCTIONS. 127 From every equation of the form n 4': + vs; + <': + . . • = 4': + 0/ + ^\^ + . . . it follows accordingly that a = « , 6 = /5 , c = ^ , . . . If now we apply the r substitutions of G to 4'\% and add the results, the sum is a function of the required kind. For in the first place w is evi- dently unchanged by G. And in the second place the properties of the equation 't') show that o> has /> distinct values, and consequently Ja> is not zero. CHAPTEE VI. THE NUMBER OF VALUES OF INTEGRAL FUNCTIONS. § 112. Thus far we have obtained only ocasional theorems in regard to the existence of classes of multiple-valued functions. We are familiar vpith the one- and two-valued functions on the one side and the 7i! -valued functions on the other. But the possible classes lying between these limits have not as yet been systematically exam- ined. An important negative result was obtained in Chapter III, § 42, where it was shown that /> cannot take any value which is not a divisor of n\. Otherwise no general theorems are as yet known to us. We can, however, easily obtain a great number of special results by the construction of intransitive and non-primitive groups. But these are all positive, while it is the negative results, those which assert the non-existence of classes of functions, that are pre- cisely of the greatest interest. The general theory of the construction of intransitive groups would require as we have seen in § 101, a systematic study of iso- morphism in its broadest sense. We shall content ourselves there- fore with noting some of the simplest constructions. Thus, if there are n = a-\-b-\-G-\- . . . elements present, and if we form the symmetric or the alternatiog group of a of them, the symmetric or alternating group of b others, and so on, then on multiplying all these groups together, we obtain an intransitive group of degree n and of order r = £ a! bid . . ., where £=:l,J,i,J,..., according as the number of alternating groups employed in the construction is 0, 1, 2, 3, ... , the rest being all symmetric. For the number of values of the corresponding functions we have then . ?i! P = salblel . . . THE SfUMBEE OF VALUES OF INTEGRAL FUNCTIONS. 129 By distributing n in different ways between a, 6, c, . . . , we can obtain a large number of classes of functions. For example, if n — 5, we may take a = 5; £ = J, /^= 2; = 10; ^5 = x^x^x^ -\- x^Xr^, a = 3, 6 = 2; e = 1 ^ = 20; ^e = {x^ — x^) (x^ — x-^) {x^ — x^) \ X^Xt^. ip^ — x^x = 40; s^8 = {x^ — x^) {x^ — x^ {x^ —x^ \ x^ a?5 . a = 3, 6 = 1, c = 1 ; £ = 1, /> = 20; ^^ = a^ja^gO^a. The imprimitive groups give rise in a similar way to the con- struction of functions with certain values. For example, for n — ^^ we may take any two systems of non-primitivity of three elements each, or any three systems of two elements each, and with these construct various groups, the theory of which depends only on that of groups of degrees two and three. § 113. General and fundamental results are not however to be obtained in this way. We approach the problem therefore from a different side, which permits us to give it a new form of statement. Oiven a />-valued function ^1 with a group G^ , we construct again the familiar table of § 41 : ^1; Sj — 1, §2, S3, . , .Sr ; G^ n\ '^2) S2 < n, Gi contains either a transpo- sition or a circular substitution of the third order, including in either case a prescribed element x^ . The same is obviously true of Any prescribed element X\. B) There are ^ ^ — ^ transpositions of the form (Xaa?^), (a == /J = 1, 2, . . . n). If therefore p <_ n — > ^^^^ if the first line of the table does not contain any transposition, then some other line con- tains at least two. If these have one element in common, as {x^x^^ {XgX^^ then, as we have seen in A), their product {XgX^Xy) occurs in (ti . If they have no element in common, as {x^x^, (XyXs), then their product (XaXp) (XyXs) also occurs in Gi . In either case Gi therefore contains a substitution of not more than four elements. C) There are (n — 1) (n — 2) substitutions of the form (x^XaXp), (a J=/? = 2, 3, . . . n). If therefore r <{n — 1) (n — 2), and if 6?i con- tains no substitution of this form, some other line of the table con- tains at least two of them. A combination of these shows that Gi contains substitutions which affect three, four, or five elements. Proceeding in this way, we obtain a series of results, certain of which we present here in the following Theorem I. 1) // the, number p of the values of a function is not greater than n — 1, the group of the function contains a sub- stitution off at the most, three elements, including any prescribed element. 2) // /> is not greater than ^ — -, the group of the function contains a substitution of, at the most, four elements. 3) If pis not greater than— ^ -, the group of the function THE NUMBER OF VALUES OF INTEGRAL FUNCTIONS. 131 contains a substitution of, at the most, six elements. ^ If p is not . .u n{n — l){n—2)...(n—k-{-l) ., ^ ... . greater than — — j^ ^ — -, the group of the func- tion co7itains a substitution of, at the most, 2k elements, b) If p is not greater than {n — 1) (n — 2) . . . {n — k-\-l), the group of the function contains a substitution of, at the most, 2k — 1 elements, including any prescribed element, so that the group contains at least ^r- such substitutions. 2k — 1 By the aid of these results the question of the number of values of functions is reduced to that of the existence of groups contain- ing substitutions with a certain minimum number of elements. § 114 In combination with earlier theorems, the first of the results above leads to an important conclusion. From Chapter IV, Theorem I, we know that the order of an intransitive group is at the most {n — 1)!. Consequently, the num- ber of values of a function with an intransitive group is at least n ' -. -rr- = n. For such a function therefore /> cannot be less than n. (n— 1)! Again, the order of a non-primitive group is, at the most, 2 ! I -^ ! I , so that the number of values of a function with a non-primitive n\ group is at least '- . For n = 4, this number is less than n; 2'— 1— » ^•2 2' but for w > 4, it is greater than n. For such a function then, if n > 4, ^ cannot be less than n. Again for the primitive groups it follows from Chapter IV, Theorem XVIII, in combination with the first result of Theorem I, § 113, that if p{n — 1) ! substitutions of G^ therefore rearrange only the p — 1 values (P2^^zi • - - ^p- Since p < n, there are at the most, only {p — 1)!^(^ — 2)! such rearrangements. Consequently among the r > {n — 1)! substitutions of G^ there must be at least two, 4 there is no such substitution (Chapter III, Theorem XIII). Consequently p^n. § 116. Passing to the more general question of the determina- tion of all functions whose number of values does not exceed a given limit dependent on n, we can dispose once for all of the less impor- tant cases of the intransitive and the non- primitive groups. For the purpose we have only to employ the results already obtained in Chapter IV. In the case of intransitive groups we have found for the maxi- mum orders: 1) r=(n — 1)!. Symmetric group of n — 1 elements, p = n. U — 1)1 2) r = - — -9-^- Alternating group of n — 1 elements, p = 2n. 3) r = 2 ! (n — 2) ! . Combination of the symmetric group of ?i — 2 nin — 1) elements with that of the two remaining elements, p = ^ . 4) r = (n — 2) ! . Either the combination of the alternating group of n — 2 elements with the symmetric group of the two remaining elements; or the symmetric group of n — 2 elements. In both cases p = n{n — 1). Etc. * L. Kronecker: Monalsber. d. Berl. Akad., 1889, p. 211. THE NUMBER OF VALUES OF INTEGRAL FUNCTIONS. 133 For the non- primitive groups we have 1) r = 2 ! I -^ ! I . Two systems of non-primitivity containing each ■^ elements. The group is a combination of the symmetric groups of both systems with the two substitutions of the systems them- selves, p —' — — ^-— . For n = 4, 6, 8, ... we have /> = 3, 10, 35, . . . 2) ?' = 3! I -^! j . Three systems of non-primitivity. The group is a combination of the symmetric groups of the three systems with n\ the 3 ! substitutions of the systems themselves, o = — :— . For Q r ^ ^^ n =r 6, 9, 12, . . . we have p = 15, 280, 5770, . . . VT V 3) r = 3 1 -^! I . As in 2), except that only the alternating group of the three systems is employed, p = ' For w = 6, 9, 12,. . . ii')- we have p = 30, 560, 11540, . . . The values of p increase, as is seen, with great rapidity. § 117. In extension of the results of § 113 we proceed now to examine the primitive groups which contain substitutions of four, but none of two or of three elements. Such a group G must contain substitutions of one of the two types The presence of Sg requires that of Sg^ = (XaXc) {xjfic^, which belongs to the former type. Disregarding the particular order in which the elements are numbered, we may therefore assume that the substitu- tion ^5 "^ \X\X2) yx^x^) occurs in the group G, We transform Sg with respect to all the substitutions of G and obtain in this way a series of substitutions of the same type which connect Xj , a?2 > ^3 > ^4 with all the remaining elements (Chapter IV, Theorem XIX). The group G therefore includes substitutions 134 THEORY OF SUBSTITUTIONS. similar to s^ which contain besides some of the old elements iCi, iCg, a?3, Xi other new elements x^,Xq,Xt, . . . This can happen in three different ways, according as one, two, or three of the old elements are retained. Noting again that it is only the nature of the connection of the old elements with the new, not the order of designation of the elements that is of importance, we recognize that there are only five typical cases : (^1^2) (^3^5)5 \^iXb) {XiX^J, {XiX^) {X2X^), {XiX^} [X^X^ff {x,x,) {x,x,). In the first case, for example, it is indifferent whether we take {X1X2) {X^X^}, {X1X2) {X^Xr,), {X^X^} {XiX^)j [X^Xi) {X2X^)j and in the last we may replace x^ by Xi^x^, or 0^4 , etc. The first and fifth cases are to be rejected, since their presence is at once found to be inconsistent with the assumed character of the group. Thus we have (^1^2) V'^3'^4) * \X1X2) [X^X^) = (X^X^X^jj Ll'^i'^a) {XsX^} • {XiX^) {^XQX^)] =■ {XiX^X2), the resulting substitutions in each case being inadmissible. There remain therefore only three cases to be examined, accord- ing as G contains, beside Sj , one or the other of the substitutions A.) {XiXs) {X2X^), B) {x,x^) {X2X^), C) (^1^5) (^3^6)> the first case involving one new element, the last two cases two new elements each. § 118. A) The primitive group G contains the substitutions S5 = (£Cia?2) (^3^4)) ^4 -^ (^1^3) (^2^5)1 and consequently also t = S5S4 = {x^x^XiX^Xi), Si = tsj;-^ = (XiX^) (XtX^). Since t is a. circular substitution of prime order 5, it follows from §83, Corollary I, that if n^l, is at least three-fold transitive. Then G must contain a substitution u, which does not affect Xi but THE NUMBER OF VALUES OF INTEGRAL FUNCTIONS. 135 replaces Xc^ by x^ and x^ by a?^ . If we transform Sg with respect to this substitution, we obtain If x„ is contained among a?2, a?3,a?4, a^g, then s' and s^ have only one element in common and if Xa is contained among x^yXg, ... then s' and S5 have only one element in common. Both alternatives therefore lead to the rejected fifth case of the preceding Section. If n>l^ G becomes either the alternating or the symmetric group. There is in this case no group of the required kind. For ti = 4 it is readily seen that there are two types of groups with substitutions of not less than four elements, both of which are however non-primitive. Groups of the type A) therefore occur only for n = 5 or n = 6. For n = 5 we have first the group of order 10, If we add to G^ the substitution (t — {x^x^x^x^, we obtain a second group of order 20 G,^=\s^,t,ff\ = \t, ^s • If ^a were X2 or x^ , then we should obtain, by transformation with respect to o-j or (T2, a substitution (XiXf,) (XcXa), SO that we may assume a = l. The possible cases are then (a) (X1X2) (x^x^), (xiX^) {X2X„,), (a^x^) (x^x^) m = 3,4, 6. (/9) (x^x^) (XnX^), m, n,p = S, 4, 6. (r) (XiX^) {x2X„)y (x^x^) {x^x„), m, n = 3, 4, 6. THE NUMBER OF VALUES OF INTEGRAL FUNCTIONS. 137 The substitutions of the first and second lines are to be rejected, since their products with ^9, at least 4-fold transitive. G contains therefore the sub- stitutions T = (xi) (x^x^) {x^x^ . . .), r-'(T,T = {x,Xs)(x2X,), so that we return in every case to the type A). For n>_9 there is therefore no group of the required type. Theorem III. If the degree of a group, which contains substitutions of four, but none of three or of two elements, exceeds 8, the group is either intransitive or non-primitive. Combining this result with those of § 113 and § 116, we have Theorem IV. If the number p of the values of a function is not greater than in{n — 1), then if n>8, either 1) p = in(n — 1), and the function is symmetric in n — 2 elements on the one hand and in the two remaining elements on the other ^ or 2) p = 2n, and the function is alternating in n — 1 elements, or d)p = n,and the function is symmetric in n — 1 elements, or 4:) p = lor 2, and the function is symmetric or alternating in all the n elements.* § 122. We insert here a lemma which we shall need in the proof of a more general theorem, f From § 83, Corollary II, a primitive group, which does not include the alternating group, cannot contain a circular substitution « •Cauchy : Journ. de I'Ecole Polytech. X Cahier; Bertrand: Ibid. XXX Cahier; Abel: Oeuvres completes I, pp. 13-21; J. A. Serret: Journ. del'Ecdle Polytech. XXXII Cahier; C. Jordan: Trait6 etc., pp. 67-75. t C. Jordan : Trait6 etc.. p . 664. Note C. THE NUMBER OF VALUES OF INTEGRAL FUNCTIONS. 139 of a prime degree less than -^ . If p is any prime number less o o than -K- , and if p^ is the highest power of p which is contained in o n!, then the order of a primitive group G is not divisible byp^. For otherwise G would contain a subgroup which would be similar to the group K of degree n and order p^ (§ 39). But the latter group by construction contains a circular substitution of degree p, n\ and the same must therefore be true of G. Consequently p = ^ must contain the factor p at least once. What has been proven for p is true of any prime number less 2n than -^ and consequently for their product. We have then o Theorem V, If the group of a function with more than two values is primitive, the number of values of the function is a multiple of the product of all the prime numbers which are less than -^. o § 123. By the aid of this result we can prove the following Theorem VI. If k is any constant number, a function of n elements which is symmetric or alternating with respect to n — k of them ha^ fewer values than those functions which have not this property. For small values of n exceptions occur, but if n exceeds a certain limit dependent on k, the theorem is rigidly true.* If (f is an alternating function with respect to n — k elements, the order of the corresponding group is a multiple of ^{n — k)\, and the number of values of the function is therefore at the most A) 2n{n—l) (n - 2) . . . (n— ^4- 1). If ^ is a function which is neither symmetric nor alternating in n — k elements, it may be transitive with respect to n — A; or more elements. But in the last case (p must not be symmetric or alterna- ting in the transitively connected elements. We proceed to determine for both cases a minimum number of *C. Jordan: Traits etc., p. 67. 140 THEORY OF SUBSTITUTIONS. values of 0, and to show that if n is sufficiently large, this mini mum is greater than the maximum number of values A) of n — k i. e., n>2k-{-l. The maximum order of the group is consequently (n— A;— 1)! (A; + l!, and the minimum number of values of is n ' n{n—l){n — 2)..,{n — k) ^ {n—k—l)\{k-j-l\)~ 1.2.3. ...(&+ 1) It appears at once that the minimum B) exceeds the maximum A), as soon as n>k-\-2{k+l)l This is therefore the limit above which, in the first case, the theo- orem admits of no exception. § 125. In the second case (p is transitive in n — x elements (z>_/f), but it is neither alternating nor symmetric in these ele- ments. The group G of (J> is intransitive, and its substitutions are therefore products each of two others, of which the one set ^1) ^2j • • ' connect transitively only the elements Xi,X2, . . . x„_^, while the other set t^^t^^ . . . connect only the remaining elements x„ _ It ^_ 1 , . . . x„. The substitutions of the group G of 2k. Consequently G contains substitutions (r^j-^ , fl-|3r|3, in which t^^t^ but 'J'a4"^^j ^"^^ therefore substitutions ffaJa ((Tprp)~^ = (TaTp~^ which affect only the elements a?j, a?2, . . .oc^-k of the first set. The entire complex of these substitutions forms a self- conjugate subgroup H of G. This subgroup is unchanged by transformation with respect to either G or S, since r^, r^, ... have no effect whatever on the substitutions of H. H is therefore a self - conjugate subgroup of the alternating group 2 , and must accord- ingly coincide with 2 (§ 92). if = S is therefore a subgroup of 6r, and 4' would, contrary to assumption, be alternating in w — x ele- ments. § 126. The maximum order of the group G is therefore equal to the product of z! by the maximum order of a non- alternating transitive group of n — /. elements. We denote the latter order by R{n — z). Then the minimum number of values of 4' is . n\ _ {n — x)! n{n — l).,.{n — x -|- 1) ^ AR{n—x)~ R{n — y)' x\ ' We have now still to determine R{n — x), the maximum order of a non- alternating transitive group of n — x elements, or -^. -A- , the minimum number of values of a non- alternating transitive function oi n — x elements. If this function is non-primitive in the n — x elements, it follows that the minimum number of values is („_.)(„_,^i),...('i_j'+i) Substituting this value in C) we obtain for the minimum number of values of 4' n{n-l). . , (n-x + l) (n-x) . . . fl-^+l) 142 THEORY OF SUBSTITUTIONS. We compare this number with the maximum number A) and examine whether, above a certain limit for n, C\) becomes greater than A), i. e,, whether >4x! !!LZ^in{n—l). , .(n— x + 1). For sufficiently large n we have I — ~ + l) <*^ — k-{-l. We have therefore to prove that n — X This is shown at once, if we write the right hand member in the form (4.![,_.]o([!^^]['i^_l]...[.-. + l]). For the first factor is constant as n increases, and the ratio of the left hand member to the second parenthesis has for its limit n-\-K 2—-'-. § 127. Finally, if the function

[2(|f-«.!]. This can be shown inductively by actual calculation, or by the employment of the theorem of Tchebichef , that if j^ > 3, there is always a prime number between v and 2v — 2. For we have from this theorem P(2v)>vP(v), P(2v) P(v) V > Now whatever value the first quotient on the right may have, we can always take t so great that the left hand member of P(2-v) P(.) f V Y increases without limit, if only v is taken greater than 2*'"". The proof of the theorem is now complete. The limits here obtained are obviously far too high. In every special case it is possible to diminish them. As we have, however, already treated the special cases as far as ^o = ^n(n — 1), it does not seem necessary, from the present point of view, -to carry these inves- tigations further. CHAPTER VII. CERTAIN SPECIAL CLASSES OF GROUPS. § 128. We recur now to the results obtained in § 48, and deduce from these certain further important conclusions.* Suppose that a group G is of order r=p'^m, where p is a prime number and m is prime to p. We have seen that G contains a sub- group H of order p". Let J be the greatest subgroup of G which is commutative with H. J contains H, and the order of J is there- fore pH, where i is a divisor of m and is consequently prime to p. Excepting the substitutions of H, J contains no substitution of an order p^. For if such a substitution were present, its powers would form a group L of order p^. But if in A) of § 48 we take for Gii Hi, Ki the present groups «/, X, jK", then since 0. These sub- groups would have only the identical substitution in common. They would therefore contain in all (p-l)(px-hl) + l=p[(p-l)x + l]>pg substitutions. This being impossible, we must have x = 0. CERTAIN SPECIAL CLASSES OF GROUPS. 147 The subgroup H contains only p substitutions; the rest are all of order q. Their number is pq—p={q — l)p. There are therefore p subgroups of order g, and consequently from Theorem I we must have that is, q must be a divisor of p — 1. Only in this case can there he any new type ^. 3) The group H ia a. self-conjugate subgroup of ^. Conse- quently every substitution t of order q must transform the substitu- tion s of H into s", where a might also be equal to 1. We write (where the upper indices are merely indices, not exponents). Then no cycle of t can contain two elements with the same upper index. For otherwise in some power of t one of these elements would follow the other, and if this power of t were multiplied by a proper power of s, one of the elements would be removed. With a proper choice of notation, we may therefore take for one cycle of t \ 1 1 1 • • • *^1 /• It follows then from t-^st = s'' that t replaces x^^ by a?„ + i* + ', x^^ by X2a + i^'^\ . . . Xa+i^ by a?a« + i^^\ . . • so that we have t = yXi Xi . . . Xi ) . , . (iCft ^ I Xaa -\- 1 ^aa^ + 1 • • • ^aa9 — 1+1 ...).•• If now the latter cycle is to close exactly with the element Xaa^-^+i% we must have aa^-{-l^a-\-l, a^^l (mod. p). The solution a:=^l is to be rejected, for in this case we should have T \P^i «^l • • • X-^ ) ( M?2 "^2 • • • '^2 ) ' ' ' \p^p '^p « • • Xp ), St ^^^ (Xi X2 . . . Xq Xq _[. I Xq _^ 2 • • • 7 • • • i SO that the latter substitution would contain a cycle of more than q elements, without being a power of s. 148 THEUJtv/ OF SUBSTITUTIONS. It follows then from the congruence a*^l (mod. p) that q* is a divisor of p — 1, as we have already shown; further that a, belong- ing to the exponent g, has q — 1 values a^,a2, . . . a^-n finally that all these values are congruent (mod. p) to the powers of any one among them. From t^^st — s" follows so that, if s is transformed by t into any one of the powers s*a, there are also substitutions in ^ which transform s into s"i,s"2, . . . s"-?-'. Accordingly the particular choice of a\ has no influence on the resulting group, so that if there is any type ^ generated by substi- tutions s and t^ there is only one. The group formed by the powers of t being commutative with that formed by the powers of s, the combination of these two sub- stitutions gives rise to a group exactly of order pg. The remaining pq — p — q-\-l substitutions of the group are the first ^ — 1 powers of the p — 1 substitutions conjugate to t s-^+Hs^-' = (cc^VV . • . ^^') . . • ('5 = 2. 3, ...p). If p and q are unequal, we have therefore only one 7ieio type 9., § 131. Finally we determine all types of groups il of degree and order pi 1) The cyclical type, characterized by the presence of a substi- tution of order pi\ is already known. 2) If there are other types, none of them can contain a sub- stitution of order pi There are therefore in every case p^ — 1 sub- stitutions of order p and one of order 1. If s is any substitution of ^, and t any other, not a power of s, then 9 is fully determined by s and t. For all the products sH'' (a,6 = 0,l,2, ...p — 1) are different, and therefore i2=z[8"f] (a,5 = 0,l,2, ...p — 1). We must have therefore ts = s«i ^^1 , fs = s^^ i%...i '' - 's ^ 6-«^' " ' t^J' - 1 . If now two of the exponents d are equal, it follows from CERTAIN SPECIAL CLASSES OF GROUPS. 149 rs=:sH% fs = sH" (a-^b, sis') that (rs)-\fs) = s-H's ^ (sH')-\sH") ■:= fy. Since for t" we may write t, it therefore appears that ^ contains a substitution t which is transformed by s into one of its powers P. The same result holds, if all the exponents S are different. For one of them is then equal to 1, since none of them can be 0, and from Ts =: st^ follows s~H"s = t^. 3) There is therefore always a substitution T — [Xi X^ . . . Xp J [X^ X2 . . . Xp j . . . (^1 X2 ... Xp ) which is transformed by s into a power of itself t"^. As in the pre ceding Section, we may take for one cycle of s (x^^x^^ . . . X^P). Then from s'Hs^^r follows S = (Xi^X,^ . . . Xf) {X2^Xa + i^Xa2 + i^ . . . XaP-^+Z' ...)... If the second cycle is to close after exactly p elements, we must have a^ + l = 2, a^=l (mod p). This is possible only if a = 1. Accordingly S — [Xi Xi . . . Xi ) yX2 X2 , . . X2 ) . . . \Xp Xp . , . p j. The p-\-l substitutions are all different and no one of them is a power of any other one. Their first p — 1 powers together with the identical substitution form the group i2. Summarizing the preceding results we have Theorem II. There are three types of groups ^, for which the degree and order are equal to the product of two prime num- bers: 1) The cyclical type, 2) one type of order pq (p>q)^ 3) one type of order pi The first and third types are always pres- ent; the second occurs only when q is a divisor of p — 1. § 132. We consider now another category of groups, character- ized by the property that their substitutions leave no element, or 150 THEORY OF SUBSTITUTIONS. only one element, or all the elements unchanged. The degree of the groups we assume to be a prime number p. Every substitution of such a group is regular, i. e., is composed of equal cycles. For otherwise in a proper power of the substitu- tion, different from the identity, two or more of the elements would be removed. The substitutions which affect all the elements are cyclical, for p is a prime number. From this it follows that the groups are tran- sitive, and again, from Theorem IX, Chapter IV, that the number of substitutions which affect all the elements is p — 1. We may therefore assume that and its first p — 1 powers are the only substitutions of p elements which occur in the required group. The problem then reduces to the determination of those substi- tutions which affect exactly p — 1 elements. If t is any one of these, then t~^st, being similar to s, and therefore affecting all the ele- ments, must be a power of s where every index is to be replaced by its least positive remainder (mod p). Since it is merely a matter of notation which element is not affected by t, we may assume that x^ is the unaffected element. It follows that If now gr is a primitive root (mod. p), then all the remainders (mod. p) of the first p — 1 powers of g G) g\cf,g\...g^-\g^-'=l (mod. p) are different, and we may therefore put m^gf^ (mod. p). We will denote the corresponding thyt^. It appears then that t^ p 1 consists of At cycles of — elements each. For every cycle of t^ closes as soon as CERTAIN SPECIAL CLASSES OF GROUPS. 151 am*+l = a+l, ^^ ^ (mod. p) and this first happens when z — . If there is any further substitution t^ which leaves x^ unchanged and which replaces every a?« + i by a?„/+i, then iy^tv^ replaces every a?a + i by a?„^aM + /3v^i. If now we take « and /5 so that aix-\-^v is congruent (mod. p) to the smallest common divisor o) of /^ and \>y we have in ^Oi — *'|U, f^v a substitution of the group, of which both t^ and f^ are powers. Proceeding in this way, we can express all the substitutions which leave x^ unchanged as powers of a single one among them t^j where g*^ is the lowest power of g to which a substitution t of the group corresponds. »— 1 The group is determined by s and t^^. Since t^^ is of order , it follows from Theorem II, Chapter IV, that the group contains in p(p — 1) all ^ substitutions. i = 0,l,2,...ff-l . * L. Kronecker; cf. F. Klein: Math. Ann. XV, 258. CERTAIN SPECIAL CLASSES OF GROUPS. 153 ^p=\z lSz\ (mo(J.p) (/5 = 1,2,3, ...p-1). The symbol t=\z fiz-\-a\ (modp) (6c = 0,l, ...p — 1; /9= 1, 2, . . .p — 1) includes all the substitutions Sa, f^p, and their combinations. Since \z I3z + a\.\z l^^z-\-a,\ = \z l3,3,z-{-a,l3-\-a\, it follows that the substitutions t form a group of degree p and of order jp(p — 1). This group therefore coincides with that of § 134. If we prescribe that the /?'s shall take only the values the products /?/?i belong to the same series, and we obtain the group vip — 1) of degree p and of order — considered above. § 136. The consideration of the fractional linear substitutions (mod. p) leads to groups of degree p + 1 and of order {p-\-V)p (p — 1). These substitutions are of the form az-{- [3 (mod. p), yz-^ where z is to take the values 0, 1, 2, . . .p — 1, oo , the elements of the group being accordingly Xq,XisX2, . . . x^_i,x^. The values «j /^j ?') ^1 determine a single substitution u, but it may happen that one and the same u results from different systems a, /?, y, d. To avoid, or at least to limit this possibility, we make use of the determinant of u D) ad — ftr. According as this is a quadratic remainder or non- remainder, we divide numerator and denominator of — ^|-^ by A^ad — ^y or by \/l3y — ad. For the new coefficients we have then D') ad — i3y=±i (mod. p). If now for two different systems of coefficients a, /5, y, S and ^i»/^i)rj^i the relation az-\-l3___a,z-^r yz-i-d y,z + (mod. p) 154 THEORY OF SUBSTIT 0TI0N8. were possiblGj it would follow from the comparison of the coeffi- cients of ^^ z^, and z^, with the aid of D', that if a, a', /5, /3', . , . are real, a 13 r 8 I ad — fir . / -. x If, therefore, we restrict the range of the values of a, /?, /', d to 0,1,2, ... p — 1, there are always two and only two different sys- tems of coefficients which give the same substitution u. With D') it is assumed that ad — i3y is different from 0. This restriction is necessary, for the symbol u can represent a substitu- tion only if different initial values of z give rise to different jSnal values of 2, i. e., if the congruence az-\ri3 az,-\-f3 is impossible. This is ensured by the assumption ad — 13}' ^0. We determine now how many elements are unchanged by the substitution u. An index z can only remain unchanged by u if E) rz'-{-{S—a)z—l3=0 (mod.p). There are accordingly four distinct cases: a) The two roots of E) are imaginary. This happens if (^') Tl {ad—^r=±l) is a quadratic non-remainder (mod. p). The corresponding substi- tutions affect all the elements Xq,Xi,X2j . . . x^_i, x^. b) The two roots of E) coincide. This happens if (^)t1=0 (mod.p) {ad-l3r=±l). The corresponding substitutions leave one element unchanged. c) The two roots of E) are real and distinct. This happens if is a quadratic remainder (mod. p). The corresponding substitutions leave two elements unchanged. d) The equation E) may vanish identically. This happens if r=0, 13=0, a = d (mod. p). CERTAIN SPECIAL CLASSES OF GROUPS. 155 The corresponding substitution leaves all the elements unchanged. Finally we observe also that M) .«^ + ^ rz + d z z {ra^-^^r,)z + {r^, + ^^,) n^ + ^i N) (ad—i^r) («i^i — /5i n) = (««i + M (r/5i + ^^0 -(«/5i+W(r«i + ^ri). We proceed now to collect our results. If we take « not ^0 (mod. p), and /5 and y arbitrarily, then for each of the (p — l)p^ resulting systems we obtain two solutions of D'). Since however there are always two systems of coefficients which give the same substitutions u, We have in all, in the present case, p^ — p^ substitu- tions. Again, if we take «^0 (mod. p) and ^5 arbitrarily, then restricting /? to the values 1,2, .. .p — 1, we obtain from D') for every system a,d,^ two values of y; but as two systems of coef- ficients give the same u, we have in this case p{p — 1) substitutions. There are therefore in all p^ — i> = (p + l)p(p — 1) fractional linear substitutions (mod. p). From M) it appears that these form a group. Among them there are ^^ ^- — — - substitutions which correspond to the upper sign in D'). From M) and N) it is clear that these also form a group. This latter group is called ^^the group of the modular equations for p^\* Both groups contain only substitutions which affect either p -\- 1, or p, or p — 1 elements, or no element. Those substitutions which leave the element x^ unchanged, for which accordingly y^O, form the metacyclic group of § 134. As the latter is two- fold transitive^ it follows (Theorem XIII, Chapter IV) that the group of order {P + 1)P {P — 1) is three-fold transitive. Theorem III. The fractional linear substitutions (mod. p) form a group of degree p + 1 and of order {p -\- 1) p [p — \\ Those of which the determinants are quadratic remainders {mod. p) form a subgroup of order — ^ ^ ^^^ group of the modu- lar equations for p. If any substitution of these groups leaves more than tivo elements unchanged, it reduces to identity. The first of the two groups is three fold transitive. * Cf. J. Gierster: Math. Ann. XVIII, p. 319. 156 THEORY OF SUBSTITUTIONS. To construct a function belonging to the group of the fractional linear substitutions, we form first as in § 133, a function V'] of the elements Xq, x^, X2, . . . Xp_-^ which belongs to the group of substi- tutions t=\z ,3z-^a\ (mod.p). (a = 0, 1, 2, ...p— 1; /?=:1,2, ...p — 1) The substitutions u, applied to ^'1 , produce p-\-l values ^11 ^2) • • • ^p + n which these substitutions merely permute among themselves. Ac cordingly, if ^' is any undetermined quantity, the function S = (¥- f\) {¥- ¥,)... {¥- !f;^,) belongs to the given group. § 137. We have now finally to turn our attention to those groups all the substitutions of which are commutative. We employ here a general method of treatment of very exten- sive application.* Suppose that 0\ d'\ 0'" . . . are a series of elements of finite number, and of such a nature that from any two of them a third one can be obtained by means of a certain definite process. If the result of this process is indicated by /, there is to be, then, for every two elements 0\ 0", which may also coincide, a third element d"\ such that f{0' 0") = d'". We will suppose further that f{o\e")=f{o",o'), f[_o'J{e",e'")-]=flf{e',d"\d'"-\, but that, if 0" and 6'" are different from each other, then f{l>',O")-]^f(0',0"'). These assumptions having been made, the operation indicated by / possesses the associative and commutative property of ordinary multiplication, and we may accordingly replace the symbol f{0', 0") by the product 0' 6"^ if in the place of complete equality we employ the idea of equivalence. Indicating the latter relation by the usual sign 00 , the equivalence O'd^'coO'" is, then, defined by the equation f{d\d") = e"'. *L. Kronecker: Monatsber. d. Berl. Akad., 1870, p. 881. The following is taken for the most part verbatim from this article. CERTAIN SPECIAL CLASSES OF GROUPS. l57 Since the number of the elements 0^ which we will denote by n, is assumed to be finite, these elements have the following properties : I) Among the various powers of an element there are always some which are equivalent to unity. The exponents of all these powers are integral multiples of one among them, to which may be said to belong. II) If any belongs to an exponent v, then there are elements belonging to every divisor of v . III) If the exponents p and ^, to which 0' and 0" respectively belong, are prime to each other, then the product 0' 0" belongs to the exponent fxr. IV) If nj is the least common multiple of all the exponents to which the n elements belong, then there are also elements which belong to n^ . The exponent n, is the greatest of all the exponents to which the various elements belong. Since, furthermore, ii^ is a multiple of every one of these exponents, we have for every the equival- ence (>"' CO 1 § 138. Given any element 0^ belonging to the exponent n^, we may extend the idea of equivalence, and regard any two elements 0' and 0" as ^^ relatively eqiiivalenV when for any integral value of k O'.O^^coO" We retain the sign of equivalence to indicate the original more lim- ited relation. If now we select from the elements any complete system of elements which are not relatively equivalent to one another, this subordinate system satisfies all the conditions imposed on the entire system and therefore possesses all the properties enumerated above. In particular there will be a number ^2 , corresponding to n^ , such that the ^2*^ power of every of the new system is relatively equiv- alent to unity, 1 e., O^'-oo (>■!'. Again there are elements 0,, in the new system of which no power lower than the n.}^ is relatively equivalent to unity. Since the equivalence ^"^ oo 1 holds for every element, and consequently a fortiori eyerj 0"- is relatively equiva- 158 THEORY OF SUBSTITUTIONS. lent to unity, it follows from I) than n^ is equal to n^ or is a multiple of Wg. If now and if both sides are raised to the power — , we obtain, writing k — = m, the equivalence From this it follows that, since 6^ belongs to the exponent nj , m is an integer and k is therefore a multiple of ng. There is therefore an element • • • which has the property that the expressions d,\d^o.O^^ . . , {h,= 1, 2, .. .n) include in the sense of equivalence every element once and only once. The number ni , rig , ng , . . . , to which the elements ^i , ^2 > <^3 ? • • . belong, are such that every one of them is equal to or is a multiple of the next following. The product Wj tig W3 . . . is equal to the entire number n of the elements ^, and this number n accordingly contains no other prime factors than those which occur in the first number Wj . § 139. In the present case the elements are to be replaced by substitutions every two of which are commutative. The number n of the elements becomes the order r of the group. We have then CERTAIN SPECIAL CLASSES OF GROUPS. 159 Theorem IV. If all the substitutions of a group are com- mutative^ there is a fundamental system of substitutions s^^s^^s^,, . . which possesses the property that the products siH*%*3... (hi = 1,2,, . .n) include every substitution of the group once and only once. The numbers r^^r^^r^, . . . are the orders of Si,S2,s^, . . . and are such that every one is equal to or is divisible by the next following. The product of these orders r^,r,), then p^ = r^, and Since (t^ belongs to 7\ , at least one of the exponents p., v, . . . must be prime to r^ . From the first remark above, we may assume that this is /x, and from th6 second it follows that the group can also be expressed by ^i^s/V=^ ... (/I, = 1, 2, . . . r,; p, = r,). Consequently the groups s^s/3 ...(h, = l,2,... r,), ,) are identical. From this it follows, as before, that p^ — r^, and 80 on. Theorem V. The numbers r^^r^^r^, . . . are invariant for a given group.* *This theorem is due to Frobenius and Stickelberger. cf. their article: Uber Gruppen mit vertauschbaren Elementen; Crelle, 86, pp. 217-262. CHAPTER VIII. ANALYTICAL KEPRESENTATION OF SUBSTITUTIONS. THE LINEAR GROUP. § 140. In the preceding Chapter we have met with a fourth method of indicating substitutions, which consisted in assigning the analytic formula by which the final value of the index of every x is determined from its initial value. Thus, if the index z of every x^ is converted by a given substitution into (p{z)^ so that x^ becomes Xa,(^), the substitution is completely defined by the symbol s^\z (p{z)\. Obviously not every function can be taken for (f{z)^ tor it is an essential condition that the system of indices ^ must not be converted into the same element unless the indices Zi,Z2, . . . z^ coincide in order with Ci, C25 • • • '*• More generally, given any system of indices Ci, C2, . . . C^., it is necessary that from a^Zi 4- b^z, -j- . . . -h Ci% EEE Ci , a.2Zi + 622^2 + • • • + ^2% = Cg , • • • (mod. m) the indices Zi, z.,, . . . z^ shall be determined without ambiguity. In other words, the m'' systems of values z must give rise to an equal number of systems of values C . The necessary and sufficient conditions for this is that the congruences ayb^ + 612:2 + . . . + c^Zj,. ^ 0, a^z^ -\- b^z^ + . . • + C2Z;,-^0 , . . . (mod. m) shall admit only the one solution ^i = 0, Z2 = 0, . . . Zj, = 0. If the determinant of the coefficients is denoted by J, these congruences are equivalent to J . 2:1^0, J . 2:2:^0, . . . J-Z),^0 (mod. m). The required condition is therefore satisfied if and only if J is prime to m. We have then Theorem III. In order that the symbol t=\z,,Z2,...z^., aiZi-i-b,Z2-Jr...+CiZk, a22:i + 622^2 + .. • +^22:^,. .. I (mod m) may denote a (geometric) substitution, it is necessary and sufficient that tti, 61, . . .Ci J __ tt2 5 ^2 ) • • • ^2 Otfc) O^, . . . C;!; should be prime to the modulus m. § 146. From this consideration it is now possible to determine the number r of the geometrical substitutions cQ^esponding to a given modulus m. We denote the number of distinct systems of p integers which are less than m and prime to m by [m, /?]. It is to be understood that any number of the /> integers of a system may coincide. 166 THEORY OF SUBSTITOTIONS. Suppose N to be the number of those geometric substitutions 1, ^2 ) ^3 J • • • which leave the first index Zi unchanged. If then r^ is any substitution which replaces z^ by ai^i + 612:2+ • • • H~Ci%, then r2, t^r^, t^r^, . . . are all the substitutions which produce this effect, and these are all different from one another. Similarly, if r^ replaces Zi by a/ Zi-{-b/z2 -{-... -\- c/z^, then '3, #2^35 te? • • • are all the substitutions which produce this effect, and these are all different, and so on. We obtain therefore the number r of all the possible geometric substitutions by multiplying N by the number of substi- tutions 1, Tg, Tg, . . . ' The choice of the systems ai, 61, . . . Cj; ai,bi, . . . c/; . . . is limited by the condition that that the integers of a system cannot have a same common factor with m. There are therefore [m, k] such systems, and an equal number of substitutions 1, T2,r3,... Consequently r= [m, k^N. The substitutions t are of the form \zi,Z2,. .,Zk Zi,a2Zi-^b2Z2-\- C.2ZJ,, ...akZi + b^2+'-- + c^k\ (mod. m). Since a2^a^, . . . a^ do not occur in the expression of the discrimi- nant /J, these integers can be chosen arbitrarily, ^. e., in m^ "^ dif- ferent ways. The 6a , • • • Ca are subject to the condition that 62 , . . . C3 6fc,. . .Cfc must be prime to m. If the number of systems here admissible is r', we have The number r has the same significance for a substitution of k — 1 indices (mod. m) as r for k indices. Consequently r = [m, k~\7n!'~^ [m, k — 1] m*"" V", and so on. We obtain therefore finally r = [m, k']m!'-' [m. A; — l]m^ -^ . . [m, 2]r(^-^), where r<^^^> corresponds to a single index, and therefore r(*~') = [m, 1] . Hence ANALYTICAL REPKESENTATION OF SUBSTITUTIONS. 167 4) r = [m, A;] m^~^ [m, k — 1] m''~'^ . . . [m, 2] m [m, 1] . The evaluation of [m, A;] presents little difficulty. We limit our- selves to the simple case where m is a prime number p, this being the only case which we shall hereafter have occasion to employ. We have then evidently 5) [p,p]=P'—l, since only the combination 0, , ... is to be excluded. By the aid of 5), we obtain from 4) 6) r = (p*-l)p'=-(p''-'-l)p'-^ . . . {p^-l)p(p--l) = ip"-^) iP'-p) iP'-p') . ■ ■ (p'-p"-').* § 147. The entire system of the geometric substitutions (mod. m) forms a group the order of which is determined from 4) or from 6). This group is known as the linear group (mod. m). If the degree is to be particularly noticed, we speak of the linear group of degree m/". It is however evident that all the substitutions of this group leave the element Xq,q, . . . o unchanged. For the congruences ci'i^i + b,Z2 + . . . -h Ci%=2;i , a^z.i + hz^ + h^2+ • • • + C2% = 22, • • • (mod. m) have for every possible system of coefficients the solution ^lE^O, z^^O, . . . Zk^^O (mod. m). We shall have occasion to employ the linear group in connection with the algebraic solution of equations. Theorem IV. The group of the geometric substitutions (mod. m), or the linear group of degree wJ" is of the order given in 4). Its substitutions all leave the element a?o,o) ...o unchanged. It is commutative with the group of the arithmetic substitutions. ♦Galois: Liouville Journal (1) XI, 1846, p. 410. PART II. APPLICATION OF THE THEOKY OF SUBSTITUTIONS TO THE ALGEBRAIC EQUATIONS. CHAPTER IX. THE EQUATIONS OP THE SECOND, THIRD AND FOURTH DEGREES. GROUP OF AN EQUATION. RESOLVENTS. § 148. The problem of the algebraic solution of the equation of the second degree 1) ^ x'^ — c^x-\-C2 = can be stated in the following terms: From the elementary sym- metric functions c^ and c.2 of the roots Xi and x., of 1) it is required to determine the two- valued function Xi by the extraction of roots.* Now it is already known to us (Chapter I, § 13) that there is always a two- valued function, the square of which, viz., the discriminant, is single-valued. In the present case we have J = (xi — x^y = (a?i + ^if — 4:XiXo = c^ — 4c2 , sjA — (x^ — X2) — \/ c^ — 4c2. Since there is only one family of two -valued functions, every such function can be rationally expressed in terms of sj ^. For the linear two-valued functions we have and in particular, for «! = 1, «2 = 0, and for «! = 0, «2 = 1 Ci , - / Tt \ Ci a?i==Y + f Vci'— 4c2, a?2=^— jVci'— 4c2. *C. G.J. Jacobi: Observatiuueulae ad theoriain aequationiim pertiuentes. Werke, Vol. Ill; p. 269. Also J. L. Lagrange: R6flexions sur la resolution alg6brique des equa- tions. Oeuvres. t. III,p. 205. ELEMENTARY CASES GROUP OF AN EQUATION RESOLVENTS. 169 § 149. In the case of the equation of the third degree /o*^ v^3 the solution requires not merely the determination of the three- valued function x^ , but that of the three three -valued functions Xi,X2, x.^. With these the 3! -valued function is also known, and conversely x^^x^^x.^ can be rationally expressed in terms of c. We have therefore to find a means of passing from the one-valued functions Cj , C2 , C3 to a six- valued function by the extraction of roots. In the first place the square root of the discriminant J =: (a?j — x^\x^ — a?3)^(a?2 — x^^ — — 27c3^ + I8C3C2C1 — 4c3C,^ furnishes the two-valued function in terms of which all the two-valued functions are rationally expres- sible. The question then becomes whether there is any multiple- valued function of which a power is two -valued. This question has already been answered in Chapter III, § 59. The six-valued func- tion 9x-^x-V ^^2 + ^'^3 ^> = y 2 J, on being raised to the third power, gives again, if ^2 is obtained from ^^ by changing the sign of \/ — 3 » we have Accordingly Xi-\-w%-{-ajx^= 4/i(;Si + 3/\/— 3^, Xi-{-wx2-{-(o^x^— \/i(Si — 3V — 3J. Combining with these the equation £Cl -j- £^2 -j- a?3 = Cj , 170 THEORY OF SUBSTITUTIONS. and observing that l + «> + .^' = 0, we obtain the following results ^3z=i[cj + «>2\^i(;Si + 3V^^^^3^ + ">^i('5i — 3V^^^^3J)]. The solution of the equation of the third degree is then complete. § 150. In the case of the equation of the fourth degree it is again only the one-valued functions Cj , C2 , Cg , c^ that are known. From these we have to obtain the four four- valued functions a?i , 072, a?3, iz^4, and with them the 24- valued function by the repeated extraction of roots. In the first place the square root of a rational integral function of Ci,C2,C3,C4 furnishes the two- valued function y'j. Again, we have met in § 59 with a six-valued function (p = {x^x^ + x^x^ + w(a?ia?3 + x = 6, r = 4. To this same group belongs the function which can therefore be rationally obtained from = 12, r = 2. may therefore be regarded as known. Finally y2 — [a^{x^ — x^) 4- a2(a?3 — X.yf, r — ^^{x^ + X^ + ^j^X^ + X^ ELEMENTARY CASES GROUP OF AN EQUATION ^^"rESOL VENTS. 171 can be rationally expressed in terms of 4'', and ;>: is a 24-valued function. All rational functions of the roots, and in particular the roots themselves can then be rationally expressed in terms of /. To determine the roots we may, for example, combine the four equations for / and r in which a^ z=z ± a^ and fti— ± 1^2 • § 151. In attempting the algebraic solution of the general equations of the fifth degree by the same method, we should not be able to proceed further than the construction of the two-valued functions. For we have seen in § 58 that for more than four independent quantities there is no multiple- valued function of "which a power is two-valued. It is still a question, however, whether the solTltioh" of the equation fails merely through a defect in the method or whether the impossibility of an algebraic solution resides in the nature of the problem. It will hereafter be shown that the latter is the case. We shall demonstrate the full sufficiency of the method by the proof of the theorem that every irrational function of the coefficients which occurs in the algebraic solution of an equation is a rational function of the roots. All the steps leading from the given one- valued function to the required wl-valued functions can therefore be taken within the theory of the integral rational func- tions of the roots. ^ § 152. We turn our attention next to the accurate formulation of the problem involved in the solution of algebraic equations. Suppose that all the roots of an equation of the n^^ degree 1) /(^) = o are to be determined. If one of them x^ is known, the problem is only partially solved. By the aid of the partial solution we can however reduce the problem and regard it now as requiring not the determination of the ?i — 1 remaining roots of the equation of the 'n}^ degree f(x) = 0, but that of all the n — 1 roots of the reduced equation We have then still to accomplish the solution of 2). If one of its roots a?2 is known, we can reduce the problem still further to that of the determination of the n — 2 roots of the equation 172 THEORY OF SUBSTITUTIONS. Proceeding in this way we arrive finally at an equation of the first degree. It appears therefore that solution of an equation of the n^^ degree involves a series of problems. All of these problems are however included in a single one, that of the detei^minatiou of a single root of a certain equation of degree n\ Thus, if the entirely independent roots a?i , .T2 , ... oc„ of the equation 1) are known, then the tz!- valued function with arbitrary constants «i , «2 , . . . «„ 4) c = «iXi -|- a^X^ + . . . + a^X^ is also known. Conversely if ^ is known, every root of the equa- tion 1) is known, for every rational integral function of a?i,a'2,...a?„ can be rationally expressed in terms of the n!- valued function c, « The function c satisfies an equation of degree n\ 5) F{S) = r'—Ar'-' + ... ± A„,=(f-c,)(|:-c.)...(f-f„,) =0, the coefficients of which are rational integral functions of those of 1) and of «!, ttg. . . . «„. This equation is, in distinction from 1), a very special one. For its roots are no longer independent, as was the case \^ith 1), but every one of them is a rational function of every other, since all the values Ij , ^2 ? • • • ^n \ belong to the group 1 with respect to a^i, 0^2, . . . x„. The solution of 5), and consequently that of 1) is therefore complete, as soon as a single root of the former is known. The equation F{^) — is called the resolvent equation, and f the Galois resolvent of 1). We shall presently introduce the name "resolvent" in a more extended sense. § 153. If the coefficients Cj, C2, . . . c„ of the equation 1) are entirely independent, the Galois resolvent cannot break up into rational factors. Conversely, if the Galois resolvent does not break up into factors, then, although relations may exist among the coef- ficients c, they are not of such a nature as to produce any simplifi- cation in the form of the solution. From this point of view the equation 1) is in this case a general equation, or, according to Kron ecker, it has no affect. ELEMENTAKY CASES- — GROUP OF AN EQUATION RESOLVENTS. 173 On the other hand, if for particular values of the coefficients c the Galois resolvent F{^) breaks up into irreducible factors with rational coefficients 6) F{^) = F,{^){F,{^)...FAr), then the unsymmetric functions Ff{:), which in the case of fully independent coefficients are irrational, are now rationally known. The equation 1) is then a special equation, or according to Kron- ecker, an affect equation. The affect of an equation lies, then, in the manner in which the Galois resolvent breaks up or, again, in the fact that, as a result of particular relations among the coefficients c,,r ,, . . . c,, or the roots ,Xi, x.,, . . . x„, certain unsymmetric irreduci- ble functions F;{:) are rationally known. The determination of that which is to be regarded as rationally known in any case is obviously of the greatest importance. As a result of any change in this respect, an equation may gain or lose an affect. If the group belonging to any one of the functions F;{c) is G,;, then every function belonging to Gi is rationally known, being a rational function of F,. It is readily seen that the groups Gi all coin- cide. For if the subgroup common to them all is J\ then the ration- ally known function aF^ -{- f^F., -{-... -\- /. F^ belongs to /', and conse- quently every function belonging to /'is rationally known. Accord- ingly the factor of F, which proceeds from the application of the substitutions of /' alone to any linear factor r — a^Xi^ — a.^Xf,^... — «„»?,•„ of F, is itself rationally known. This is inconsistent with the assumed irreducibility of F,-^ unless J'= Gi^= G.>=^ ... = G^. All the functions F^ , F.,, . . . F^, therefore belong to the same family, and this is characterized by a certain group G or by any function '/* {x^,x.2, . . . x^ belonging to G. Every function belong- ing to G is rationally known and conversely every rationally known function belongs to G. Theorem I. Every special or affect equation is character- ized by a group G, or by a single relation betiueen the roots (/^{x,,x,,...x,,) = 0. The group G is called the Galois group of the equation. Every equation is accordingly completely defined by the system ->A = ^\ , ->Ai^V = C2, . . . ; ^ (a?, , ^2 , . . . x,) = 0. * *C/. Kronecker; Grundziige einer arithmetischen Theorie der algebraisclien Gros- ser!, §§ 10, 11. 174 THEORY OF SUBSTITUTIONS. For example, given a quadratic equation the corresponding Galois resolvent is ^'^ — 2(ai -f- a^c^^ -}- («! — a^'^Cc^ + \a^a^c^ — 0. In general the latter equation is irreducible, a*id the quadratic equa- tion has no affect. But, if w^e take 2ci — 771 -j- ?i, C2 — m ?i, the equation in c becomes (c — a^m — o.^ri){^ — a^n — a2m)=0=(l — a^X^ — a^x^{^^ - - a^x^ — ag^j) and the given quadratic equation has an affect. Again, if c^ — Ci^ = 0, we have (c — a^C^ — a^c^^-= 0= (c — a^x^ — a^x^{'^ — a^X^ — «2^i)- But if C.2 — 2ci^ = 0, we obtain .^^ — 2(a, + «2)c,.^ + 2(ar + a2V-0, and this equation has no affect, so long as we deal only with real quantities. If however we regard i= s/ — 1 as known, the equa- tion has an affect, for the Galois resolvent then becomes (- — («i + «2) Ci + («i — "2) c,i) ($— («! + flg) Ci — («i — «2) c,i) = 0. § 154. It is clear that every unsymmetric equation ^3 » • • • ^« of tlie first equation are also roots of the second. Consequently /(a?) = is satisfied by is exceptional. For general equations and resolvents this reduction is not possible. We shall see later that it is possible for the biquadratic equation and the particular series of resolvents employed above only because the family of every resolvent was a self- conjugate subfamily of that of the succeeding one. § 159. Given any /^-valued resolvent (p{x^, x.>^ . . . ir„), this sat- isfies an equation of the //^ degree To determine the group of this equation we adopt the same method as in the preceding Section. Suppose that are the //-values of cr. Every substitution of the group G of th« ELEMENTARY CASES GROUP OF AN EQUATION RESOLVENTS. 179 equation f{-)c) = produces a corresponding substitution of these fj values. The latter substitutions form the required group of the resolvent equation for (f. This group is isomorphic to G, and it is transitive, since G contains substitutions which replace cpj by every (Px. If, in particular, the group of , (p, as usual, always denoting a prime number), is called the cyclotomic equation. It is of the form 1) ''^^=x^-'-i-x^-'-j- ... ■i-x--\-x-irl = 0, audits p — 1 roots are 2) oj,aj\a}\..ioP-\ We prove that the left member of 1) cannot be expressed as a pro- duct of two integral functions i may be any root of 1), since the series 2) is identical with the series ui^^M^^u)^^ . . . u)^^~^. The equation s'^~^ — (a, u)). From § 129 the resolvent (a, u))^~^ is unchanged by s and its pow- ers, that is, by the group of the equation. It can therefore be expressed as a rational function of a and the coefficients of 1). If we denote a {p — 1)*^ root of this rationally known quantity by Tj we have 5) (a, w) = ri. The quantity r^ is a {p — 1)- valued function of the roots of 1). It is changed by every substitution of the group, for the substitution s converts it into (a, w*') = a~^(a, w) = a~"'ri. 182 THEORY OF SUBSTITUTIONS. It follows from the general theory of groups that every function of the roots can be rationally expressed in terms of r^. We will however give a special investigation for this particular case. The group of the cyclotomic equation leaves the value of 6) {a\w){a,wy-'-^ unchanged; for the effect of the substitution 8 is to convert this function into i. e., into itself. If now we denote the rationally known value 6) by Ta, where in particular Tj^ "^ = Tj, we obtain for / = 1, 2, . . .p — 2, the following series of equations : («, <„) = r„ («^ ,o) = ^r,% (./, <«) = ^ r,^ . . . {a--\ o.) =^rf-\ Combining with these the obvious relation among the roots and the coefficients of 1) (1, «.) = -!, we obtain by proper linear combinations It is evident that a change in the choice of the particular root a p — 1 or of the particular value of Tj = V T^ only interch«,nges the val- ue lo among themselves. Theorem II. The solution of the cyclotomic equation for the prime number p requires only the determination of a primitive root of the equation z^~^ — 1 = 0, <^^ + * s*^ '^ ) P (cos '9 — isin '^) = /)l Again it can be shown, exactly as in the preceding Section, that (a,aj){a-\oj-') belongs to the group of the cyclotomic equation and is conse- quently a rational function of a and of the coefficients of 1). If we denote its value by U we have Accordingly for any integral value of k (s aj)=^/u [cos -jzrr + * ^*^ p _i J Since U and ''> are both known, we have then Theorem III. The solution of the cyclotomic equation requires the determination of a primitive root of the equation z^~^ — 1=0, the division into p — 1 equal parts of an angle which is then knotvn, and the extraction of the square I'oot of a known quantity. The latter quantity, i/, is readily calculated. We have To reduce this product we begin by multiplying each pair of cor- responding terms of the two parentheses together. The result is Again, if we multiply every term of the first parenthesis by the fc*^ term to the right of the corresponding term in the second par- enthesis, we obtain K) a-^^(^-^^- + i_|_^-^^- + '+?_^.,,,-p^ + 2 + <,2^ .y Now w~^*+^ is a p*^ root of unity m^ different from 1; for if 184 THEORY OF SUBSTITUTIONS. then — gr*+1^0, gr*E^l (mod. p), i. e., fc = or p — 1. The quantity K) is therefore equal to and consequently ?7 = p—l — («-' + «-'+ . . , -j- a-^ + ')=p — l — {—l) = P Theorem IV. The quantity of Theorem III, of which the square root is to be extracted, has the value p. §164. The resolvent 5) was {p — 1)- valued, and consequently the preceding method furnished at once the complete solution of the cyclotomic equation. By the aid of resolvents with smaller numbers of values, the solution of the equation can be divided into its simplest component operations. Suppose that p, is a prime factor of p — 1, and that p — 1—Piqi. We form then the resolvent where «i is a primitive root of the equation ^^1 — 1 = 0. The values a^ , a^^, . . . a/\ are all dilBPerent, and the higher powers of «i take the same values again. It follows that, if l_|.^«^2^1-l_j_^p3^1-l_^ ^ ^ ^ ~j-<0»^l 9,-1 the resolvent above can be written {fo + «1^1 + «lV2 + . . . + «l''^ ~^'^pi- iV', or, again in Jacobi's notation, By the same method as before we can show that this resolvent is unchanged if to is replaced by w^, that is, that it belongs to the group of 1) and is consequently a rational function of a^ and of the THE CYCLOTOMIC EQUATIONS. 185 coefficients of 1). We denote its value by iVj — v/'i, and have accordingly If then we write precisely as before, it appears that N\ is rationally known, and that These several functions are all unchanged if co is replaced by w"^', that is, they are unchanged by the subgroup .s^\ s-'^'i, s'^\ . . . .S'^^'i. We have therefore Tlieoreiii V. The p^-valued resolvents s^o, ^i , . . . <^j,i-i of the cyclotomic equation belong to the group formed by the powers of s^'i. They can be obtained by determining a primitive root of z^\ — 1 = and extracting the p^'' root of a quantity which is then rationally known. * If p.2 is a second prime factor of p — 1, and if p — -i =2^iP2Q2i then the resolvent in which o..^ is a primitive p/*^ root of unity, is reducible to the form where This resolvent is unchanged if oj is replaced by o)^^\ that is, it is unchanged by the substitutions s^'i, S"^i . .». Consequently it 12a 186 THEORY OF SUBSTITUTIONS. can be rationally expressed in terms of ^o? if «2 is regarded as known. A^ain if we write then Ma is also a rational function of • • • 9'j>] — 1 are rationally known, since they all belong to the same group. Similarly the coefficients of r {X — O^) {x—io^''^) {x — io'P^) , . . {x — co^^^--'^''^) = 0, 7) J {x — io9){x — a>^'^^') (cc — 0.^2^1 + 1) . . . (^_,,.(^x-i)^.+i) 3=0 , are all rationally known. Accordingly after the process of Theo- rem V has brfen carried out, the equation 1) breaks up into p^ fac- tors 7). Since the group belonging to each of these new equations is transitive in the corresponding roots, the factors 7) are again all irreducible, so long as only the coefficients of 1) and = 3, 5, 17, 257, 65537, and in these cases the corresponding regular polygons can there- fore be constructed. But for /j. = 5 we have 2^ + 1 = 4294967297 = 641 . 6700417, 188 THEORY OF SUBSTITUTIONS. .,^ SO that it remains uncertain whether the form 2-4-1 furnishes an infinite series of prime numbers.* § 167. We add the actual geometrical constructions for the cases p = 5 and p = 17. For p = 5, we take for a primitive root g—2, and obtain accord- ingly g'^-l, ^^-2, ^^ = 4, ^^ = 3 (mod. 5). Consequently (T^—lU-f- io\ Cy = (0^ -f co^ — 1+\/5 — 1— V5 fo = 2 ' ^^ = 2—- If oj- is substituted for w, the values ctq and c>i are interchanged. The algebraic sign of V^ is therefore undetermined unless a par- ticular choice of 0, c?i < 0, and the V5 in the expression above is to be taken positive. Furthermore /.o = ^, Zi = ^*; X > = '''', )^3 = <^^ — 1+ V5 + *VlO + 2V^ /o = «> = ^ , , — 1-h V5-iVlO + 2\/5 /i=^ = ^ * cy.'Gauss: Disquisit. arithni., § 362. The statement there made that Ferinat sup- posed all the numbers 22'* +1 to he prime, is corrected by Baltzer: Crelle 87, p. 172. At present the following exceptional cases are known : 2^^^ +1 divisible by 641 (Landry), ol2 2 +1 divisible by 114689 (J. Pervouchine), 2^^ + 1 divisible by 167772161. . . (J. Pervouchine; E, Lucas), 2*^ + 1 divisible by 274877906945 (P. Seelhoff ), C/. P. Seelhoff: Schlomilch Zeitschrift, XXXI. pp. 172-4, THE CYCLOTOMIC EQUATIONS. 189 the sign of i being so taken that the imaginary part of oj is positive legative. For the construction of the regular pentagon it is sufficient to know the re- solvent c?o = 2 gos-^ . ^ ^ Suppose a circle of radi- us 1 to be described about O as a center. On the tan- gent at the extremity of the horizontal radius OA a distance AE = lAO — i is laid off. Then . 0E= \/l+i If now we take EF= E O, we have /\/5 — 1 AF = EO--EA = -^^ ^' -{-lO^ -f c«l«-(-o.l5_|_^13_^^9^ ^^ =:, ^6 _|_ ^12 _|. ^7 _|_ ^U _^ ^11 _|_ ^5 _^ ^iO _^ ^3. 190 THEORY OF SUBSTITUTIONS. To find cTq V^i ? we multiply every term of s^o into the k^^ term to the right of that immediately below it in ^j, taking successively A; = 0, 1, ... We obtain then "Po -\- ^1+ ^ ;^, = a>« + «>^ + a>" + «>^«, ;^3 := a>i2_^ «,h + ..^ -f «,3. ;^0/l=/3+/l+/0+/2= —1, /2/3=/0 + /3+Z2+/l=— 1; /o,/i= 2 =*" Y — 4""' /2,/3- 2 =^Y — 4 — • The algebraic signs are again easily determined. We have THE CYCLOTOMIC EQUATIONS. 191 2r ;,„ = (<„+ <„») + («« + ,o») = 2 (cos ^ + COS 1^) > 0, X, = («,» +) = 2 (cos ^ + COS^) < 0, Oonsequently With /o as a basis we proceed further: ■f- ^S = ^05 V''oV\ = /3 = ^ + a/^ + 1; ^'o, V'l = 9_ Q Since now v''o = '^ cos -^ , v''i ^2cos-^, we have 0o > ^^? and there- fore These results suffice for the construction of the regular polygon of 17 sides. Suppose a circle of radius 1 to be described about O as a center, and a tangent to be drawn at the extremity of the hori- zontal radius OA. On the tangent take a length AE = iOA = ^', then Vi7 OE=\/l-{-j\= ^ Further, if EF=EF' = E O, we have V ,j,_ vi7-i _^o ,^._ vn+i _ ^1. AF- - _^, AF- ^ --2"' 192 THEOEY OF SUBSTITUTIONS. OF=^l^ + l, OF' = ^/^^t —M Taking then we have FH^FO, F'H' = F'0, AH=AF-{-FO = ^ + J^+l = Xo: Air = — AF' + FO fi I=;f3. We bisect A if in F; then AY=ix,. THE CYOLOTOMIC EQUATIONS. 198 We take now AS^l, and describe a semicircle on H'S as a diam- eter; if this meets the continuation of OA in K, then AK^ = AS.AH'=x^- Again if we take LK=AY and KL = LM = LN, and describe a circle of radius L K about jL, we obtain AN-\-AM^2KL = 2AY = Xo = 4'^ + ^'i AN.AM=: AK^ = AS.AH'=x^ = M\. The greater of the two lengths A 2V, A3f is equal to c^o; we may write therefore 2- AM=:0o = 2cos-^. • * If P is the middle point of AM, and if ^P ;' AG, and OD , A O, then Q and D are two successive vertices of the required reg- ular polygon of 17 sides. § 169. We consider now the case Pi=2, under the assumption p>2. If gr is a primitive root, then ip^= o> -fwfl^+^/>^-j- . . .+-^" , >^2 <-"] 4- m^ -^ - , ?/ii -j- wio + »^3 = — s— » where m^ , mo , m^ represent the number of brackets of the several kinds. If gr'^+^ + l^O (mod. i?), then 2(2a + l)=p — 1 and accord - p 1 ingly — ^— is an odd number. The third case occurs therefore only when p = 4Ar-f-3, and then only once, !'*>. f or a = — - — . Con- sequently m. = or m-^ — X according as is even or odd. Since cr„ ctj is rational aad integral in the coefficients of the cyclotomic equation 1), this product is an integer, and we may therefore take c'„ if, — m, — ^ — n = — n{ ^„ -f ^){x-~o)9^) {x — io^') . . . (iC — o>^^"-) = 0. The roots and consequently the coefficients of these equations are un- changed if (D is replaced by (o^\ If therefore, when the several factors are multiplied together, any coefficient contains a term mw« it must also contain terms m^^*^', m.(o°-^\ . . . , that is, it con- tains (Pq or ^1 , according as « is a quadratic remainder or non- remainder (mod. p). Accordingly every coefficient will be of the form m! if,, + m" i?, = — ! 2~^ ' and on introducing these values we have __X-{-Y\(-l)'\ where X and Y are integral functions of x with integral coefficients. Again since z^ is obtained from z^ by replacing co by at^, or % by i different roots x^, 0{x^\ 0\x^\ . . .0>'-\xo). But since the equations 4) 0-^{y)—y = O 0f'(z) — Z = O have each one root y=^Xi,z = X2 in common with the irreducible equation 1), they are satisfied by all the roots of the latter. The former equation of 4) is therefore satisfied by a?2, the latter by a?,, consequently m is a multiple of /^ and ,0. is a multiple of m, i. e. m — [i. Again all the roots of the second series are different from those of the first. For if 0\x,):=0^{x,) (a,b systems of non-primitivity of m ele- ments each. The number of admissible permutations of the v systems is not as yet determinate; in the most general case there are v! of them. If any .r^ is replaced by 0^(xa), then every 0''(xa) is replaced by O^' + ^ixa); there are therefore m possible substitutions within the single system. The order of the group of 1) is therefore a multiple of m'' and a divisor of vim*'. Theorem II. The group of the equation 1) is non-primitive. It contains > systems of non-primitivity, which correspond to the several lines of 5). The order of the group is r=^r^ m^, where r^ is a divisor of )^\. § 173. In the following treatment we employ again the notation of Jacobi where w is a root of the equation w*** — 1 = 0. Similarly we write We form then the following resolvents: ^1 is symmetric in the elements of the first system and is changed in value only when the first system is replaced by another; it is therefore a v-valued function, its values being s^i, ^2? • • • S^*" 200 THEORY OF SUBSTITXTTIOIfR. Every symmetric function ot the ^'s is a rational function of the coefficients of 1) . The quantities that is, the coefficients of the equation of the v*^ degree of which e\x)+... (o^-'e^{x)] = {. The m}^ root of this known quantity T^ we denote by r^ . Again if we write {w\d{x))( 1 ; consequently m=p and v = 1. § 175. If all the coefficients of f{x) are real, the process of the preceding Section admits of further reduction. The quantity T^=i{a), d{x)f' = p {cos T? + ^ sin ^9 ) can be rationally expressed in terms of ^ and the coefficients of /, Tlie latter being real, the occurrence of i= ^ —1 in T^ is due eat jjely to the presence of (o . Consequently. T/ = {(o-\ d{x)Y = p{cos^'^ — isin^), T^T^' = p\ s/l\f;'=^/p':^(^"i ~ '^"'^ (^i), and with coefficients which are rational functions of a resol- vent /i = a?i + 0"'i (cci) + • • • /i is a root of an equation h^ (x) = of degree m^ . ( g,{x)=0, with the roots .^i , 6^'" = ^i-\- ^""^ (^i) + • • • Z=SS- Again and a comparison of the two expressions for 6''"(a?) shows that «„, = «i««._i + n/^m-l , /5m = /5i«m-] + \['^m-\, THE ABELIAN EQUATIONS. 205 From these equations we obtain at once the characteristic relation 7) aj,n — /^,. r»» = {'h'\ ~ 1^1 Tl) («.« - I'^m - 1 —P^m-xYm-l) By dividing every coefficient of f^{x) by ^/ ad — j3y or by \/ f^y — «^^i according as ad — jSy is ppsitive or negative, we can arrange that for the new coefficients of. 0(x), and consequently for those of every 0^(x), the relation shall hold 8) ad — flr=:±l. We determine now under what conditions it can happen that 0'^{x) = X. The values of x which are unchanged by the operation satisfy the equation _ ax-j-ft yx^ + (d — a)x — /S = 0. For these fixed values we have therefore, according as ad — j3y = ± 1, and consequently A) We assume in the first instance that x' and x" are distinct, that is that iV-j-1. We have then 0(x) X' _ j^3C — x' O'^jx) — x' _ -^x — x' 0{x) — x"~^j^^^" 0''(x) — x"~~ x — x'"'" 0"ix)—x' ^r^x—x' 0^'^{x)—x" x — x"' The necessary and sufficient condition that 6^*"(x) = a? is there- fore that 206 THEORY OF SUBSTITUTIONS. ..=<,„.[^'_^(f+.')V.]-.,. This condition can be satisfied by complex or by real values of the quantity in the bracket. In the former case the upper alge- braic sign must be taken, and further so that we may write a-\-d N*^ = (^cos f — i sin ipf"*" = cos 2 m ^ — i sin 2 mz^{x — z^ . , . {x — 2;^), then for every Za 0{Za) = Za, and consequently d'^(Za) = ^^(Za) = . . . = 6\Za) = Za . Moreover ^^+i(aj)_^^(a;) = l0^(x)—z,-] ld\x)—z,-] . . . l0\x)—z;], and consequently d'' + \x)—e\x) _ d\x)—Z, e\x)—Z^ \Xy) — Z^ -T> { \ " \X\ X X ~~^ Z-y X Z^ X Zy where P is a rational integral function of x and of the coefficients of 0, since it is symmetric in the roots z-^^z^^ • - -Zy. If now we take k successively equal to 0, 1, 2, . . . m — 1, and add the resulting equations, we have as asserted d-^{x) — x = [0{x) — £c] Q{x\ where ^ is a rational integral function of x. From this equation it follows that for every root of Q{x) = Q we have 0'^{x) = x, and conversely that every root of THEORY OF SUBSTITUTIONS. which is not contained among z^, z^, . . . Zt, also makes Q{x) vanish. Every root ^ of Q{x) = therefore gives and consequently also so that d{^), and likewise 0\^), e\^\ ... are all roots of Q{x) = {). Again since ^ is different from the z's, 0{^) if, and 0{^) =Za. Theorem IX. If 6{x) is a rational integral function of x of degree v , then the roots of the equation of degree (v — l)m [X) X can be arranged as in Theorem I, 5). If m is a prime number then each of the v — Irows of 5) contains m roots * § 180. Conversely if the equation f(x) = has the roots x^, x^ = d{x^\ x^ = 0\xo\ . . . x^^i =e^-\xy, \0^{x^ == io] , every one of these roots will also satisfy the equation ^{x)- x = 0, but no one of them will satisfy e{x) — a? = 0; consequently / {x) is a divisor of the quotient The restriction that e(x) shall be an integral function is unessen- tial. For if d{x) is fractional where gx and grg are integral functions, then in r Qi^ ^ ^ gi{^o)iQ2{x,)g^{x,) . . . g^jx^-i)'] 92(^0) g2{xi)92(x,) . . . g2(xm-i) the denominator, being a symmetric function of the roots of /(a;) = 0, is a rational function of the coefficients of f{x); and the THE ABELIAN EQUATIONS. 209 second factor of the numerator, being symmetric in x^, x^, . . . x^_i is a rational integral function of Xq. Consequently 0(xq) is a rational integral function of x^, which can be reduced to the (m — 1)'^ degree by the aid of /(xq) =0. We have therefore Theorem X. Every polynomial of the equations treated in § 174 is a factor of an expression or(x) — x . d,(x)—x ' where 0^{x) is an integral function of the (m — Vf^ degree. For example, if we take 6^ — x^ -\-hx-\-c, we may reduce this by the linear transformation y = x-\-aio the form O^^^x^-^- a. Then 0,\x) — x = {0,{x) — x) [x' 4-ic' + (3a+ l)a;* + {2a-\-l)x' + (3a^ + 3a + l)a)^ + (a' + 2a + l)a; 4- (a=^ + 2a^ + a + 1) J- The discriminant of the second factor on the right is jz=r_(4a + 7)(16a2H-4a + 7)l If now we take ■ 4a + 7=— fc^ a= — ^^^, the second factor breaks up into two, and this is the only way in which such a reduction can be effected. We have then 6'i(a;)— a? [8x' + 4(l + A;K— 2(9 — 2A; + A;> — (1 + 7A:— A;' + A;^)] [8i«'4-4:(l — fcK— 2(9 + 2A; + fc>— (1 — 7fc— A;^— A;^)] or, for A; = 2x + 1, a = — /^ — / — 2, ^(nr\ 'T ^yz:^ = [^^ + (A + l)^^-(A^+2)^-(A^ + A^ + 2A + l)] [a^' — Aa^^ — (A2 + 2A + 3)a^ + (/l=* + 2P + 3/l-f 1)]. In this way we obtain the general criterion for distinguishing those equations of the third degree the roots of which can be expressed by £c, 0{x), 6^\x). In the first place must [be reducible to the form d=:x'—{X'-{-X-\-2) 14 210 THEORY or SUBSTITUTIONS. that is, 6^ — 4c must be of the form 4 (/^ + ;. + 2) ^ (2A -f 1)2 -|_ 7 . then to every d there correspond two equations of the required type ^' + (>^ + lK-(A^ + 2)x — (/^ + /2 + 2/ + l)==0 x'—Xx" - (A2 + 2A + 3)^ + {X' + 2A^ + 3>^ + 1) = 0. It appears at once, however, that is unchanged if X is replaced by — {X 4- 1), and that the first equation is converted into the second by this same substitution. It is sufficient therefore to retain only one of the two equations. § 181. We introduce now the following Definition. If all the roots of an equation are rational f mic- tions of a single one among them, then, if these rational relatio7is are such that in every case OaO^ix,) = Op0,{x,), the equation is called an ^^Ahelian Equation^ * We have already seen (§ 173) that, if the roots of an equation are defined by 5), the resolvents cp, =: (1, 0,{X,) ), the coefficients of which are rationally known. We noted further that this equation is solvable only under special conditions. These conditions are realized in the present case. We proceed to prove Theorem XI. Abelian equations are solvable algebraic- ally. ** In the first place we observe that since cr^ is symmetric in x^ , ^i{xi), . . . '^s > • • • those which are distinct. It is clear that from SaS^ = s^Sa follows also o,...hh 214 THEORY OF SUBSTITUTIONS. and form the cyclical resolvent 71,2(^1) = in which (o., is a primitive yi^^^^ root of unity, the function /j .^(x^) is unchanged by the group G2 , and is therefore a rational function of ^''1. For the substitutions of the group leave (/'^ , o unchanged and the powers of s.y convert v^'i , 2 iuto %''l,2, f^2V'],2,. . -^"-"^''1,2, respectively. Applying Theorem IV again, we obtain ^'j ^ 2 from (/\ by the solution of a second simplest Abelian equation of degree ^2- In general, if we write V],2,...|/— ^j 'V + i ...'a; (,a^ij, /] , 2 , . . . V =^ [0l.2,...v+^Av^,2....v+^.V0^2....v^-...+^/•'-'^.''•'-v^.2....J"^ the value of ^''1,2,... v is determined from that of the similar func- tion ...-.=2'''''"'---'''''(^')- by the aid of a simplest Abelian equation, as defined by Theos rem IV. By a continued repetition of this process we obtain finally * Theorem XIII, If the n roots of an Abelian equation are defined by the system ,ho,h . . . e.'^^ix,) (/I, ^ 0, 1, 2, . . . n,— 1) n^n^n^. . .nj, = n^ the solution of the equation can be effected by solving successively k '''' simplesV^ Abelian equations of degrees ni,n2,ns, . . . n,,. *L. Kronecker: Berl. Ber., Nachtrag z. Dezemberheft, 1877; pp. 846-851. THE ABELIAN EQUATIONS. 215 § 184 The solution of irreducible Abelian equations can also be accomplished by another method, to which we now turn our attention. Theorem XIV. The solution of an irreducible Abelian equation of degree n = p^^'-^p^'^- . . . , where Pi,P2-, • . . cife the differ- ent prime factors of n, can be reduced to that of k irreducible Abe- lian equations of degrees Pi"i, P2^, . . . The proof * is based on the consideration of the properties of the group of the equation. For simplicity we take n ^Pi'^ip^"-^ Since the order of the group is r = n, the order of every one of the substitutions is a factor of n, and is therefore of the form Pi"ip/2. Every substitution of the group can accordingly be con- structed by a combination of its (^2"^)'^ power, (which is of order Pl^) and its {p"^y^ power (which is of order pa^O- Consequently we can obtain every substitution of the group G by combining all the the orders of which are a power of Pi , with all the /" /" t" i" ^ 1)'' 2)*' Z-> • ' • ^ r2) the orders of which are a multiple of p^ . Since the f s are all com- mutative, the substitutions of G are, then, all of the form s = {t'j'^t\...){r^r',ri,..). The order of the product in the first parenthesis is a power of Px , and therefore a factor of Pi*i. For we have Two substitutions {t'J'^'-'){i"^t"^'--\ {t'J\..,){t"at'\...) are different unless the corresponding parentheses are equal each to each. For if the two substitutions are equal, we have {rj'^ . . .) {t'J\ ...)-' = {t",t'\ . . .)-'{f'at"e . . .), and since the order of the left hand member is a divisor of Pi\ and that of the right hand member a divisor of p^^ each of these divi- sors is 1. * O. Jordan ; Trait6 etc. § 405-407. 216- THEORY OF SUBSTITUTIONS. The number of substitutions s is equal to n — p^'^\p2°-^. And since the substitutions t' form a group and every substitution of this group is of order p{^\ the order of the group itself must be p{^^, (§ 43). Similarly the Order of the group formed by the t" 's is equal to p^.. It follows then from n = Pi°-^P2^ = p^^p.^'i' that mi = aj , m ^^^ the theorem is proved. § 185. Theorem XV. The solution of an irreducible Abe- lian equation of degree p" can be reduced to that of a series of Abelian equations the groups of which contain only substitutions of order p and the identical substitution. If G is the group of such an equation, the order of every sub- stitution of (r is a power of p. Suppose that p^ is the maximum order of the substitutions of G. Then those substitutions which are of orders not exceeding p^~^ form a subgroup iJ of G. For if t^ and ^2 are two of these substitutions, then from the commutative property it follows that ^ so that t^t^ is also of order not exceeding p^~\ If the group H is of order p% any function (f belonging to H will take p*"** values and will therefore satisfy an equation of degree p*^""". If we apply to

2, suppose that s^ is a substitution of G not contained among the p^ substitution Sj^Sa^. Since s^s^ = s^s^ and S2S3 = s^s.2 , the group H^— {si,S2, s^\ contains at the most p^ substitutions. And it contains exactly this number, for if Si^s.^^s^" = s^'^s./s^y, then Sg'y"'; ==:si"~"S2^~^, and so on, as before. Proceeding in this way, we perceive that all the substitutions of G can be written in the form s,^^s,\ . . . s/% (A.- = 0, 1, ... p — 1) 218 THEORY OF SUBSTITUTIONS. where every substitution occurs once and only once (c/. § 183). If now we take for the resolvents and the corresponding groups ^a V «^1 5 "^2 ) • • • '^n) '■) -^ a "^ ] ^1 ) ^2 ? • • • ^a — 1 f ) S-'^a — 1 (, ^1 ) ^2 ) • • • '^n)\ -" a — 1 ^^^ '(^1^ ^'2i • ■ • ^'a — 2 ? '"^a C ) then every, resolvent depends on an Abelian equation of degree p^ The roots of the given equation of degree p'^ are rational functions. of 7: COS^ — -, COSrz — - , . . . cos. 2,-1-1' ^^"2v + l' 2v4-l This equation is of the form 220 THEORY OF SUBSTITUTIONS. -j- 5^^ ^ ^ U^ ... — U. With the notation 2- ^^^ o — T~T = cosa = X Zv-j- 1 we have then 2m- cos-^ — TT ~ ^v^/ ~ cos ma, so that the equation Ci) has also the roots e{x\ d\x), d\x\ . . . that is cos a, cos m a, cos in^ a, cos m^a, . , . cos m^ a, . . . If now g is any primitive root (mod. 2v + 1) then the v terms of the series Ri) cosa, cosga, cosg'^a, . . . cosg*'~^ a, are distinct, For from the equation cos g*^ a = cos g^ a (« > ; a, < v) it would follow that g'^a=±g^a-{-2k7:, or, replacing a by its value „ ' , Zv "p JL g<^^:g^ = g^(g<^-^zfl) = k{2v-\-l). Dividing both sides of this equation by g^, and multiplying by g,a-^ _j_ i^ ^Q obtain the congruence gKa--^)^l (mod. 2v + l). But, since since 2(a — /S) < 2v, this congruence is impossible. Con- sequently cosg'^a is different from cosg^a. Again cos g^a = cos a. For since f""'" — 1 = {g" — l)(g*'+ 1) is a multiple of 2v + l, one of the two' factors is divisible by 2v-|-l, so that ^v=±l + fc(2n + l), and consequently the relation holds THE ABELIAN EQUATIONS. 221 Gosg^a — cos[± 1 + ^(2'' + l)]a = cos{±: a-|-2fcr:) = cosa. It follows then that the v roots of the equation CJ are all contained in the series R^, or again in the series X, d{x\ b\x\ ...e^-'{x), while e''{x) = 1. The equation Oj) can therefore be solved algebrai- cally. We have an example of § 174. If we have v = iii n^ . . . n^ , it appears that we can divide the cir- cumference of a circle in 2v-}-l equal parts by the solution of to equations of degrees Ui, n2, . . . n^^. It n^,7i2, . . . n^ are prime to each other, the coefficients of these equations are rational numbers. (§176). In particular if v = 2'^, we have the theorem on the construction of regular polygons by the aid of the ruler and compass. CHAPTER XII. EQUATIONS WITH IIATIONAL RELATIONS BETWEEN THREE ROOTS. § 190. The method employed in § 183 is also applicable to other cases. We will suppose for example that all the substitutions of a transitive group G are obtained by combination of the two substitu- tions s, and 80, which satisfy the conditions 1) that the equation ,s^« = s/ holds only when both sides are equal to identity, and 2) that SoSi = 8i^s<, (Cf. § 37). If, then, the orders of Si and S2 are Ui and n.,, all the substitutions of G are represented, each once and only once by s/'is/"-' (h,= 0,l,2, ...n, — 1). Suppose now that G is the group of an equation f{x) — 0. We construct aresplvent ^ = 4'o belonging to the group 1, Si , s^, . . . s"^~\ and denote the functions which proceed from 9'-^ on the application of S2, So", . . . s/'2 "^ by (J'l, (/'2^ . . . ~^SiS2 = Sj^. From Sy we can therefore obtain every possible 6\, by the method of § 4G. We have only to write under every cycle of s^ a cycle of Sj^" of the same order, and to determine the substitution which replaces every element of the upper line by the element immediately below it. This substitution will be one of the possible s/s. We consider separately the two cases 1) where 8^ consists of two or more cycles, and 2) where 8^ has only one cycle. In the former case the transitivity of the group is secured by So. Consequently every cycle of S]^^ must contain some elements different from those of the cycle of Sj under which it is written. It is clear also that all the cycles of Sj must contain the same number of ele- ments. Otherwise the elements of the cycles of the same order would furnish a system of intransitivity. The order of the cycles can then obviously be so taken that the elements of the second cycle stand under those of the first, those of the third under those of the second, and so on, so that with a proper notation the following order of correspondence is obtained 224 THEORY OF SUBSTITUTIONS. Si = {XiX.2 X^ . . . ) (t/j^2 2/3 •..)•••. ^1 — (2/12/1 +i 2/1 +2A; • • • ) (^1^1 +4^^1+2* •••)••• It follows then that S2 = (x,y,z, . . . ){x2yi + j,z, + j,2 ...)••• The group is therefore non- primitive, the systems of non-primitivity being a^j , cr, , . . . ; 2/1 5 2/2 ? • • • ; ^ > ^2 ? • • • The substitutions Sj* leave the several systems unchanged, the substitutions Si% permute the systems cyclically one step, Si%^' two steps, and so on. Accordingly every substitution of the group except identity afPects every element. The group is, in fact, a group i2 (§ 129). The adjunction of any arbitrary element x^ reduces the group to the identical substitution. Consequently all the roots are rational functions of any one among them. The following may serve as an example: Si = {x,x,x^) (2/12/22/3), s, = {x,y,) {x,y.) {x,y,), . s,s, = (X1I/2) {x,y,) (xgi/g) = s,% . § 193. In the second case, where s^ consists of a single cycle, the transitivity is already secured. We may write then, as in Chap- ter YIII, Si = \z Z-\-l\, s^'^=\z Z-{- a\ To construct the s./s we proceed as before and obtain from S] = (X1X2 a?3 ' ' '^^ (k~> 1) ^1 —-^ \^i^i -f k^i + 2k • • • ) the series of substitutions Jc^ 1 §2 = 1-^^ kz + i — k\, s/=\z k^z^—-—{i—k)\. Now, in the first place, it is easily shown that the group contains substitutions different from identity, which do not affect all the ele- ments. For among the powers of s^ there is certainly one s^'* which has a sequence of two elements in common with S2. Then Sj'^Ss"^ does not affect all the elements. Again, it can be shown that there is no substitution except iden-^ tity which leaves the elements unchanged. For* we have hP 1 s,'^s/ = \z mzJra)-{--j^—^ii-k), • RATIONAL RELATIONS BETWEEN THREE ROOTS. 225 and if X\ and Xx^i were not affected by the substitution we should have m 1 fc?(;, + «) + |_i(i_fc)=;, ^/3 1 /fc^(^ + 1 +«) + ___ (;_fc)=/i + 1, and consequently The substitution then becomes s,'^s/=z\z z-\-a\, and since Xx and x^^i are unchanged, a^O, and the substitution is identity. The following is an example of this type: , Sj := \XiXo) is rational within the rational domain. Since p is a prime number^ it is possible to find an integer /j. such that the congruence or the equation ma :z= -yp-j- 1 shall be satisfied. Then the quantity (± Xi"uJ-'Y = ± iC^^ + ^wMT— ^ A''XiOjl^-'= ± A^x, and consequently x/, is rational. From the reducibility of the equation would therefore follow the rationality of a root, which is certainly impossible. The group of the equation is of order p{p — 1). For if we leave one root Xi unchanged, any other root ojXi can still be converted into any one of the p — 1 roots (ox^j oj^x^ , oj'^Xi . . . w^~'a?i. RATIONAL RELATIONS BETWEEN THREE ROOTS. 229 Theorem IV. The binomial equation in which A is not the p^^ power of any quantity belonging to the rational domain, belongs to the type of § 196. Its group is of order p(p — l). § 198. Remark. By Theorem III every irreducible equation the roots of which are rational functions of two among them is alge- braically solvable. At present we have not the means of proving the converse theorem. It will however be shown in the following Chapter by algebraic considerations, and again at a later period in the treat- ment of solvable equations by the aid of the theory of groups, that every equation of prime degree, which is irreducible a)id algebrai- cally solvable, is either an equation of the type above considered, or an Abelian equation. Before we pass to such general considera- tions, we treat first another special case, characterized by rational relations among the roots taken three by three. § 199. An equation is said to be of triad character, or it is called briefly a triad equation* if its roots can be arranged in tri- ads Xa,Xp, Xy in such a way that any two elements of a triad deter- mine the third element rationally, i. e., if Xa and Xp determine Xy, x^ and Xy determine Xa, and Xy and Xa determine x^. Thus the equations of the third degree are triad equations; for Xi -j- X> -J- £^3 ^^= Cj . Of the equations of the higher degree, those of the seventh degree may be of triad character. In this case the following distribution of the roots x^ , Xo, . . . Xj is possible : Xi , X2 ) "^3 j X\, X^ , X^ I J?i , Jyg , Xi \ X2 , X4 , Xf, ', X2 , .^5 , X'j ', "^3^ Xif Xf', X^ , X^, X^, ♦ If the degree of an equation is n, there are „ — pairs of roots Xa,Xp.. With every one of these pairs belongs a third root Xy. Every such triad occurs three times, according as we take for the original pair of roots x^^ Xp-, x^,Xy', or a:-^? '^a- There are there- fi(fi I ) fore ^ triads, and since this number must be an integer, it ♦Noether: Math. Ann. XV, p. 89. 230 THEORY OF SUBSTITUTIONS. follows that the triad character is only possible when n = 6i)i + 1 or n = 6m -|- 3 . The case n = Qin must be excluded, because n must be an odd number, as appears at once if we combine .^i with all the other elements, which must then group themselves in pairs. The general question whether every n = Qm-\-l, n — Qm-^-S- fnTnishes a triad system we do not here consider. It is however easy to establish processes for deducing from a triad system of n elements a second triad system of 2n -j- 1 elements, and from two- triad systems of >?i and Wg elements a third system of n^ n.2 elements. From the existence of the triad character for 7i = 3 follows therefore that for n = 7, 15, 31, . . . ; 9, 19, 39, . . . ; 21, 43, . . . These do not however exhaust all possible cases. There are for example triad systems for n ■= 13, etc. § 200. We proceed to develop the two processes above men- tioned. In the first place suppose a triad system of n elements a\, Xo, . . . x„ given. To these we add n-f 1 other elements x^; x\, ic'a, . . . x' „. We retain the ln{7i — 1) triads of the former ele- ments, and also construct from these — ^-^ new triads by accent- ing in each case every two of the three .t's. Finally we form n further triads Xq, Xi, x\', Xq, x'r,, x'2', . . . , and have then in all 4.n(n—l) _ {2n + l)2n 6 '^''~ 6: triads, which furnish the system belonging to the 2n -\- 1 elements. For example, suppose n = 3. We obtain then the following sys- tem X^ y X2 ^ X-^', -l^j , J? 2 ) "^ 3 5 ^ 1 ) ^2 5 '^* 3 ) ^ 1 ? ^ 2 > "^S 5 '^'0 5 •^^H '^ 1 5 '^'(1 ? '^2 ? "'^ 2 ? ^0 ? ^^3 ? '^ 3 > which agrees, apart from the mere notation, with the triad system for seven elements established in the preceding Section. § 201. Again, suppose two triad systems of degrees n, and r/g to be given. The indices of the first system we denote by a , 6 , c , . . . , those of the second by a, fi , y, . . . We may designate a triad by the corresponding indices. Suppose that the triads of the first system are RATIONAL RELATIONS BETWEEN THREE ROOTS. 231 ^^i) a, b, c; a,d,e; b ,d,g; . . . and those of the second We denote the elements of the combined system by a?«a, a?«/3, x^a, .. . and form for these a triad system as follows. In the first place, we write after every index of T^) the index a. In this way there arise _i}^-L____Z triads of elements with double indices. D In the same way we write /?, then y, then ^, . . . after every index of T^). We obtain then in every case \ and in all 6 h(ni — 1) 6 triads of the elements x„ 1 '^a /3 ) '^ay ) ^6 a J <^6^! All of these are different from one another. They are aa, da^ ea\ ba^ da^ ga', nf:t, dfi, ey9; bfi, d^ , giS-, cf'r, "^3 5 *^i J '^2 > ♦^4 ? Xi, X2, x^; Xi, X2f X(^', x^, X2 , x^j ; KATIONAL RELATIONS BETWEEN THREE ROOTS. 233 •it appears that only tlie second and the fourth cases give rise to a triad distribution of the required character, viz. Ml) X'l, X2, x^; x.2f x^, x^j x^j .^4, Xqj X4, ii?5, x-;', X-,, x^j x^] .1*6 , X-J , X.) 5 Xjy Xly X^\ Xf, , x~, , x^ ; Xf , Xi, Xr^ ; The two distributions are not essentially different, each being ob- tained from the other by interchanging x>, d?^; x^, x^', and Xi, Xr^. We may therefore assume that Tj) is given, and that s^ belongs to the corresponding group. If there are other substitutions of the 7*^ order belonging to the group, a proper power of every one of these will contain ic, and x.^ in succession. We may write the sub- stitution therefore "^ (xiX2Xa.^x„^x„^x,,^x„^) = (1 2 tta cti a^ cif, a-,) To this substitution correspond, as in the case of s^ , only two triad systems, which proceed respectively from 1, 2, a^ and 1, 2 a^. The indices a^, . . . a^ must be so taken that the new systems coin- cide with T,). In this way we obtain seven new substitutions s. For example, if the seven triads 1, 2, a,,; 2, ag, a^; ag, a^, 1; a^, a.,, 2; a,-, ae, ag; a^, a^, a^ ; a^, 1, a,, " are to coincide respectively with 1,2,4; 2,3,5; 3,7,1; 7,6,2; 0,4,3; 4,5,7; 5,1,6, we must have a^ = 3, a^ — 7 , a^ = 6, a^ = 4, a^ = 5, and accord- ingly s = {x^X2XQX^x^x^X:). Similarly we obtain for the seven new s's .S'2 = (XiX2Xr^X^X.^-,X(i), 6*3 = {x^X^X^X^X-jX^Xr,), S4 = {XiX^X-.X^X^X^X^, 05 — yXyXqijC^X'^XQ^x^Xi^f^ og — \XiX2XrpCffiC'jX ^x^j ^ s-j ^^^^ {XiX^x^Xepc^x^x-j)^ og — yXiXoX-^x^XKpCj^x^). Beside the powers of .s, , 82 , . . . Sg there can obviously be no other substitutions of the V^ order in the group. We note, without further proof, that it follows from this by the aid of § 76, Theorem XII, that the required group is I tS*] , §2 , . . . Sg j- The same result has been obtained by Kronecker from an entirely different point of view. 234 THEORY or SUBSTITUTIONS. Theorem V, The roots of the most general in^editcible triad equation of the 7'^ degree can be arranged as follows: j's = '^,(iro, Xi), Xi = >'^,{x, , x^), X-, = ^^Xo, x^), x^ — //i(u'3, x,), The group of the equation is tlie Kronecker group of order 168^. defined by \z az-i-b\, \z aO{z-\-b) + c\ (a = 1, 2, 4; 5, c = 0, 1, ... 6; 0{z) = —z\z' + 1) ) It is doubly transitir>e. Those of its substitutions which replace Xq j Xi by a?; , a?3 are yXQXiX^) {X.iX^X^), [XqXiX^) {X2P0qX^), [XqXiX^) {^X2X^X^), (XqXiX^) {X^X^X^f.. All these also replace x^ by Xq. Consequently ive have also and similarly x^ = '?i(a?2 , Xi), X2 = ^^x, , X,) ; etc. All the substitutions of the group ivhich interchange Xq and x^ are yXQXi) (X^Xr^)^ yX^fXl) {X^XQ), (^S^q^jJ {^X2XiX^XQfj yX^Xif yX2XQXQX^), and since these all leave x^ unchanged, it folloivs that and the same property holds for all the other triads. Every sym- metric function of the roots of a triad is a 1 -valued resolvent. § 203. We examiDG also the triad equations for n = 9. In the construction of the triads it is easily recognized that there is only one possible system, if we disregard the mere numbering of the ele- ments. We can therefore assume the system to be that constructed in §201, and designate the elements accordingly by two indice8> each 00, 10, 20 00,01,02 00, 11, 22 00, 12, 21 A characteristic property of every such triad pq, p'q\ p"q" is the condition 01,11,21; 02,12,22; 10,11,12; 20,21,22; 01,12,20; 02, 10,21: 01,10,22; 02,11,20: RATIONAL RELATIONS BETWEEN THREE ROOTS. 235 B) p -h p' -^p"^*! + q' + q" ^- (mod.8) From this it follows that every substitution s -—p,q ap -i- bq + a , a'p + h'q -j- a \ transforms the triad system into itself. For the indices p, g; p\q'r p", q" become respectively ap + bq -{- a , a'p -\- b'q + a' ; ap' -f hq' -\- a , a'p' + '^'q' + *'^' \ ap" + bq" + a, ay + b'q" + «": and if the condition B) is satisfied by p,p',p"', q, q', q'\ it is also satisfied by the new indices. Conversely, every substitution that leaves the triad system un- changed can be written in the form .). RATIONAL RELATIONS BETWEEN THREE ROOTS. 237 That conversely all these substitutions satisfy these two conditions- is obvious. Their number is 3" -2^, since ah' .'lO (mod. 3). It is further required that t shall also leave the third and fourth lines unchanged. The third line has the property that in every triad (pq, p'q', xj"q") the three sums p-\-q, p'-\-q\ p"-\-q" have respectively the values 0, 1, 2 (mod. 3); the fourth that P + ^- P'+g':-p"+^"^ (mod. 3). If now we apply ' + o. + a' (mod. 3), that is, a b' (mod. 3). The linal form of (j is therefore n= p^q ap-\-a^ aq-\-a \. Conversely all the substitutions of this type convert every one of the four lines into itself. The substitutions form a subgroup H of G of order 2 • 3", since a can only take the values 1 and 2. if is a self -conjugate subgroup of G. For if r is any substitution of (r, then r ^ i/r leaves every line unchanged, /. e., r^^ Hr =z H. The group H of order 2 • 3', being a self- conjugate subgroup of G which is of order 2* • 3^, we can construct, by §86, the quotient T= (t\H of order 2^- 4 = 24 and of degree 4 (corresponding to the four lines of triads). This group is of course the symmetric group of 4 elements. If therefore we construct a function y? of the 9 elements x, which belongs to the group ii, this four-valued function is the root of a general equation of the fourth degree, the group of which is T. If this equation of the fourth degree has been solved, the group of the triad equation reduces to H^ of order 2 • 3^, as is readily apparent. The systematic discussion of this class of questions is however reserved for Chapter XIV. 238 THEORY OF SUBSTITaTIONS. § 205. We consider further the subgroup of H which leaves every single triad of the first line of our table unchanged. In order that \p,q ap-i-a, aq-{-a' \ may have this property, the values q = 0, 1, 2, must give again g = 0, 1,2. Consequently «e^0, a = 1, and we must take r= p, q pJ^a, q\. The r's form again a self-conjugate subgroup of I of H of order 3. We construct by j^ ^^"^ the quotient U — H: I. U is of order 3 • 2 and of degree 3, corresponding to the three triads. U is therefore the symmetric group of three elements. If, then, we construct a function c'' of the 9 elements ,r, which belongs to the group /, this (after adjunction of cr) three-valued function depends on a general equation of the third degree. If the latter has been solved, the group of the triad equation reduces to /. Accordingly the symmetric functions of are known, and therefore those three values depend on an equation of the third degree, the coefficients of which are rationally express- ible In terms of (x, , .r,), ■ .i\ = 0(,x', , x,) = 0(x, , ,r,,), .r,= 0{x^,.r,) = 0{x,,x,), in which is a rational function of its two elements, the?) the c(/ua- tion can be solved algebraically. We consider the group of the equation. It is transitive, and it replaces the three roots, x^^ , x_ , x. by three others between which the same relation must exist as between a?i,,r2,.r.j themselves. Suppose the new roots to be x\, x'.,, x'... If the two systems RATIONAL RELATIONS BETWEEN THREE ROOTS. 239 ,Ti , ,r. , .r;i , and x\,x'2, x'^, have two roots in common, then they have also the third root in common. For, if Xi = x\ , .r^ = x\ , it follows that ^nd if x'.. and Xj are not the same root, the given equation, having «qual roots, would be reducible. If x^ is a root different from a',, x.,, .r.,, there is a substitution in the group which replaces x\ by .r^. If this substitution leaves no element unchanged, we obtain an entirely new system x^ , x-, , ^^. But if one element, for example x., , remains unchanged, we have for anew system x.,, .r^, x^. Proceeding in this way, and examining the possible effects of the substitutions, it is seen that all the roots arrange themselves in the triad system of 9 elements. Comparing this result with Theorem VI, it appears that the equation is exactly one of the triad equations just treated. It is known * that the nine points of inflection of a plane curve of the third order lie by threes on straight lines. These lines are twelve in number, and four of them pass through every point of inflection. Any two of the nine points determine a third one, so that the points form a triad system, as considered above. The abscissas or the ordinates of the nine points therefore satisfy a triad equa- tion of the 9th degree, and this equation, belonging to the type above discussed, is algebraically solvable. It can, in fact, be shown that if ^'i,x,, X:~ are the abscissas or the ordinates of three points of inflection lying on the same straight line, then X; = 0{xi,X2), x, = 0{x2,x.^, x, = 0(x.„Xi), where is a rational and symmetric function of its two elements. The discussion of this matter belongs however to other mathemati- cal theories and must be omitted here. *0. Hesse: Crelle XXVIII, p. G8; XXXIV, p. 101. Salmon: Crelle XXXIX, p. 365, CHAPTER XIII. THE ALGEBRAIC SOLUTION OF EQUATIONS. § 207. In the last three Chapters various equations have been- treated for which certain relations among the roots were a priori specified, and which in consequence admitted the application of the theory of substitutions. In general questions of this character, however, a doubt presents itself which, as we have already pointed out, must be disposed of first of all, if the application of the theory of substitutions to gen- eral algebraic questions is to be admissible. The theory of substi- tutions deals exclusively with rational functions of the roots of equations. If therefore in the algebraic solution of algebraic equa- tions irrational functions of the roots occur, we enter upon a re- gion in which even the idea of a substitution fails. The funda- mental question thus raised can of course only be settled by alge- braic means; the application to it of the theory of substitutions, would beg the question. To cite a single special example, proof of the impossibility of an algebraic solution of general equations above the fourth degree can never be obtained from the theory of substitutions alone. § 208. In the discussion of algebraic questions it is essential first of all to define the territory the quantities lying within which are to be regarded as rational. We adopt the definition* that all rational functions with integral coefficients of certain quantities 9i', ^J{", 91'", . . . constitute the- rational domain (9f{, 9f{" 3??'", . . . ). If among any functions of this domain the operations of addition, subtraction, multiplication, divis- ion, and involution to an integral power are performed, the result- ing quantities still belong to the same rational domain. The extraction of roots on the other hand will in general lead *Ii. Kronecker: Herl. Ber. 1879, p. 205 If.; cf. also: arithm. Theorie d. algeb. Grossen. . THE ALGEBEAIC SOLUTION OF EQUATIONS. 241 to quantities which lie outside the rational domain. We may limit ourselves to the extraction of roots of prime order, since an {mnY^ root can be replaced by an m^^ root of an n^'' root. All those functions of 9i', 91", 9^'", . . . which can be obtained from the rational functions of the domain by the extraction of a sinojle root or of any finite number of roots are desifrnated, collect- ively, as the algebraic f unctions of the domahi (91', 91", 91"', . . .). In proceeding from the rational to the algebraic functions of the domain, the first step therefore consists in extracting a root of prime order p„of a rational, integral or fractional function i^^v(91', 9t", 91"' . . . ) which in the domain (^R', 9t'', 91'", . . .) is not a peifect p„*^ power. Suppose the quantity thus obtained to be V^ so that yf'' = i'V(9^',9l",9t"',..,). We will now extend the rational domain by adding or adjoining to it the quantity Vy, so that we have from now on for the rational domain (Vv\ 91 , 9t", 9t"', . . .), i. e., all rational, integral or fractional functions of F^, 9^', 91", 9i"', . . . are regarded as rational. The present domain includes the previous one. With this extension goes a like extension of the property of reducibility. Thus the function xP — Fv{^\ 9t", . . .) was originally irreducible: it has now become reducible and has, in the extended domain (F„-, 9i', 91", . . .), the ra- tional factor X — Vy. The new domain can be extended again by the extraction of a second root of prime order. We construct any rational function which is not a perfect {py_^,^'^ power within (Fv;9i', 91", . . ., and denote its (p^_i)^*^ root by Fv_i, so that It is not essential here that Vy should occur in -P„_, If now we adjoin Vy_^, we obtain the further extended rational domain (Fj,_i, Vy\ 91', 91", . . .). Similarly we construct V''L-'=t\_,(V,_„ n_„ n; SR', 3!", . . .), 16 242 THEORY OF SUBSTITUTIONS. where the F's denote rational functions of the quantities in paren- theses, and Pi,P2) • • ■ Pv-2 are prime numbers. Any given algebraic expression can therefore be represented in conformity with the preceding scheme, by treating it in the same way in which the calculation of such an expression involving only numerical quantities is accomplished. § 209. The Fa's are readily reduced to a form in which they are integral in the corresponding F's, that is Va^i, Va+2 • • • ^v? and are fractional only in the 9ft', 91", . . . Thus, suppose that where Gq, Gi, Gg, . . . ; H^.H^.H^, ... are rational in F^ +2? Fa+3,... Vv\ 91', 91", ... If now i<^ is a primitive (pa + i)'^root of unity, the product i'a + l-^ \ = o is a rational function of Va+2, Va+s^ . • • V^; 91', 91", . . . For on the one hand the product is rational in the if s, and on the other it is integral and symmetric in the roots of n;v =^«+i(^a+2, . . . T^v; 9t', 91", . . .) and is therefore rational in the coefficient Fa 4-1 of this equation. Again, if we omit from the product P) the factor Hq-\-Hi Va + i 4-H2 F\+i + . . . , the resulting product Pa+l — ^ is integral in V^+i and rational in 'Fa+2? • • • ^v'^ 9i'? 9i"» • • • More- over, since (o does not occur in P) or in the omitted factor, it does not occur in P,). » If now we multiply numerator and denominator of Fa by Pj), the resulting denominator is a rational function of Va+2» • . • Vpi 91', 9^", . . . alone, while the numerator is rational in these quan- THE ALaEBRAIC SOLUTION OF EQUATIONS. 243 titles and in Va+i- Dividing the several terms of the numerator by the denominator, we have for the reduced form of Fa where the coeffiaients Jq) «^n <^2» • • • are all rational functions of Va+2i ' ' ' ^v\ Sfi', 31", ... On account of the equations we may assume that the reduced form of F^, contains no higher power of Fa + i than the (pa+i — l)^^ The several coefficients J can now be reduced in the same way as Fa above. By multiplying numerator and denominator of their fractional forms by proper factors, all the J's can be converted into integral functions of Fa + o of a degree not exceeding Pa+2 — 1? ^^^ with coefficients which are rational in Ta+sj • . • yv\ «R'j 3fl", ... In this way we can continue to the end. § 210. We have now at the outset to establish a preliminary theorem which will be of repeated application in the investigation of the algebraic form peculiar to the roots of solvable equations.* Theorem I. J/ /,,/,,.. ./^.i; i^ are functions within a definite rational domain, the simultaneous existence of the two equa- tions B) iv^—F =0, requires either that one of the roots of B) belongs to the same rational domain with /o,/i, . . . fjj-i'-, F^ or that If all thp/o,/j, . . ./^j_] are not equal to 0, the equations A) and B) have at least one root w in common. In the greatest common divisor of the polynomials A) and B) the coefficient of the highest power of w is unity, from the form of B). Suppose the greatest common divisor to be C) (Pq -f (f^w 4- c^w"^ -jr . . .-Jrwr Equated to 0, this furnishes v roots of B). If one of these is deno- *Tliis tlieorein \v is orij^iaaily ijiveii by AiujI: ()c;iivie-i coiiiiJiCies II, l which doos not vanish. we determine a new quantity Wa + i by the equation annex to this the equation of definition for Va^-i and fix for the rational domain It follows then, if A) and B) of Theorem I are replaced by A^) and Bj), that, since the possibility W^j^^ — 0, J^ = is excluded, we must have where «> is a (pa^-i)*^ loot of unity. We can therefore introduce into the expression for Fa in the place of Fa+i the function T^a + 1, provided we adjoin the (/>a+i)"'root of unity, ">, to the rational domain. From A,) and C^) it is clear that (TFa + r, V«+2,...; 9l'JR",...)and(F«+,,F«+„ ...; 9^'. 91", . . . ), define the same rational domain, and the equation . THE ALGEBRAIC SOLUTION OF EQUATIONS. 245 can be taken in the scheme of § 208 in place of Bj). The equations of definition for Va^ Va-u - - ■ V-i are not essentially affected by this change. We have only to substitute in the func.ions Fa-, i^a-n • • • -^i in the place of Va+i the value taken from Cj). The expression for Fa then becomes simplified We may ^ippose this reduction to have been effected in the case of every Fa. § 212. We pass now to the investigation of the form of the roots of algebraically solvable equations. Criven an algebraic equation 1) fix) = of degree n, the requirement that this shall be algebraically solv- able can be stated in the following terms: — from the ratioLal domain (9^', ^If",...), which includes at least the coefficients of 1), we are to arrive at the roots of 1) by a finite number of algebraic operations, viz. addition, subtraction, n^ultiplication, division, raising to powers, and extraction of roots of prime orders. One of the roots of 1 ) can therefore be exhibited by the following scheme: v:r,'=F..,(v.',^\w\...) 2) If these powers of x^ are substituted in 1), we have A) f{xo) = Ho-{-H,v,-^H,v;'+ . ... +H^,-iy^i-s where the jff's are formed additivily from the G^""s and the coeffi- cients of 1). Joining with A) the equation of definition of Vi 246 THEORY OF StJBSTITTJTiONS. and applying Theorem I to A) and B), we have only two possibili- ties: either a root of B) is rational in the domain (F2, V^, . . . V ; iR', 9{", . . . ), or Both cases actually occur. In the former the scheme j), by which we passed from the original rational domain to the root Xq, can be simplitied by merely suppressing the equation ' and adding the Pi*^ root of unity to the rational domain. § 213. As an example of this case we may take the equation of the third degree f{x) = x' — dax — 2b = 0, the rational domain being formed from the coefficients a and b. "By Cardan's formi.ila This algebraic expression can be arranged schematically as follows: V^^ = b'' — Q; V^:=^b^-V,^ V,' = b—V,, The expression for J/(a:o), formed as in the preceding Section, then becomes y(x,)^--aV, + (F/-a)F, + V,V,' = 0. Comparing this with and determining Vx from the last two equations, we obtain ^' " a^+(6+F3)F2 — aF/ so that Fi is already contained in the rational domain (F2, F3; a, b). If we now transform Fj into an integral function of Fj by the pro- ceis of § 20y, we obtain from the relations THE ALGEBRAIC SOLUTION OF EQUATIONS. 247 [a'+ (6 + V,)i is adjoined to the rational domain. § 210. In the construction of the scheme 2) it is not intended to assert that Fa necessarily contains Fa_i, Fa_.2, • • . If Va-i is missing in Fa, another arrangement of 2) is possible; we can replace the order V:XV=Fa + dVa + ,,...), V'^ = i^a(F« + „...), Ff^^ = F. (F., . . .) by the order v:-=F.^,{Va+„ . . .), n»+r (^«+2' • • •). vi'-'=F^_,(v^, . . .). It is therefore possible, for example, that different F's occur at the end of the series 2). In this case different constructions 3) for the THE ALGEBKAIC SOLUTION OF EQUATIONS. 24© root Xq are possible, aiid the theorem proved in the preceding Sec- tion holds for the last V of 2) in every case. To prove the same theorem for all F's which occur, not in the last, but in the next to the last place in 2), we will simply assume that i^i actually contains V2. The proof (§ 215) of the theorem for Vi was based on the fact that an expression satisfied an equation with rational coefficients. We demonstrate the same property for an expression If we suppose all the permutations of the roots of the equation 1) to be performed on "-[jiS---]': fc = the product of the resulting expressions is an integral function of y, with coefficients which are symmetric in the x^s and are therefore rational functions of 91', 91", . . . If we denote this function by 2, there must be a rational function Vy_yof the roots, which is (2j)t,_i)- valued, and of which the(p^_i)^^ power is two-val- ued. But such a function does not exist if n > 4 (§ 58). Conse- quently the process, which should have led to the roots, cannot be continued further. The general equation of a degree above the fourth therefore cannot be algebraically solved. § 218. We return now to the form of the roots of solvable alge- braic equations 3) x,= G,-\- F,4- <^.Fr-h . . . -\-G,^_,V,^^^-\ We adjoin to the rational domain the primitive Pi"', p.^-^, . . . roots of unity, and assume that the scheme which leads to a\, is reduced as far as possible, so that for instance Va is not already contained in the rational domain (Fa_, . . . V^^ 9i', 9i", . . . ; Wj, <'a,, . . . ). We have seen that the substitution of o>,^V, {k = l,2,...p,-l) for V in 8) produces again a root of f{x) = 0. We proceed to prove the generalized theorem: Theorem IV. // in the scheme 2), ii'hich leads to the expres- sion 3) for a^o, any V^ is multiplied by any root of unity, the values Fa-M ^^a- J? • • • 1^2) ^1 ^^**^^ *^* general be converted into new quan- tities Va, Va-i, • • • ^'21^1- ^/ ^'^^ latter are substituted in the place of the former in the expression for x^, the result is again a root offix) = 0. We may, without loss of generalit}^, assume that f{x) is irredu- cible in the domain {W , 3t'', . . . ). Starting now from 3), and denoting by cUr ,a primitive r*^^ root of unity, we construct JJ{x-x,) = JJlx-iGo-\-^2) is the lowest V that actually occurs. Then 4) JJ (^ — ^\)='fa{00',Va, Fa + i,.. ••) = «0 + «1 "^a + «2 "^a' + • -. A = The a's which occur here belong to the domain (Fa + n • • •)• ^® construct further ra - 1 ' \ = where Vj, {b^a-{-l) is again the lowest V that actually occurs. Similarly let Pb-l A. = and finally, supposing the series to end at this point, Pc — 1 A = where /d is rational in 91', W, . . . , all the F^+i) - • .V^, disappearing with V,. We assume now, reserving the proof for the moment, that the functions Mx; n, . . .), Mx', y„ . . .), fXx; K., . . .), Ux', 3^', . . .) are irreducible in the domains (T-«,n+i . . .), (n, n+i . . •), (K., y.+,, . . .), (r, r'. . .), respectively. Then fa{x; 91', . . .) = and /(a?) = have in the do- main (91', 9t", . . .) one root x = Xq in common, since x — Xq occurs as a common factor of fd{x) and f(x). Both these functions being by assumption irreducible, it follows that '/,(a:;i»',r',...)=/H- If now we assign to V^ any arbitrary value tv consistent with Vl" =^ F^,{'3i' , . . . ) and to V^_i any value tV-i consistent with V^'/S^^ = Fv_^{Vy', 91', . , •), and continue in this way, we have thej series 2') THE ALGEBKAIC SOLUTION OF EQUATIONS. 253 3') -- = 9i> + Q'l ^1+ 92Vi^ + . . • + (/in - 1^''' ' '^ 3') being obtained from 3) by putting the r's in place of the F's. The product A = <» will only differ from those obtained above by the introduction of the gr's and?y8 in place of the 6r's and F's, since in all the reductions 2') replaces 2). Consequently this product is equal to/„(^'; r,,, r„+i, . . .) and similarly A=0 Pb—\ A.= , Pc—l JJMx', coM; ^.+,, . . .) = .«^; '^i". ^)^"', • • •) -^/(^•)- A = This furnishes the proof that r^ is a root of f{x) = 0. We have still to prove the irreducibility of fa{x), fh{x), ... in the rational domains (F,,, F„ + i, . . .)? (^/o Vh + i, • • •)• • • ? respect- ively. Assuming the irreducibility of f„{x) in the domain ( V,, , y„ + 1 , . . .), we proceed to demonstrate that of fo{x) in the domain (Vi, F^,^ j , . . .). The method employed applies in general. If (f{x', Vb, . . .) is one of the irreducible factors of fi{x), so chosen that it contains fa{x', F„, . . .) as a factor, then we have in the domain V„ , F^ + 1 , . . . the equation 8) „"' F,, , . . . Again, f„{x; iof^V„, . . . ) is different from fu{x; ojf^V,,, . . .). For if we write faix; F„, . . .) - s,-i-,,V,, + .,y/+ . . . , it would follow from the equality of the two functions /„ that and consequently, from the equation of definition that Va must be rational in the domain (F„ + 1, . . . ^}t', . . . (Wj, . . .), since "^py^. Accordingly /„(ir; y„, . . . j, fa{x; w„F„, ...),.. . are all divisors of (f. All these functions are different from one another, and they are all irreducible in the domain (F„, F^, ^j, . . . ). Consequent!}^ e con- tains their product, which, on account of the degrees of /„ and fi in A', is possible only if (f and/^ coincide. Since the foregoing proof holds for every irreducible factor of 1), it still holds if we drop the assumption of irreducibility. § 219. At the beginning of the preceding Section we remarked that in the product construction with F, other T"'s might vanish. This possibility is however excluded in the case of certain T's, as we shall now show. We designate any Vr of 2) as an external radical when the fol- lowing Ft + i? Ft 4 2, . • . i- (i-, Fr-^i,F^ + :, . . . do uot contain Vr- Every such external radical can be brought to the last position of 2), and the expression of .r,,, as given in 3), can be arranged in terms of every external radical present. We shall see that in the product construction with V^ no other external radical can be missing. Thus, if F/ is missing in ?', - 1 THE ALGEBEAIC SOLUTION OF EQUATIONS. 255 then /„ cannot be changed if we replace Vr in the fundamental radical expression by oj^'^Vj, without thereby changing V^. If, as a result, the (xs are converted into the g's, we should then have also A = Every linear factor in x of this last expression must therefore be equal to some factor of the preceding expression A) S'o+ V,-\-g,V{+ ...= Go-i-t, which may be excluded, since otherwise 2) could be reduced fur- ther on the adjunction of o^, or that In some one of these equations Vr must actually occur. Develop- ing this equation according to powers of F^, we have A,) K,-i-K,Vr-{-K,Vj'-{- . . . =: K,^K,co,Vr + K,co,'Vr'-h . . . , and combining with this B,) V/r — Fr(Vr+,,...)=0, the impossibility of both alternatives of Theorem I appears at once. Consequently Vr could not have been missing in the product con- struction. If we consider only /„ (§218, 4)), the series 2) ending with V„ can also contain external radicals, in fact possibly such as are not external in respect to the entire series. These also cannot vanish in the further product construction. The irreducibility of /« being borne in mind, the proof is exactly the same as the preceding. Theorem V. In the pi-oduct construction of the preceding Section no external radicaU can disappear from /„ except V^.. The same is true for fi, in respect to the external radicals occurring among F^, F^+u • • • V^, and so on. If several external radicals occur in x^ or in one of the ex- pressions f a, fh ,fci ' ' • y tf^ product of all the corresponding expo- nents is a factor of n. 256 THEOKY OF SUBSTITUTIONS. Theorem VI. // an irreducible equation of prime degree p is algebraically solvable, the solution icill contain only one exter- nal radical. The index of the latter i^ equal to p, and if lo is a primitive p"' root of unity, the polynomial of the equation is V - 1 Theorem VII. // the algebraic expression is a root of an equation f(x) — 0, ivhich is irreducible in the domain (9t', 9t", . . . )> <^^^ ^f ^^ construct the product of the p^ factors, in which Fj is replaced by oj^V^, ^o^V^ . . . f,{x; Y,,, . . )=jl {^ — Xj,), where V„ is the lowest V present, and again the product fb(x; Vj,,...) of the p„ factors f„(x; o)„^V,,, . . . ), and so on, ive come finally to the equation f{x) — 0, the degree of which is n = p^p„Pb . ■ • The functions f„ ,fi,... ar^e irreducible in the domains {V„,V„^i,.. .), § 220. We examine now further those radicals which vanish in the first product construction. The remaining V^, F« ^ j , . . . are not altered in the product construction. We may therefore add these to the rational domain, or, in other words, we may consider an irreducible equation /(a?) =/„(x; F,,, .. .) in the rational domain (F„,y„ + „...;3{',r',...)- Here all the V^, Vo, . . . V„__^ already vanish in the first product construction. We examine now what is the result of assigning to F„_i any arbitrary value consistent with its equation of definition, then with this basis assigning any arbitrary value of V„_2 consistent with its equation of definition, and so on. Suppose that the functions Va-ly Va-2i • • ' V2i Vl'l ^0, G^2 J • - • ^Tp-l are thereupon converted into THE ALGEBRAIC SOLUTION OF EQUATIONS. 257 The new value assumed by Xq is then ^0 = 90+ g^^i + g^vi' + Qzv;' + . . . + gp-iv,^"'. From § 218 Cq is again a root, and this together with the system $1, ^2- • • • ^i)-i) which arises from Cq when v^ is replaced by w^i, '^i + g2«>V+ . . . = G^o + «>'"^i + G^2«>'"'T^i'-f . . . , where w', w", a»'" , . . . are the p^^ roots of unity w, «>^ w^, . . . , apart from their order. By addition of these equations we obtain 9o= Go, so that Gq is unaffected by the modifications of l^«_i, T^a-2> • • • ^2- Also p Gq is the sum of all the roots, and is therefore a rational function in the domain (91^, 9t", . . . ). Again we obtain from the system above the equation + G2F/K + ^'"^-'+^'"'">-'+ . . .) + ... Here the first term on the right vanishes. We denote the paren- theses in the following terms briefly by p^i,p^2i P^si • • • > ^^^ write 9) v,=P.,V, + P.,G,V,' + P^,G,v;'+... On raising this to the p^^ power A) v,^ = F,{v„v„ . . . v„_,',U\ ...) = [P^V,-j-P,G,V,'-i- . . .7 and annexing the equation of definition B) V,^ = F,{V„ V„... F„_,; 91', . . .), it follows from Theorem I that either Fj is rational in ^2, F3,...K_,; v„v,,...v„_,; 91', 9t",..., or that Vf = A,, A, = 0, A, = 0, ...A^_, = 0. 17 258 THEOBY OF SUBSTITUTIONS. We consider now the first of these alternatives. In the rational expression of V^ in terms of T^2 > ^3 ? ♦ • • ; ^2? ^3 » • • • ; 9^' Ut" , . . . all the ^2,^3,. . .Va-i cannot vanish; otherwise V2 should have been sup- pressed in 2). If then we define Fj , as in §§ 208 and 212 by a system of successive radicals, some V^ will occur last among the v^s and some V\ last among the v^s. If we substitute the expression for Vi in Xq, we have x,=R{V„.. . n.i;^,. . .) = R,(V„ . . . V,_,',v,, . . . v„_,;^', . . .) Here all the v's cannot vanish, as we have just seen. For the same reason all the T' s cannot vanish, since we might have started out from ?o- S^t V^ and V\ are two external radicals, and the product of their exponents must therefore be a factor of p (Theorem V). This being impossible, the first alternative is excluded. Accordingly we must have in A) The question now arises what the form of 9) must be in order that its p*^ power may take the form Vf = Aq . The equation A) is The result just obtained shows that the left member is unchanged if Vi is replaced by wFi, oj'^V^^, . . . Consequently and on the extraction of the p^^ root we have But from 9) follows also P.OJ'^V, + P2G2 oj'^Vi' + . . . = w'^^i, and equating the two left members and applying Theorem I as usual, we have i2,=:0, P,G2 = 0, . . . ^K_iG^._i = 0, P, + ,G, + , = 0, . . . , that is, 9) reduces to the single term 9') V, = PM.V,\ THE ALGEBRAIC SOLUTION OF EQUATIONS. 259 Substituting this result, together with Qq = Gq in the expression for ^oj we have On the other hand the root ^o, which is contained among Xq, Xi, . . ., oan also be expressed in the form ^0= Go + — 2), where /As < p — 1 and is defined by the congruence e^^A (mod. p). Consequently the quantities 1) JXq, ill, -^2> • • • -t^p~.2 are converted in order into -L^Ki -lCk + 1^ -rtK + 2> • • • ■^K+p — 21 and, if the same operation is performed a times, I) is replaced by -t^aK^ -f^aK-[-lj •^a/c-i-2) • • • -\--'^aK-\-p — 2') where the indices are of course to be reduced (mod. p — 1). If there is another modification of the radicals, which converts Rq into Rfj,, this on being repeated /9 times converts the series I) into Finally if we apply the first operation « times and the second /5 times, I) becomes Here a and /? can be so chosen that ax -\- iS/j. gives the greatest common divisor of /- and /j-. Consequently if i^//is the R of lowest index which is obtainable from R^j by alteration of the radicals, every other R obtainable from Rq in this way will have for its index a multiple of /?, so that the permutations of the i^'s take place only within the systems Ro, Rk, R9i- • ' ■ R(^ p-i i\j. Here k is sl divisor of p — 1. There are then alterations in the meaning of the radicals which produce the substitution (-Ro ^k Rih • • • ) (^1 Rk -f 1 R2k + 1 ...)... § 222. The preceding developments enable us to determine the group of the irreducible solvable equations 1) of prime degree p. Every permutation of the x's can only be produced by the alter- THE ALGEBEAIC SOLUTION OF EQUATIONS. 261 atiohs in the radicals V^, V2, . . . Va-x, and consequently only such permutations of the a?'s can occur in the group as are produced by alterations of the y's. From the result of the preceding Section Vi can be converted into W'G^V^'' , and the possible alterations in Vi do not change this form. Substituting this in the table of § 215, we have We examine now whether any root cc^ can remain unchanged in this transformation. In that case we must have Go4- . . . +^'^^G^/T^f + . . . = G^o+ <+''G,^Vf+ . . . , and from the method which we have repeatedly employed it follows, as a necessary and sufficient condition, that /^e ^fy--\-T (mod. p). If e^^l (mod. p), then for r4=0 there is no solution /x, and therefore no root cc^ which remains unchanged. But for r = 0, every /j. satisfies the condition, and the substitution reduces to iden- tity. If e^dsl, then for every r there is a single solution //, and the corresponding substitution leaves only one element unchanged. Theorem IX. The group of a solvable irreducible equation of prime degree is the metacyclical group (§ 134) or one of its sub- groups. . § 223. Since now, as we saw in § 221, all the substitutions of the group permute the values i?o,-Rt,-R2fc?- •• ^^^J among themselves, the symmetric functions of these values are known, and the values them- p — 1 selves are the roots of an equation of degree — —- . The latter is an rC Abelian equation since the group permutes the values Bq, Rk,R2k, • • • only cyclically. Consequently eyery Kk, Bok, ^sk, • • • is a rationa function of i?o- 33ut the same is true of every B^. For the form of the substitution at the end of § 221 shows that after the adjunc- 262 THEORY OF SUBSTITUTIONS. tion of Ro all the other BaS are known, since the group reduces to 1. Finally it appears that if Ra = ^a(-Ro) then Ba + km ^^ Fa\Bkm)- f For the application of properly chosen substitutions of the group converts the first equation into the second. We consider now all the substitutions of the group of f(x) = which leave Rq = F/ unchanged and accordingly can only convert Vi into some w^Fj. Then Xq is replaced by x^. But since R^ is a rational function of Rq, it appears that R^ = GJ'V^ is also unchanged, so that GeVi" is converted into some (r^fy'^Vi*. The power w^^ can be determined from Xi,-, for the expression for Xj, contains the term GeVi^j^''% and this must be identical with G^oji^Vi^ Consequently /x = ve, and GeV^^ becomes G,co-^V,% while at the same time V^" becomes so that the factor Ge remains unchanged. That is, every substitu- tion of the group, which leaves Rq unchanged, leaves Ge unchanged also. Accordingly Ge is a rational function of Rq. The same is true of all the other (r's. We can therefore write 12) x,= G, + V,+ ^,{Vf)- V,'-^n(V,-). v;'+...+ / Ri becomes If now we apply these transformations to the equations above, we obtain . We can therefore also write x,^G,+ ^R, +\/^ +/C/^,+ ... + ^2 W • ^Rf + v''2 W • ^Rf + 02(i^2.) • \^i^2/ + Theorem X. The roots of a solvable equation of prime degree p can be tvritten in either of the two forms 12) or 13). In 13) wRk, ^R2k, • • . «^e rational functions of \/Ro. The values Rq, Rj,, R2k, . . . R(pzlL-iY k I are roots of a simplest Abelian equation, the group of which is com- posed of the powers of a — (Rq Rf, R^j, . . .) Its roots are connected by the relations 14) ^S;=/(i?o)•^/57, ^Rr,=f{R,)-s/Rf, \/R;,=f{R,,)^- From 14) we have H.. =/^(-Ra) -/""{R,) -r^iR,) ■ R/\ and since R^j, = Rqj it follows from these equations that i=-Bo-""'-[r"'"'w-/-""'-^(i?.).../(-B,-.-.)]^ 1 = R.-'-'-'lf'-'-'iB,) -f '-'-'" (R,,) . . ./(i?„)] . Now the primitive congruence root e for p can be so chosen that €^~^ — 1 is divisible by no power of p higher than the first. For if e^-'=l (mod.p') then (p — e)^-'=e^-'' — (p — l)pe^-''^e''-'-{-peP-^=l-\-pe^-^ * (mod. p'), 80 that (p — e)^~Ms divisible only by p, and we can therefore take p — e in place of e. In e^~^ — l=p.q, then, q is prime to p. Consequently we can determine t so that tq-\-1^0 (mod. p). Suppose that tq = sp — 1. Substituting this in equation 15), and taking the p*^ root of the t^^ power, we have 1 = R,-"-' If -'■-'-'■ (i?„) ./'^-'-*(iJ,) ...]', THE ALGEBKAIC SOLUTION OF EQUATIONS. 265 Again, if. we write 16) f{R^,)=aa and take again the p*^ root we have Since by equations 16) the tta's are rational functions of the roots of an Abelian equation, the aja are themselves roots of an Abelian equation. The substitution (T= {EQRj,R2k . . .) of the former corresponds to the substitution r = (tto tti tta . . . ) of the latter. If the roots aQ^a^a^, . . . are different from one another, then Bak is a function of aa and this function is, in fact, the same for all values of a (§ 189). Theorem XI. The quantities ^ R can he reduced to the form ^R,= ->(^ll5 ^12 J • • • ^iMijj 2/2 *^ \'^21 5 ^^22) • • • '^2w/j 2/3 — ^ ^-^(^315 ■^32 5 • • • ^3»ij) 2) Consequently y is a root of an equation of degree v 3) v{y)=0, the coefficients of which are unchanged by all the substitutions of G, and which are therefore, from Theorem I, rationally known. " If = — factors 10) *" F.{S)=0, every one of which can serve as the Galois resolvent of the special equation. All the roots of 10) are rational functions of every one among them, and in terms of these all the roots of f{x) =0 can he rationally expressed. The transition from f{x) = to F^{^) — has its counterpart in the transition from G to the simply isomor- phic group ^ (§ 129) of F,{$). 272 THEORY OF SUBSTITUTIONS. Since the construction of 10) depends only on'the group G, and not on the particular nature of 9), this same resolvent belongs to all equations which are characterized by functions of the same family with y). If one of these equations has been solved, then x^, X2,. . . £c„ and consequently ^^ are known. The equation 10) is therefore solved, and with it every other equation of this sort. We have then the proof of the theorem stated in § 226: Theorem V. Given an equation f{x) = 0, the coefficients of ivhich belong to any arbitrary rational domain, the adjunction of either ^1 = or ^2 = 0, where (p^ and (p^ belong to the same family of the roots x^, X2, . . . x„, leads, as regards solvability, to the same special equation. § 230. We have treated in earlier Chapters cases where such relations between the roots either were directly given or were easily recognized as involved in the data. Frequently, however, the conditions are such that, instead of a known function, 4'{^\ ,oc2... x„) being directly designated as adjoined, (J> presents itself implicitly as a root of an equation which is regarded as solvable. For example, in the. problem of the algebraic solution of equations the auxiliary equation is of the simple form y^ — ^(^1, X2, . . . X,) = 0. Here y is regarded as known, i. e., we extend the rational domain of f{x) = by adjoining to it every rational function of the roots of which any power belongs to the domain. The actual solution of the auxiliary equations does not enter into consideration. It is a natural step, when an irreducible auxiliary equation is- regarded as solvable, to adjoin not one of roots 4'-, but all of its roots to the domain of f{x) — 0. These roots are the different val- ues which 4'{^\', X2, . . . x^ assumes within the rational domain. For to find the auxiliary equation which is satisfied by 4\ we apply to 4\ all the r substitutions of the group G and obtain, for example, m distinct values 11) ^'l,V''2,V''3,...V''.. The symmetric functions of these values, and therefore the coeffi- cients of the equation THE GROUP OF AN ALGEBRAIC EQUATION. 273 12) g{4') = {4'—4\)(4'—4'2) • • • {m) = ^ are known within the rational domain of f{x) = 0, and 12) is the required auxiliary equation, the solution of which is regarded as known. Now given the equation f{x) = 0, characterized by the group G, or by any function 4 the alternating group is simple. If there is an m- THE GROUP OF AN ALGEBRAIC EQUATION. 275 valued resolvent v'', its values 4\^ <,''2, . . .4>m are obtained by the solution of an equation of the w}^ degree. On the adjunction of these values, or of the group of the given equation reduces, by Theorem VI, to the identical substitution. The equation /(x) = is therefore solved; for all functions are known which belong to the group 1 or to any other group. The investigations of Chapter VI show, however, that no reduction of the degree of the equation to be solved can be effected in this way, since if n > 4, the number m of the values of 4) is solved^ as soon as any arbitrary resolvent equation of a degree higher than the second is solved. There are, however, no resolvent equations the degree of ivhich is greater than 2 and less than n. Moreover, if n==6, there is no resolvent equation of the n*^ degree essentially different from f(x) = 0. For n = 6 there is a distinct resolvent equation of degree 6. One other result of our earlier investigations, as reinterpreted from the present point of view, may be added here: Theorem VII J. The general equation of the fifth degree ■has a resolvent equation of the sixth degree. § 232. We return now, from the incidental results of the pre- ceding Section, to Theorem VI, and examine the group of the equation 12) flr(9^) = (0 — 00 {4' — 4',) ... (0— 0.) =0, the roots 4\ ? •••/>' ^,11 belong to the same group r, (§ 109, Theorem VIII). The group of 12) is therefore a group ii\ for it is transitive, since g{x) is irreducible. We have therefore Theorem X. If the group G of the equation f{x) — is THE GROUP OF AN ALGEBRAIC EQUATION. 277 of order r, and contaiyis a self- conjugate subgroup F of order r', and if y^ is a function of the roots x^, X2, . . . x^, belonging to F, then an irreducible resolvent equation of degree v = r:7'' =ir\r'. We assume that G' contains a self- conjugate subgroup /'', of order r^. From Theorem IX G' is r'-fold isomorphic to G. From the results of § 73 it follows that the subgroup J of G, which corresponds to the group F', is a self- conjugate subgroup of G and is of order v'r'. J is, then, like F, a self- conjugate subgroup of G, and their orders are respectively i-' r' and r'. We show that F is con- tained in J. This follows directly from the construction of G' (§ 232), in accordance with which the substitution 1 of G' corresponds to all the substitutions of G which leave the series 11) unaltered. F in. G therefore corresponds to the one substitution 1 of G'. Accordingly if G' is compound, then F is not a maximal self -conjugate subgroup of G. The converse theorem is similarly proved from the properties of isomorphic groups. In these last investigations we have dealt throughout with the group of the equation, but never with the particular values of the coefficients. If therefore two equations of degree 7i have the same group, the reductions of the Theorem X are entirely independent of the coefficients of the equations. The coefficients of h{x) will of course be different in the two cases, but the different equations h{x) = all have the same group, and every root of any one of these equations is a rational function of every one of its roots. This com- mon property relative to reduction, which holds also for the further 278 THEORY OF SUBSTITUTIONS. investigations of the present Chapter, is the chief reason for the collection of all equations belonging to the same group into a family. § 234. We observe f urtlier that with every reduction of the group there goes a decomposition of the Galois resolvent equation, v^hile the equation f{pc) =■ need not resolve into factors. Collecting the preceding results we have the following Theorem XII. If the group G of an equation f(x) = is compound, and if G, Gi, G2, . . . Gy, 1 is a series of composition belonging to G, so that every one of the groups Gi, G21 . . . Gy, 1 is a maximal self- conjugate subgroup of the preceding one, further if the order of the several groups of the series are then the problem of the solution of f{x) = can be reduced as foU lows. We have to solve in order one equation of each of the de- grees r r^ r^ iV-_i ^i' n' rs' ••• r, ' '''" the coefficients of which are rational in the rational domain deter- mined by the sohdion of the preceding equation. These equations are irreducible and simple, and of such a character that all the roots of any one of them are expressible rationally in terms of any root of the same equation. The orders of the groups of the equa- tions are respectively ^ ^ ^2 ^V-i n ' ^2 ' ^3 ' ' " '^v ' The groups are the quotients G : Gi, Gi'. G2 5 G2 '. G3 5 . . . Gy _ 1 : G^, , Gy : 1. The equations being solved, the Galois resolvent equation, ichich was originally irreducible and of degree r, breaks up successively into /V» /V» «J^ ^ ' 1 '2 '3 factors. After the last operation f{x) — is therefore completely solved THE GEOUP OF AN ALGEBRAIC EQUATION. 279 § 235. The composition of the group G of an equation f{x) = is therefore reflected in the resolution of the Galois resolvent equa- tion into factors. We turn our attention for a moment to the ques- tion, when a resolution of the equation f{x) = itself occurs. It is readily seen that, in passing from Ga to Ga+i in the series of com- position of G, a separation of f{x) into factors can only occur when Ga-^-i does not connect all the elements transitively which are con- nected transitively by Ga. The resulting relations are determined by § 71. Ga is non-primitive in respect to the transitively con- nected elements which Ga+i separates into intransitive systems. Starting from G, with an irreducible f(x) = 0, suppose now that Gi, G^, . . . Ga are transitive, but that Ga+i is intransitive, so that by § 71 Ga is non- primitive. Then at this point f(x) separates into as many factors as there are systems of intransitivity in Ga + n But (again from §71), all the elements occur in Ga+i- We arrange, then, the substitutions of Ga in a table based on the sys- tems of intransitivity of 6ra + i- Suppose that there are // such sys- tems, so that f{x) divides into fj. factors. Then we take for the first line of the table all and only those substitutions of Ga, which do not convert the elements of the first system of intransitivity into those of another system. The substitutions of this line form a group, which is contained in Ga as a subgroup. Its order is there- fore kVa+i. The second line of the table consists of all the substi- tutions of Ga which convert the first system of intransitivity into the second. The number of these is also kVa + i. There are n such lines, and they include all the substitutions of G%. Consequently kr, a + l i. e., the number ix of the factors into which f(x) divides is a divi- r sor of the number of the factors into which the Galois resolvr ''"a+ 1 ent equation divides at the same time. A similar result obviously occurs in every later decomposition. The decomposition can therefore only take place according to the scheme of Theorem III. The several irreducible factors are all of the same order. § 236. Thus far we have adjoined to the given equation /(a?) = 280 THEORY OF SUBSTITUTIONS. the root 4' of a second irreducible equation only when the <^'''s were rational functions bi x^^x^, . . .x^. This seems a strong limitation. We will therefore now adjoin to the equation f{x) = all the roots of an irreducible equation 18) g{z) = Q without making this special assumption. The only case of interest is of course that in which the adjunction produces a reduction in the group G of f{x) = 0. In the first instance we adjoin only a single root z^ of g{z) — 0. Suppose that G then reduces to its subgroup H^. If the rational function o = -K(/>), where i^ is a rational function; and since w is rational in the a?'s, it follows that 17) (O — Ro{<0o) or, which is the same thing, 17') p^RoiPo)' From 16) and 17') it follows that p and Pq belong to the same family. The adjunction of all the roots of /=0 to g = there- fore gives rise to the same rational domain as the adjunction of all the roots of ^ = to / = 0. It is obvious at once that the first adjunction, since it made 9ii ^2y ' ' ' • • • ^r? and form fir(?)s(l-f,) {?-?,) . . . (l-f,.). It is characteristic for the solvability of G that g (?) can be resolved into linear factors by the extraction of roots. If now H of order r is a subgroup of G, and if the applica- ALGEBRAICALLY SOLVABLE EQUATIONS. 289 tion of H to c, gives rise to the values Cj, ^3? • • • ^/j? then these are all contained among ^1, ^2, . . . ^,.. Consequently ft(c) = (?- ?,)(?-?,)... (?- = ,) is a divisor of g (c). Then h (1) is also resolvable algebraically into linear factors, i. e., H is a solvable group. We might also have proved this by showing that all the factors of composition of H occur among those of G. Theorem VIT. If the order of a group G is a power of a prime number p, the group is solvable. The group G is of the same type as a subgroup of the group which has the same degree n as G and for its order the highest power p-^ which is contained in n ! (cf § § 39 and 49). That the latter group is solvable follows from its construction (§ 89), all of its fac- tors of composition being equal to the prime number p. It follows then from Theorem VI that G is also solvable. Theorem VI JI. // the group G is of order r=p,''p/p^yp,^ ... where Pi,P3, Ps, Pi, . . . are different prime numbers such that Pi > Ih^Ps'^Pi^ . . ■ , P2> Pz^Pi^ • . • , P3 > i?/ . . . , then G is solvable.^ We make use of the theorem of § 128, and write r = Pi'^q, where then Pi> q. G contains at least one subgroup H of the order p^\ If we denote by kp^ -\- 1 the total number of subgroups of order pi" contained in G^ and by PiH the order of the maximal subgroup of G which is commutative with H, then r = p^°- i{kp^ + !)• Since r = p^'^q and q2)P3) • • • Conse- quently the same is true of f{x) — 0. All prime factors of the degree n of the solvable equation f{x) = are factors of composi- tion of the group G, and in fact each factor occurs in the series of composition as often as it occurs in n. To avoid a natural error, it must be noted that if in passing from G to G\ the polynomial f{x) resolves into rational factors one of which is f\{x), this factor does not necessarily belong to the group G),. It may belong to a family included in that of G\. The number of values of f'\{x) is therefore not necessarily equal to rirx. It may be a multiple of this quotient. And the product f')Sx) ■ f"\{x) ... of all the values of f\{x) is not necessarily equal to f{x), bat may be a power of this polynomial. We will now assume that n is not a power of a prime number p, so that n includes among its factors different prime numbers. Then different prime numbers also occur among the factors of composi- tion of the series for G, and consequently (§ 94, Corollary I) G has a principal series G, H,J,K,.., M, 1. Suppose that in one of the series of composition belonging to G other groups ALGEBRAICALLY SOLVABLE EQUATIONS. 291 3) H', H", . . . if W occur between H and J. Since n includes among its factors at least two difPerent prime numbers, f{x) must resolve into factors at least twice in the passage from a group of the series of composition to the following one. Since the number of the factors of f{x) is the same as the factor of composition, and since the latter is the same for all the intermediate groups 3), the two reductions of f{x) cannot both take place in the same transition from a group H of the prin- cipal series to the next following group J. It is to be particularly- noticed, that all the resolutions of f{x) cannot occur in the transition from the last group M to 1, that is, within the groups following M in the series of composition. At least one of the resolu- tions must have happened before M. Suppose, for example, that the first resolution occurs between H' and H". Then it follows from § 235 that H' is non-primitive in those elements which it connects transitively, and that H" is intransitive, the systems of intransitivity coinciding with the system of non- transitivity of H'. The same in- transitivity then occurs in all the following groups H'", . . . H^^\ and likewise in the next group J of the principal series, which by assump- tion is different from 1. * Suppose that J distributes the roots in the intransitive systems 1 5 2 5 • • • *^ * ) •^ 1 ? *^ 2 • • • * 5 • • • tJU J J *JU 2 5 • • • *^ 1 5 these systems being taken as small as possible. Then the expression f\{oo) = (x—x\) (x — x'^) . . . x — x\) becomes a rationally known factor of /(x), which does not contain any smaller rationally known factor. Since from the properties of the groups of the principal series G-'JG = J, all the values of f\{x) belong to the same group J. They are there- fore all rationally known with f\{x). Of the values of f\{x) we know already f\{x) = (x — x\) (x — x'2) . . . {x — x\), f\{x) = {x—x'\) {x—x",) ...(X— a^",), fx^^){x) = (x — x,^"^^) (ic— ^2^'")) . . . (a;— aj/*")). 292 THEORY OF SUBSTITUTIONS. If there were otker values, these must have roots in common with some /a^*^ (x). Then /x^*' (x) and consequently /a(.^') would resolve into rational factors. This being contrary to assumption, /\(a?) has 71/ only m = - values, and is therefore a root of an equation of degree m. If this equation is 4) ^(2/) = (2,-A)(2/_/'\)...(2,-/,C")) = 0, then f{x) is the result of elimination between 4) and 5) f\{x) = af-- = ait. + « follows necessarily a::^!, «i^0, and the substitution becomes iden- tical: \ — \z z\. If a substitution of G leaves one root Xx unchanged and if it con verts a?A + i into x^, then every x^, becomes a?(^_A.)(v-A) + A- For from A ^a / + « , !'- ^ a(A + 1) + a , follows a^// — / , a ^ /(/ — jj. — 1), and the substitution is of the form \z ( // — A)z-]r'^^ Q- ■-:>■ + ! • If a substitution of G leaves no root unchanged, and if it con- verts Xx into a?^, then every Xy is converted into x^j^^^^x- For only in this case is there no solution I of the congruence /^^ax-l-a, when aEE; 1. If / + 1 is to become ,a, then we must have /j-. = A + a. This gives a = // — A, and the substitution is | 2: z-\- ji. — >^^ | . These are precisely the same results which the earlier algebraic method furnished us. ALGEBBAICALLY SOLVABLE EQUATIONS. 297 Theorem XII. The general solvable equations of prime degree p are those of § 196. Their group is of order p(p — 1) and consists of the substitutions of the form s^\z az-\-a\ (a = l, 2, ...p — 1; a = 0, 1, . . .p — 1) (mod. p). Its factors of composition are all prime divisors of p — 1, each fac- tor ocurring as many times as it occurs in p — 1, and beside these p itself. § 246. We pass to the general solvable primitive equations of degree p^ As a starting point we have the arithmetic substitutions t=\Zi,Z2 2:, + «i,2?2 + «2| (mod.p), which form the last group M of the corresponding principal series. To arrive at the next preceding group, we must determine a substi- tution s which has the following properties. Its form is s^\zi,Z2 ai2ri-f-^i'2^2) «2^i + ^2 2^2 1 (mod.p), and the lowest power of s which occurs in M, and is therefore of the form t, must have a prime number as exponent. Since now all the powers of s are of the same form as s itself, the required power must be \zi, z.2 ^i , 2^2 1 = 1. That is, the order of the substitution s must be a prime number. From these and other similar considerations we arrive at the fol- lowing results, * the further demonstration of which we do not enter upon. Theorem XIII. The general solvable, primitive equations of degree p^ are of three different types. The first type is characterized by a group of order 2p^(p — 1)^, the substitutions of which are generated by the following : \z,,Z, Z, + a^, Z,-{-a,\ (ai,a2 =0,1, 2,.. .p — 1), I / 100 i\ (mod. p), \Zi,Z2 a.z.^a^z^] (ai, a2, = l, 2, 3, . . .p — 1), pl » ^2 ^2 ? 2^1 I . The groups belonging to the second type are of order 2 p^i^p^ — 1), and their substitutions are generated by the following : •C. Jordan: Liouville, Jour, de Miitb. (2) XIII, pp. 111-J35, 20 298 THEOBY OF SUBSTITUTIONS. \Zi,Z^ Zi-\-a,,z,-\-a,\ («,, a2=0,l, 2, ...p— 1), \zi^Z2 aZi-{- bez2<, bz^ + az» 1 -» 2 . . • S A is not ^0 (mod. p), then the x systems Tj), Tg), . . . T^) each of X -|- 1 congruences with the unknown quantities a,b, , . . c; a T,) {a, — 1) C/^) + &i C^^"^ + . . . + c, C.(^) + «i =0, T.) a. C/^) + &. C/") + . . . + (c.- 1) C.W + «<^0, have only one solution each, viz: Li) «! = 1, &i = 0, . . . Cj = 0; «i = 0, La) ' ttg = 0, 62 = Ij . • . C2 = 0; flg = Oj jL^) CTk^O, 6^ = 0, . ..c« = 1; aK = 0, and these solutions furnish together the identical substitution 1. We designate now a system of x-\-l roots of an equation for which E^O (mod. p) as a system of conjugate roots. • We have then Theorem XV. If a substitution of a primitive solvable group of degree p" leaves unchanged x -f- 1 roots which do not form a conjugate system, the substitution reduces to identity. ■» 1 * 2 • ft ftt fit ■» 1 ^ 2 • ftt :(") c/*^) ; . . c('>. ALOEBHAICALLY SOLVABLE EQUATIONS. 301 If therefore we adjoin ^ + 1 such roots to the equation, the group G reduces to those substitutions which leave y.-\-\ roots unchanged, i. e., to the identical substitution. The equation is then solved. Theorem XVI. All the roots of a solvable primitive equa- tion of degree p" can be rationally expressed in terms of any y.-\-l among them, provided these do not form a conjugate system. If we choose the notation so that one of the ^ + 1 roots is ^0 ,0, ... 0, the determinant becomes ±E. If the roots are not to form a conjugate system, then ^ = (mod. p). The number r of systems of roots which satisfy this con- dition is determined in § 146. We found r =: (p" — 1) (p^—p) (p^—p') . . . (p^—p"'^). Theorem XVII. For every root ^«i,^2,...^^ ^^ (^(^'^ deter- mine (p'^ — l) (p'^—p) . . . jp'—p"-') 1, 2, ... X systems of x roots each such that these x-\-l roots do not form a con- jugate system, so that all the other roots can be rationally expressed in terms of them. The system composed of the z -f- 1 roots •^'O , , , . . . > "^l , , . . . > "^O , 1 , , . . ) ... '^0,0,0,...! is appropriate for the expression of all the roots. These results throw a new light on our earlier investigations in regard to triad equations, in particular on the solution of the Hes- sian equation of the ninth degree (cf § § 203-6). It is plain that we can construct in the same way quadruple equations of degree p^, and so on. ^DOK MU5T SB CHA-RGEP OOT WITH CASE >V' 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, ot on the date to which renewed. Renewed books are subject to immediate recall. Ri£<.0 LD AUG U '65 -12 M PE.B 5^96854 N3\r2ni973 4 7 m fiQ\2^''^'^^*' LD 21A-60m-4,'64 (E4555sl0)476B General Library University of California Berkeley P.?of?ir?5 QA 17 THE UNIVERSITY OF CAUFORNIA UBRARY . «y_*. t