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 ENGINEERING & 
 
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 ON THE IMPRIMITIVE SUBSTITUTION 
 GROUPS OF DEGREE PIFTEEII AM) THE 
 PRIMITIVE SUBSTITUTION GROUPS OP 
 DEGREE EIGHTEEN 
 
 by 
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 ' ON THE IMPRIMITIVE SUBSTITUTION GROUPS 
 OF DEGREE FIFTEEN AND THE PRIMITIVE 
 SUBSTITUTION GROUPS OF DEGREE 
 
 EIGHTEEN 
 
 A DISSERTATION 
 
 PRESENTED TO THE FACULTY OF.BRYN MAWR COLLEGE FOR THE DEGREE OF 
 
 DOCTOR OF PHILOSOPHY 
 
 Bv EMILIE NORTON MARTIN 
 
 iqoi 
 Z^ £orb (§attimott (prcee 
 
 The Friedenwald Company 
 baltimore, mu., u. s. a.
 
 ON THE IMPRIMITIVE SUBSTITUTION GROUPS 
 
 OF DEGREE FIFTEEN AND THE PRIMITIVE 
 
 SUBSTITUTION GROUPS OF DEGREE 
 
 EIGHTEEN 
 
 A DISSERTATION 
 
 PRESENTED TO THE FACULTY OF BRYN MAWR COLLEGE FOR THE DEGREE OF 
 
 DOCTOR OF PHILOSOPHY 
 
 By EMILIE NORTON MARTIN 
 
 IQOI 
 
 The Friedenwald Company 
 baltimore, md., u. s. a.
 
 Engineering & 
 Mathematical 
 
 Sciences 
 
 Library 
 
 MS4><5-' 
 
 Oil the Imprimitive Substitution Groups of Degree 
 Fifteen and the Primitive Substitution 
 Groups of Degree Eighteen. 
 
 By Emilie Norton Martin. 
 
 The following work is, with some slight modifications, the same as that of 
 which an abstract was presented at the summer meeting of the American Mathe- 
 matical Society in 1899. With regard to the imprimitive groups of degree fifteen, 
 which form the subject matter of the first part of this paper, it should be stated 
 that I have added two new groups to the list as originally presented, namely, the 
 groups with five systems of imprimitivity simply isomorphic to the alternating 
 and symmetric groups of degree 5, and that Dr. Kuhn reported at the February 
 meeting of the Society, 1900, that he had carried the investigation further, adding 
 28 to the 70 groups that I succeeded in finding. 
 
 In the second part of this paper the determination of the primitive groups 
 of degree 18 depends to a great extent upon the lists of transitive groups 
 of lower degrees already determined. Any new discovery of groups of degree 
 less than 18 would necessitate an examination of such groups to determine 
 whether they can be combined with others in such a way as to generate a 
 primitive group of degree 18. This list, therefore, cannot claim to be abso- 
 lutely complete, since omissions are always possible. 
 
 Imprimitive Svhstitution Groups of Degree Fifteen. 
 
 Every imprimitive group contains a self-conjugate intransitive subgroup 
 consisting of all the operations that interchange the elements of the systems of 
 imprimitivity among themselves without interchanging the systems. Therefore, 
 the problem of the determination of all imprimitive groups of degree 15 falls 
 into two parts: 1st, the determination of all intransitive groups of degree 15 
 
 21643.3
 
 2 Martin : On the Imprimitlve Substitution Groups of Degree 
 
 capable of becoming the self-conjugate subgroups of such imprimitlve groups ; 
 2d, the determination of substitutions that will interchange the systems of 
 imprimitivity and at the same time fulfill other conditions depending upon the 
 particular group under discussion. The intransitive self-conjugate subgroup is 
 called for shortness the head, the remaining substitutions of the imprimitive group 
 are designated as the tail, a terminology that has been adopted by Dr. G. A. 
 Miller in his papers on imprimitive groups. 
 
 The elements of an imprimitive group of degree 15 may fall into three sys- 
 tems of five elements each, or into five systems of three each. For the first of 
 these cases, certain theorems given by Dr. G, A. Miller (Quar. Jour. Math., vol. 
 XXVIII, 1896) are useful. With a slight modification in notation in order to 
 adapt them to the notation of this paper, they are as follows, where G^ repre- 
 sents a group in the elements with index 1, while G^ and G^ represent precisely 
 the same group in the elements with indices 2 and 3. 
 
 Theorem I. — All the substitutions that can be used to construct tails are 
 
 {a\al .... a\) all {ajal . . . . a^) all {alal .... al) all 
 
 \ (a\alal . alajal aWnCil) . («i«i • «2«2 «n«n) f 
 
 — {a\al . . . . al) all (afal .... a^) all {alal .... a') all. 
 
 Theorem II. — If G^ = ((A^l • • • • «n) all, there are three imprimitive groups 
 with the common head {G^G~G^) pos, and two with the common head G^ pos GF pos 
 G^ pos -+- G^ neg G^ neg G^ neg. 
 
 Theorem III. — If G^ =. (ala] .... a\) pos, there are three imprimitive groups 
 luith the common head G^G~G^, and three with the common head {G^6r^G^)i^ j^ i . 
 
 Theorem IV. — If the head w G^G^G^, there is only one group which corresponds 
 to (abc) eye. 
 
 The possible heads for these groups are got either by the direct multiplica- 
 tion of transitive groups of degree 5 in the three systems of elements, or by the 
 establishment of isomorphic relations between such groups. 
 
 The transitive groups of degree 5 are five in number, and fall naturally into 
 two categories, the first containing the symmetric group and its self-conjugate 
 subgroup, the alternating group, the second containing the metacyclic group, 
 together with its two self-conjugate transitive subgroups. These five groups are 
 represented respectively by 
 
 (aia^asa^a^) all, {aia^a^a^a^) pos, ((ha^a^a^a^).;^, {a^a^asa^a^^o, {aia^asa^a^)^ .
 
 Fifteen and the Primitive Substitution Groups of Degree Eighteen. 3 
 
 P>om the first two groups come the following heads : 
 
 I. {a\alalalal) all (alalaldlal) all {alalalalal) all = A7280oo- 
 11. \(a\alalalal) all (alalalalal) all (afa^c4«4«l) all| pos = Esmm- 
 
 II I . (alalalalal) pos (a?a|a|a|a^) pos (af<i|a^a^a|) pos 
 
 + (a}c4r4«l«^5)neg(aja|«^a^a^)neg(afa|a|a|«i) neg = ^/432ooo- 
 
 IV. {a\cdala\al) pos (a]a^a|a|al) pos {alcdaldldl) pos = //gieooo- 
 V. {a\alalalal . alalalalal . alalajalal) all = Hi2o- 
 
 VI. (alalalalal. dfdlalalal. ajalalalaf) -pos = H^q. 
 
 From the three remaining groups come the heads : 
 
 VII. (alalalalaiy^o («i«If'l«4^5)2o («i«i«3«'4«5)3o = ^sooo- 
 
 VIII. {{alalalalaDzo («?«!' 3^' 4« 5)30 («i«2«!«4«5)3o f pos = i?4ooo- 
 
 IX. J (alalalalal) ^f, («^''>«|al«I)2o («i«3«Ift!(''5)2o } 10, 10, 10 = ^sooo- 
 X. (alalalalaWo («i«a«3«4«5)io («i«l«3(«4«5)io = ^looo- 
 
 XI. -I (alala]alal)oo (alalalalaDsa (oi^i^l^^^D^ol 5, 5, 5 = ^soo- 
 
 XII. ] (alalqlalaDiQ (aia|«^a^a|)io (alahlalafji^i \ 5, 5, 5 = ^250- 
 
 XIII. (alalalalal) eye (alafalalal) eye (ajalaga^al) eye = H^z^. 
 
 XIV. (aja2<^3c4«5 • (^i(«|«3«l«5 . ai«|«|«|«|)oo = -Sao- 
 XV. (alalalalal . a^ala^alal . «i«3<^<3'/4«5)io = -^lo- 
 
 XVI. (alalalalal . afalalapl . alalalaldf) eye. = H^. 
 
 The groups corresponding to these heads may be isomorphic either to (a^d\i^) 
 eye or to (a^a^a^) all. To generate a group isomorphic to (a^a^a^) eye a substitu- 
 tion with the following properties must be added to the head : it must have its 
 cube in the head, it must interchange all three systems, and it must transform 
 the head into itself. Calling the group so found G, the groups isomorphic to 
 (a^a^a^) all may be found by combining with G any substitution that has its 
 square in the head, that interchanges two of the systems leaving the third 
 unaffected, and that transforms H into itself, and G into itself. 
 
 As all the heads given above are symmetric in the three sets of elements, each 
 head furnishes two groups by means of the symmetrically formed substitutions 
 
 s = a{aiai . alar,al . a^ayil . a\aiai . atfliai , t = a\ai . a\a% . a^a^ . aial . a'^a^ . 
 
 The letters s and t are used throughout this section of the paper to denote these 
 particular substitutions, other substitutions fulfilling the same conditions being 
 denoted by the same letters with suflQxes.
 
 4 Martin : On the Imprimitive Substitution Groups of Degree 
 
 According to Theorem I, anys„ or ^„ must be the product of some substitution, 
 (7„, of the most general head, ^17289001 by s or t. Therefore a^ must be a substitution 
 of a subgroup of ^j^ogooo that contains the special 5" under consideration as a self- 
 conjugate subgroup. 
 
 We may now proceed to the determination of the groups to be derived from 
 the various heads taken in order. 
 
 I. ^1728000 gives us, according to Theorem I, only the two groups, 
 
 j^i728ooo, s, i of order 5184000i, 
 and J5i728ooo, «, t\ of order 10368000. 
 
 II. j5864ooo gives us, in accordance with Theorem II, three distinct groups. 
 Of these, two are the groups, 
 
 j^eeiooo, A of order 2592000i, 
 ) 5864000. ^^ i\ of order 51840002, 
 
 A a that transforms the head into itself without belonging in the head is 
 a =: a\a\ . This cannot be combined with s , as {psY is an odd substitution ; it may, 
 however, be combined with t . The remaining group is therefore 
 
 •| i?864ooo. '5, «i«2 . ^[ of order 5I840OO3 . 
 
 Of these two groups of order 5184000, the first contains both odd and even 
 substitutions, the second only even. 
 
 III. jS432ooo gives, by Theorem II, the two groups 
 
 ]^4320oo>«t of order 1296000i, 
 1^432000, s,t\ of order 25920000. 
 
 IV. 5216000 gives us, by Theorem III, three distinct groups, a = a\a\ trans- 
 forms the head into itself, but when combined with s it gives an odd substitu- 
 tion whose cube cannot be found in the head. The substitution at furnishes us 
 however, with a new t^. The three groups are, therefore, 
 
 l^oieooo, s\ of order 648000, 
 \Rmm^ s,t\ of order 1296OOO2, 
 1^216000. «. «i«2-^l of order 1296OOO3. 
 
 The two groups of order 1296000 are distinct, since the one contains both odd 
 and even substitutions, the other only even.
 
 Fifteen and the Primitive /Substitution Groups of Degree Eighteen. 5 
 
 V. ff^oQ is not contained self-conjugately in any larger subgroup of 5"i728ooo> 
 therefore only the two following groups can be formed from it : 
 
 {^120, s\ of order 360i, 
 \Hi2o,s, t\ of order 720. 
 
 VI. ^eo gives, in accordance with Theorem TIT, three groups : 
 
 J^eo, s] of order 180, 
 
 \ITqo, s, t\ of order 3602, 
 
 \ H^Q, s, a\al . «!«! . a^al . ^[ of order BGOg. 
 
 The last of these groups consists entirely of even substitutions. 
 
 The remaining heads are all composed of substitutions of the type 
 
 VliW^^V^a^U^vl.ut, (1) 
 
 where v,,i^=- aldlala], u^i= a\alala\al, while v^i, Ua2, v^^, w^s denote the same sub- 
 stitutions written in elements with the indices 2 and 3 respectively. The substi- 
 tutions Va'y Ua^ generate the metacyclic group in the five elements with index 1, 
 these substitutions being subject to the conditions 
 
 The most general s^ is given by 
 
 From this we find 
 
 ,9f = <+^ + 'n,^.< + * + ''7^S.<3+'' + '^iC, (3) 
 
 where X=2'{ 2^ -f Z^i) -f Cj , ^ 
 
 v= 2'^(2''ci + ai) -f ^. J 
 
 Transformation of the general substitution (1) by s^ gives us 
 
 Sa^K^ u-a^ v^^, iC vl^ ut s^ = vl ulx vl, ul. v^^.u''^,, ( 5 ) 
 
 where ;^ = «! -f 2"a' — Ta-^ , -j 
 
 l, = h, + 2'^'-2%,^ (6) 
 
 1. = Ci + 2 Y' — 2^Ci , ) 
 
 The general substitution of the group G= \TI, s^\ is 
 
 T=slv:.u^:.vl.u^vl.itt.. (7)
 
 6 Martin : On the Imprimitive Substitution Groups of Degree 
 
 The most general t^ is given by 
 
 t^ = vluOviluilv%u'^,t. (8) 
 
 Upon squaring this substitution, we get 
 
 tl = v'',^ + "' M^ vll + '^ K. v'J^ K,, (9) 
 
 where 2,= ^^'Gs + bs,^ 
 
 fi=2''^b,+a,l (10) 
 
 On transforming the general substitution T by the general t^, we have, after a 
 straight-forward calculation, the following expression for the case x= I: 
 
 tj'T^^,t^ = s^^v-.''^ + ''' + ^+''K.vl\-'^-^'' + UCv-.'^ + '^ + '' + ''ii':,^, (11) 
 
 where ^= — 2''^-"» + ^+V3 + 2' + '^ai + 2*^/3' + h,, ^ 
 
 r = — 2'^'-*' + ^ + ''Z>3 + 2^+'^^&i + 2'y' + Cg. j 
 
 We may now return to the consideration of special groups. 
 VII. i/gooo gives only the two groups formed with s and t, as any a that 
 might be used is already contained in this head. The groups are, therefore, 
 
 {jSgooo, s\ of order 24000i, 
 ^^eooo, s, t\ of order 48000. 
 
 YIII. iTiooo ^^s the general substitution (1) subject to the condition a -\- (3 
 + y= (mod 2). From (3), it is evident that s„ is subject to the condition a-{-b 
 -}- c = (mod 2). Therefore, s^ is already in the group generated by ^4000 and 
 by s, and there is only one group isomorphic to (a^a^a^) eye. We find by (9) 
 that every t^ has its square in the head, and by (11), that every f^ transforms 
 the head into itself, therefore, we may take as a new t^ the simplest substitution 
 for which ^3 -\- b^ + Co= 1 (mod 2), viz. : 
 
 Vat = a\a{ . cqa^a^a^rr^aia^iao. 
 The three groups with this head are, therefore, 
 
 ]^40oo> -^f of order I2OOO1, 
 jFjooo. 5. ^f of order 24OOO2, 
 J-&4000' *' ^a^|- of order 24OOO3.
 
 Fifteen and the Primitive Substitution Groups of Degree Eighteen. 7 
 
 Of these groups the first and third consist of even substitutions, the second of 
 even and odd. 
 
 IX. ^2000 has the general substitution subject to the condition a = ^ = y (mod 
 2). From (3) and (5), it is phxin that every s^ can be used to generate a group of the 
 kind required. The only possible form for the cofactor of s, if it is not to give 
 the group generated by s and the head, is Vaivl-iV^i, where a, b, c do not fulfill the 
 condition a = b = c (mod 2). The simplest form for such a cofactor, and a form 
 to which all others reduce, is found by making two of the exponents vanish and 
 the third become equal to 1, e. g., Si== VgiS == a\(ila\ .alalalalalalaldlalalalal. 
 Now, 81=: Si .VaxVa^Vas aud sl =z sf . vliV^ivls ] we may, therefore, take s\ as the s. 
 in the place of §1 and still have the same group. But si^= (vl-^vlz)'^ s {vliV%)^ 
 therefore the group we have now found is merely the transformed of the group 
 generated by s with respect to the substitution v^vlz. Consequently, there is 
 but one group corresponding to the cyclic group of degree three. 
 
 If, in addition to the group given by t, we have a group given by t^, then 
 according to the relations derived from (11), a.^^b.y^c^ (mod 2), i. e., the pos- 
 sible values of ^^ are already present in the group generated with the help of t. 
 The two imprimitive groups with this head are, therefore, the groups 
 
 ]^2ooo. s\ of order 6000. 
 j/^Mooo, s,t\ of order 1200O2. 
 
 In this, and all following work, the terms u in the cofactors of s and t are 
 taken as unity, unless the contrary is expressly stated. 
 
 X. ^jooo has its general substitution subject to the condition a = l3 = y = 
 (mod 2). By Theorem IV, this head gives only one group isomorphic to 
 {abc) eye. If, in addition to the substitution t, there is a substitution fp, the 
 relations satisfied by the exponents of the v^s in (11) reduces to a^^ib-i — c^ 
 (mod 2). We have, therefore, two distinct groups according as ag is even or odd. 
 The three groups with this head are 
 
 j ^1000 ' * ) of oi"der SOOOj , 
 j^iooo. s, t] of order 6OOO2, 
 ]^iooo> '5, Va^Va^VaJ] of Order 6OOO3. 
 
 XI. ^500 subjects the general substitution to the conditions a=^ ^ =■ y, where 
 a =: 0, 1, 2, 3. Since every substitution s^ satisfies the necessary conditions, the 
 following independent types of s„ must be examined: Va^s, vl.s, vliS, Va^vl^s. The
 
 8 Martin : On the Iinprimitivc Substitution Groups of Degree 
 
 fourth power of these substitutions is in every case the transformed of s with 
 
 respect to some combination of the ^-'s; therefore, they give nothing new. The 
 
 possible forms for t^ are derived from the equation easily deducible from (11); 
 
 — Co + b^^Oi — c.^ = — ^2 + ^2 (mod 4), which, taken in conjunction with the 
 
 limited range of values of ao' ^2- ^z^ gives org =: Z>2 = c^. That is, every possible t^ 
 
 is alread}^ included in the group generated by t. This head gives accordingly 
 
 only the two groups, 
 
 ]^5oo, s\ of order 1500i, 
 
 ^iJgoo, s,t\ of order SOOOg. 
 
 XII. i?25o subjects the general substitution (1) to the conditions a == /? = }/=0 
 (mod 2). To determine an s^, we have from (3) the condition a + 6 + c = 
 (mod 2). An examination of the four apparently distinct types of s„, v\s, v^v.^s, 
 vfv^s, vlvls, shows that just as in the last set of groups, these each give a group 
 that can be derived from the group generated by s by means of an easy trans- 
 formation. 
 
 The possible forms f^ must fulfill the conditions, deducible from (1 1), — « g + 
 b2 = — 60 + 02 = — C2-{- a2 = (mod 2) and also — 02 + 62 = ^2 — ^i (mod 4). 
 These reduce to the simple condition a^ = bo = Co, which furnishes the substi- 
 tution t^ = Va^ Va7 v^z t. Thls hcad gives therefore the three groups, 
 
 ]^2oo» ^1 of order 750i, 
 {Eo^Q, s, t\ of order 150O2, 
 ]J525o,s, Va^Va-iVai t\ of ordcr ISOOg. 
 The second group alone contains odd substitutions. 
 
 XIII. ^125 gives in accordance with Theorem IV only one group in which 
 the systems are interchanged cyclically. The general substitution of this head is 
 subject to the condition a = /? = ^ = 0. Applying this condition to (9) and (11) 
 we find (^2 ^= ^2 =^ <^2 ' while a^ lies under the further restriction of being even. 
 Therefore we have in addition to t the substitution, 
 
 J "> 2 5 J 1" 12 1'' A '> 12 ^3 S3 
 
 i^ =. Va^v^^K^ t = a\ai. aias.agaj.ojag. aiai. aoal.alal. 
 The three groups given by this head are, 
 
 1^125, s\ of order 375, 
 
 \Ei2bi ^) i\ of order 75O2, 
 
 {^i25» s,VaiVa2Va3t\ of ordcr 750g. 
 
 XIV. ^20 imposes upon the exponents of the general term the conditions
 
 Fifteen and the Primitive Suhstitation Groups of Degree Eighteen. 9 
 
 a = 13 = '/, a'=-P'=.y'. Making use of this in (5) and (6) we find 2"a' = 
 2V = 2'^ a' (mod 5) , which gives at once az=.h=ic. Using this latter equaUty in 
 the equations that are deduced from (3) and (4) we find a^ = 5^ = Ci with the 
 single exception of the case a = 0, where the equations become indeterminate, 
 being satisfied by every value of a^, Sj, c^. An examination of all of the appa- 
 rently independent sets of value for %, 6^, c^ shows that in every case the group 
 is transformable into that generated by .9 alone. In order to determine all sub- 
 stitutions t^ we use the equation, derived from (11), — a^_ + So^ag — ^3 = '^2 — ^3 
 (mod. 4), from which follows at once t/g = b^ = c.,. From (12), by making use of 
 the special case a = (3 =^'y =^ 0, can be derived the relations — a^-\- 1)^ = — Cg 
 + «3 = — ^ + C3 (mod. 5); i, e, ag = ^3 = Cg. The only groups with this head are 
 therefore the two groups, 
 
 ]5oo, s\ of order 6O1, 
 
 1 530, s, t\ of order 120. 
 
 XV. HiQ has the general term (1) subject to the conditions a = ;5 = )/ = 
 (mod 2), a' = /?' = y'. By precisely the same line of argument as that laid down 
 in the preceding case we arrive at the conclusion a=ib = c, a^ = b^z=c^, ^3 = 
 S3 = C2, ag =: &g =: Cg. In this work, too, the indeterminate values of a^, b^, c^ 
 require a careful examination that leads to no new group. From this head come, 
 therefore, the three groups, 
 
 \HiQ, s\ of order 30i, 
 
 \HiQ, s, t\ of order GO,, 
 
 jiTio, ^. v„^Va, Va^ t\ of ordcr 6O3. 
 
 Of these three groups the second alone involves odd substitutions. 
 
 XYI. H^ imposes upon the general term the conditions a=: (3 = y =: 0, a' = 
 /?' = y, By arguments similar to those used in the last two cases, with the 
 further addition of the condition imposed by (3), a + ^ + c = (mod 4), we find 
 a = 6 = c = 0, «! = &i = Cj. In the determination of t^ we see at once from (9) 
 that C3 must be even, while from (ll) we find a3=:Z>3=C3, and from (12) 
 a3=bs = Cs. 
 
 The groups given by this head are as follows: 
 
 j^g, s\ of order 15, 
 
 jiTg, s, t\ of order 3O2, 
 
 ] Ilr,, s, vl^ v% vis t\ of order 3O3.
 
 10 Martin : On the Lmprimitive Siihstitution Groups of Degree 
 
 Passing now to the case of five systems of three elements each, there are 
 seven heads considered in this paper, six involving all the systems symmetrically, 
 the remaining head being unity. 
 
 I. {a\a\a\) all (ala|a|) all {a\alaf) all {a\a\af) all {a\ala\) all = Hm^, 
 
 II. \n^m\ POS= ^3888, 
 ill. |i27776j 3,3,3,3,3 ■— -"486 > 
 
 IV. {a\a\a\) pos {a\a^xv^^ pos {a\alaf) pos {a\a\a\) pos (aicisal) pos = ^^243 . 
 
 V. {a\a\a\ . a\a\a\ . alalaj . a\ala\ . a\ala\) all = H^ , 
 
 VI. {a\a\a\ . alaldi . alalaj . a{alal . alalal) eye = II3 , 
 
 VII. Unity. 
 
 Denoting the system with index r by A,, it is evident that these systems 
 must be interchanged according to the five groups (^.1^2^.3^14^.5) eye, (^.1^.3^.3^.4^.5)10, 
 (^i^^3^4^)2o, (A^A^A^A^A^) pos, (A^A^A^A^A^) all. 
 
 The order of procedure in each case is as follows : 
 
 1. If the group is to correspond to (^i^2^3-^4^5) ^y^^ -^ substitution s must 
 be found that will interchange the systems cyclically, transform the head into 
 itself, and have its fifth power in the head. The imprimitive group so generated 
 may be called G. 
 
 2. If the group is to correspond to (J.i^2A^4^5)io. it must contain (tj as a 
 self-conjugate subgroup. In addition, therefore, to the s of case 1, a substitution 
 t must be found that will interchange four of the systems in two pairs, as 
 AzA^ . A^Ai, while leaving the remaining system unaltered, and that will, at the 
 same time, transform the head into itself and G into itself. This substitution t 
 must also have its square in the head. This imprimitive group shall be 
 called G^. 
 
 3. If the group is to correspond to (A^ A. A3 A^ A^\o , it must contain both Gi 
 and 6r3 as self-conjugate subgroups. In addition, therefore, to the s of case 1, a 
 substitution u must be found interchanging four of the systems cyclically, accord- 
 ing to A2A3A^Ai for instance, transforming Gi and G2 into themselves and hav- 
 ing its fourth power in the head. 
 
 4. If the group is to correspond to (A^AoA^A^A^) pos, two substitutions, 
 V and v', must be found corresponding to ^, A Jg and Ji J^iA • These sub-
 
 Fifteen and the Primitive Suhstitiition Groups of Degree Eighteen. 11 
 
 stitutioiis must, therefore, each interchange three systems, leaving two mialtered, 
 they must have their cubes in the head, and must transform the head into itself. 
 This group may be called &. 
 
 5. If the group is to correspond to [A^A-.A^A^Ar^ all, two substitutions, w 
 and w', must be found corresponding to ^^^-2^-3^4 and A^A^. G' is to be con- 
 tained in this new group as a self-conjugate subgroup, therefore w and to' must 
 transform the head into itself and G' into itself. The fourth power of w and the 
 square of to' must both be contained in the head. 
 
 I. Hti^q is the largest possible intransitive group with the given systems of 
 intransitivity, and, consequently, only one group with this head corresponds to 
 each of the transitive groups of degree 5. For each of these groups a substitu- 
 tion or pair of substitutions can be found fulfilling all required conditions and 
 involving the elements of the systems symmetrically. A second set could be 
 found only by multiplying this first step by some substitution belonging to the 
 largest group that contains the head self-conjugately without interchanging any 
 of the systems. But this group is the head itself. The required groups are, 
 therefore, the following : 
 
 {57776, 6-} of order 38880, 
 1^7776, s , t\ of order 77760, 
 ] 5777s, s, u\ of order 155520, 
 J57776, v,v'\ of order 466560i, 
 J//7776, w, 10' \ of order 933120, 
 
 where s = alalala\al . alajalalal . alalalalal, 
 
 . OS Ql 25 3 4 an 3 4 
 
 t = a-{a\ . ayq . a^^^l . cdat . a^cq . a^cq , 
 
 o Q r A a;-!")! 2 3 54 
 
 u ^=. aiaiciial . a2ar,alal . aicqcqal , 
 
 193 l^q I"!? 
 
 V = a{a^ai . aUir/i^ . a^a^ct^ , 
 v' =■ a\alai . alcdal . ala'^al , 
 w = alalala\ . alalalal . alalalal , 
 2o' =:a\al.alal.alal. 
 
 These letters shall be kept throughout this section of the paper to denote these 
 symmetrically formed substitutions, other substitutions with corresponding prop- 
 erties being denoted by the same letters with suffixes.
 
 12 Martin : On the Imprimitive Suhstitution Groups of Degree 
 
 11. //gggg gives only one group isomorphic to {A^A.A^A^A-^ gjq, viz., the 
 group generated by s. Any new s^ must have as cofactor an odd substitution 
 belonging to H^^-^, but the fifth power of such a substitution is not contained in 
 the head. There are, however, two groups isomorphic to (^1^.2^3^4^5)10, since 
 both t and t„ = a\al . t fulfill the necessary conditions. The former generates a 
 group (rggggo containing only even substitutions, the latter generates a group 
 ^38880 containing both odd and even substitutions. There are likewise two 
 groups isomorphic to (J.1J.2 -43^.4^.5)20, one generated by u, the other by a]al.u. 
 The first of these groups contains odd substitutions, the second only even, (xggggo, 
 is contained self-conjugately in both. 
 
 Only one group G' can be found for this head, as no new v^ or vl fulfills the 
 necessary conditions. Such a substitution would necessarily be of the form gv 
 or Gv', where g would belong to the group -07776- If cr were even, the group so 
 generated would be a repetition of the group generated by v and v'. If cr were 
 odd, the cubes of at-, gv' would not be contained in the head. 
 
 Two groups can be found isomorphic to (Ai^A^A^A^A^) all, the substitutions 
 ic and lu' generating one group, the substitutions alal.iv , o}ai. ic' generating the 
 other. This latter group contains only even substitutions. 
 
 From this head we have, therefore, derived eight groups : 
 
 j^gggg, s\ of order 19440, 
 
 {^jggg, s. t\ of order 38880., 
 
 ^Sgggg, s, a\al.t\ of order 3888O3, 
 
 {Hgggg.s, u\ of order 7666O2, 
 
 ^^gggg.s, a\al.u\ of order 7666O3, 
 
 ■J-S3888. ^. "y't of order 233280, 
 
 jSgggg, w, iv'\ of order 4665602, 
 
 ■IjBgggs, a\al.w, a\al.w'\ of order 46656O3. 
 
 III. 5486 furnishes us with only one group isomorphic to (^1^.2 ^3 ^4^5) eye, 
 for an examination of the groups given by all possible types of substitutions s^ 
 shows that each of these groups is merely the group generated by the help of 
 s and transformed with respect to some easily discovered substitution. More- 
 over, there is but one group isomorphic to (J.i ^12^3^4^5)10' viz., that generated 
 with the help of t. Any cofactor of t must be of one of the types a\a}2, a\al . a^al , 
 a\al.alal.a\al, ajal . alaj . a{al . a\al , but any t^ got by means of these, transforms
 
 Fifteen and the Primitive Substitution Groups of Degree Eighteen. 13 
 
 s into 5^ (a substitution not in the head). Precisely the same reasoning shows 
 that there is only the one group isomorphic to (^A-^A.;^A^A^A^).,(^. 
 
 In addition to the group isomorphic to (^1^2^.3 ^4^.5) pos generated by 
 means of the substitutions v and v', we must examine groups generated with the 
 help oi v^ and v'^, substitutions which contain as cofactors of v, v' respectively 
 the products of transposition, one transposition from each system. A number 
 of these may be rejected at once, but we are left with the possible forms : 
 ■yj = a\a\ . a\a\ . v = alala^ . alalcil . alalal , 
 V2 == ala] . a^ai . alal . v = <i\alal . alalal . a\a\ . afa| , 
 v[ == a\a\ . a\a\ . v' = alajal . cdalal . alalal , 
 vi = a\al . a^dl . afal . v' = a\aial . alalal . a\dl .a\a\. 
 
 But, since vi, -yg ^-i'^ transformable into v, and v[, v'^ are transformable into v', it is 
 impossible to generate any group by means of any combination of these four 
 substitutions excepting a group that can be transformed into the one generated 
 by means of v and v' . 
 
 A similar examination of all groups isomorphic to (^1^0^-3-44^5) all, shows 
 that, in addition to the group generated with the help of id and w\ there is one 
 other group generated by means of w^ = alal . lo and iv'. 
 
 From this head are therefore formed the six following groups : 
 
 ] 1/486, «1 of order 2430i , 
 {ffise, s,t\ of order 4860i, 
 {^^486) s , u\ of order 9720, 
 )/7,8c,v, v'\ of order 29160i, 
 jZ/jse, w, iv'\ of order 58320i, 
 ) £?436 , alal .w,io'\ of order 58320o. 
 
 IV. 7/243 gives one group isomorphic to {A^AoA^A^A^) eye, by means of s. 
 The only other permissible forms of 5„ are of the type 
 
 §1 = ala] . alal .s, 53 = a}aj . a^al . alal . a\al . s . 
 
 But s^ and s^ are each the transformed of s with respect to some substitution 
 that transforms the head into itself; therefore, there is only the one group of this 
 type. On the other hand, there are two groups isomorphic to (J-j Jo ^3^4^5)10, 
 since both t and t^ = a\al . ajal . alal . alal . alal . t fulfill all necessary conditions and 
 generate, one a group of even substitutions, the other a group containing odd
 
 14 Martin : On the Imprimitive Substitution Groups of Degree 
 
 substitutions. There are also two groups isomorphic to {A-^A^A^A^A^oq, one 
 containing both odd and even substitutions, the other only even. These are 
 generated respectively by means of u and of 11^:= a\a\.aial.a\al.a\cd.a\a.\.u, 
 and each contains as a self-conjugate subgroup the group isomorphic to 
 (^1^.0^3^4^5)10 that consists entirely of even substitutions. 
 
 Only one group can be found isomorphic to {A-^A^A^A^A-^ pos, and this is 
 the one formed by the help of v and v'. An examination of the various substi- 
 tutions v^ and v'^ corresponding to various types of cofactor of i? and v' shows 
 that all groups formed by means of these substitutions are transformable into 
 the one group. 
 
 On the other hand, we have two distinct groups corresponding to 
 {A^AzA^A^A^ all, the one consisting of both odd and even substitutions and 
 generated by the aid of ic and u:', the other consisting entirely of even substitu- 
 tions and generated by the aid of a\al . ic and a\al . iv'. 
 
 From this head we have, therefore, the eight following groups : 
 
 {Has, s\ of order 1215, 
 
 1^243, s, t\ of order 2430o, 
 
 ] ^043 , s, a\al . rqal . alal . a\cd . a\c4 .t\ of order 243O3 , 
 
 Jj^243> *> ^^( of order 486O2, 
 
 ] 5o43, s, alal . a\al . alal . alai . a^al .u\ of order 486O3, 
 
 \R2i3,v, v'\ of order 14580, 
 
 \Eziz, w, w'\ of order 291600, 
 
 "I ^043 , alal . ic, alal . w' j of order 29 1 6O3 . 
 
 V. R^ furnishes one group corresponding to each transitive group of degree 
 5. These groups are generated respectively by the substitutions s, t, ic, v, v', ic, to', 
 and can readily be seen to be indentical with those of orders 30o, 6O2, 120, 360o, 
 720 included among the groups with three systems of imprimitivity. An inter- 
 change of suffixes and indices in the one set of groups gives the generating sub- 
 stitution of the other set of groups. 
 
 VI. j^g furnishes groups corresponding to the transitive groups of degree 5 
 by means of the substitutions s, t, u, v, v', ?r, iv'. As in the last case, however, 
 these correspond to the groups of orders 15, 30i, 6O1, 180, 360i included in the 
 groups with three systems of imprimitivity. By the use of the cofactor a = a]al .
 
 Fifteen and the Primitive Suhstitution Groups of Degree Eiglitcen. 15 
 
 alal . a\al . a\al . a\al three more groups can be found generated respectively by the 
 help of t]^ = at, ill = <7^*. ^^1 = cr^^) ^ = cr^^'- These groups, however, are seen 
 to be identical with those of orders SOg, 60g, and SGOg included in the groups 
 with three systems of imprimitivity. This head gives, therefore, no group essen- 
 tially new. 
 
 VII. In the discussion of the head unity a useful theorem is the following 
 given by Frobenius (Crelle t. ci, p. 287): 
 
 The average number of elements in all the substitutions of a group is n — a,n 
 being the degree of the group, and a the number of its transitive constituents. 
 
 The only transitive groups of degree 5 containing 15 as a factor of the order 
 are the symmetric and alternating groups. We have therefore to find an imprira- 
 itive group of degree 15 with 5 systems of intransitivity simply isomorphic to 
 the alternating (symmetric) group in 5 letters. 
 
 In determining the imprimitive group corresponding to {A-^A^A^A^A^) pos, 
 we make use of the following facts: (1) the 15 conjugate substitutions corres- 
 ponding to terms of the type A^A^. J.3j44must be of degrees 12 or 14; (2) the 20 
 conjugate substitutions corresponding to terms of the type A^ A^ A^ must be of 
 degrees 9, 12, or 15; (3) the 24 conjugate substitutions corresponding to terms 
 of the type A^A^A^A^A^ must be of degree 15. It must, therefore, be possible to 
 solve the equation 
 
 15 (12 + 2a) + 20 (9 + S(3) -f 24.15 = 14.60 
 
 where a =0, 1 ; /3 = 0, 1, 2. The only solution is a = 0, /3 = 2. 
 
 Therefore the imprimitive group we are seeking contains among its substi- 
 tutions 15 of degree 12 and order 2, 20 of degree 15 and order 3, 24 of degree 15 
 and order 5. Making use of the relations among the generating substitutions of 
 such a group of order 60 as given in Burnside, Theory of Groups, p. 107, we find 
 that the two substitutions corresponding to ^.1-43^3^.^^.5, A^^A-yA^A}, substitutions 
 which will generate (^1-43^.3^4^5) pos, are respectively , 
 
 s =z a\alalalal . alalalalal . alalala^al, 
 
 12 3 4 12 34 12 34 
 
 p = a\a^ . alal . at^ai . a^a} . a^a^ . ala] ; 
 
 s and p are therefore the generating substitutions of an imprimitive group simply 
 isomorphic to the alternating group of degree 5. 
 
 In determining a group simply isomorphic to {AiAoA^A^A^) all, we argue as 
 before in regard to the various sets of conjugate substitutions. The 15 substitu-
 
 16 Martin : On the Imprimitive Substitution Groups of Degree 
 
 tions corresponding to terms of the type A^A^ . A^A^ are of degrees 12 or 14, the 
 20 corresponding to the type A^A^^A^, are of degrees 9, 12, or 15, the 24 corres- 
 ponding to the type A^A^A^A^A^ are of degree 15, the 10 corresponding to the 
 type A-^A^ are of degrees 6, 8, 10, or 12; the 30 corresponding to the type A-^A^ 
 ^3^4 are of degrees 12 or 14; the 20 corresponding to the tj^pe A-^AoA^A^A^^yq of 
 degree 15. The equation to be satisfied is therefore 
 
 15 (12 + 2a) + 20 (9 -f 3,5) + 24.15 + 10 (6 + 2/) 4- 30 (12 + 2.^) -f 20.15 
 
 = 14.120 where a = 0, l;/3 = 0, 1, 2; ^ = 0, 1, 2, 3; ^ = 0, 1. 
 
 The only solution is a = 0, /3 = 2, ^ = 3, (5 = 1. The substitutions A^A^A^A^A^, 
 A^A^A^Ar^, A^A^A^A^ will generate the group {A^A^A^A^A-^ all, and corresponding 
 to these as generators of the imprimitive group we have the three substitutions, 
 
 i^SiT 1-^34'; i-'t4'; 
 
 5 rr a{aiala\al . aMp,Uqal . a^aia'^alcq , 
 G = alal . alalalaj . ao«|a|rt| . alalalai, 
 p =r a\al . alal . alal . alal . ajal . ajaf . 
 
 To sum up the results of the preceding work, the 16 heads with three sys- 
 tems of intransitivity give 41 groups with three systems of imprimitivity. The 
 7 heads with five systems of intransitivity give 42 groups with five systems of 
 imprimitivity, but of these 13 groups contain also three systems of imprimitivity. 
 Therefore there are 70 imprimitive groups of degree 15 as determined in this 
 paper. 
 
 Primitive Substitution Groups of Degree Eighteen. 
 
 The main theorems employed in this investigation of primitive groups are 
 the following, in which p is always to stand for a prime number. 
 
 I. The order of a primitive group of degree n cannot exceed ' , where 
 
 2, 3, . . . . p are the distinct primes ichich are less than f n. (Burnside, Theory of 
 Groups, p. 199). 
 
 n. A group of degree p -\- x or of degree 2p -\- x, x >» 2, cannot be more than 
 X times transitive. (Miller, Bull. A. M. S., v. IV, pp. 142, 143). 
 
 III. If a primitive group of degree n contains a circidar substitution of prime
 
 Fifteen and the Primitive Substitution Groups of Degree Eighteen. 17 
 
 order p, the group is at least {n — p -\- \)-fold transitive. (Cole's tr. of Netto's 
 Theory of Substitutions, p. 93). 
 
 IV. A self-conjugate subgroup of a primitive group must he transitive. (Burn- 
 side, 1. c, p. 187). 
 
 V. A self -conjugate stibgroitp of a x-ply transitive group of degree n{2<C x <Cn) 
 is hi general at least (x — ^)-p^y ti'arisltive. The only excepticm is that a triply tran- 
 sitive group of degree 2"' may have a self-conjugate subgroup of order 2™. (Burn- 
 side, 1. c, p. 189). 
 
 VI. A group G ivhich is at least doubly travisitive either must be simple or mus 
 contain a simple group H as a self-conjugate subgroup. In the latter case no opera 
 tion of G except identity is per mutable ivith every operation of II. The only exceptions 
 to this statement are that a triply transitive group) of degree 2'" may have a self 
 conjugate subgroup) of order 2"\ and that a doubly transitive group of deyree p^ may 
 have a self -conjugate subgroup of order p'". (Burnside, 1. c,, p. 192). 
 
 VII. The substitutions of a transitive group G which leave a given symbol 
 unchanged form a maximal subgroup G^ , ivhich is one of a set of 7i conjugate sidy- 
 groups, each leaving one of the n elements unaffected. (Burnside, 1. c, p. 140). 
 
 VIII. The number of substitutions of degree l<C,n contained in a transitive 
 group) of degree n is equal to the number of substitutions of this same degree I contained 
 
 in the maximal subgroup G^ of degree n — 1 multipliedby ■ j- . (Stated by Miller, 
 
 Quar. Jour. of. Math. v. XXVIII, p. 215.) 
 
 IX. The average number of elements in all the substitutions of a group is n — a, 
 n being the degree of the group and a the number of its transitive constituents. (Fro- 
 benius, Crelle, t. ci, p. 287.) 
 
 X. Sylow's theorem, as stated by Burnside, 1. c, p. 92, or by Sylow, "Theo- 
 remes sur les groupes de substitutions," Math. Ann., v. V (1872), pp. 584 et seq. 
 
 XI. The class of a primitive group of degree n is the same as the class of its 
 maximal subgroup that leaves one element unaffected.
 
 18 Martin : On the Imiyrimitive Substitution Groups of Derjree 
 
 While the preceding theorems are used throughout the work on primitive 
 groups, the following are used mainly in the determination of simply transitive 
 primitive groups. 
 
 XII. A simply transitive primitive group) G of degree n cannot contain a transi- 
 tive subgroup of degree Jess than n. (Miller, Quar. Jour, of Math., v. XXYIII, 
 p. 215. 
 
 XIII. When Gi contains a self -conjugate suhgj'oup H of degree n — a, R must 
 be intransitive, and it must be the transform with respect to substitutions of G of any 
 one of a — 1 other subgroups of G^ (S[, HL, .... Hl_-^ . (Miller, Proc. Lon. 
 Math. Soc, v. XXVIII, p. 534.) 
 
 XIV. All the prime numbers which divide the order of one of the transitive con- 
 stituents of Gi divide also the orders of each of the other transitive constituents. 
 
 Corollary I. If one of the transitive coivstituents of Gi is of a prime degree, 
 each of its other transitive constituents is of the same or a larger degree, and the 
 order of G^ is the same as the order of the group formed by these other transitive 
 constituents. 
 
 Corollary II. If the order of Gi is not divisible by the square of a prime 
 number, all its transitive constituents are of the same order, and G^ is formed by 
 establishing a simple isomorphism between them. (Miller, 1. c, p. 536.) 
 
 XV. If a transitive constituent of Gx is of a prime order, the order of G^ is the 
 same prime number, and G is of class n — 1 . 
 
 Corollary. If Gi contains a constituent of degree 2, its order is 2, and the degree 
 of G is a prime number. (Miller, 1. c, p. 536.) 
 
 The above theorems are given in the form and with the symbols most con- 
 venient for use, and so are not always exact quotations from the papers and 
 books referred to, while the references given are not always references to the 
 original paper in which the theorem appeared. 
 
 Applying these theorems now to the special case in ^vhich 7i = 18 , we pro- 
 ceed as follows : 
 
 Since 18 = 2. 7 + 4, by Theorem II a primitive group cannot be more than 
 4-ply transitive.
 
 Fifteen and the Primitive Substitution Groups of Degree Eighteen. 19 
 
 By Theorem I the order is seen not to exceed 
 
 18! 
 
 2.3.5.7.11 
 
 = 2^13^517.13.17 
 
 If the group included circular substitutions of orders 2, 3, 5, 7, 11, 13, it would 
 be at least 17, 16, 14, 12, 8, 6-fold transitive respectively according to Theorem 
 III. This is impossible; therefore circular substitutions of these orders are not 
 present, and consequently we see at once that 11 and 13 cannot be factors of 
 the order. 
 
 If the order includes the factor 7, then, by Theorem X, there is a subgroup 
 of order 7. This must consist of the powers of a substitution composed of two 
 cycles of 7 elements each, and it must be contained self-conjugately in a group 
 of order 7 . 4m that interchanges transitively among themselves the four remain- 
 ing elements. (Cf. Burnside, Theory of Groups, p. 202.) It is quite possible 
 to establish a (7a, 1) isomorphism between an imprimitive group of degree 14 
 with the systems of imprimitivity 7, 7 and a transitive group of degree 4; there- 
 fore 7 may be a factor of the order. 
 
 A subgroup of order 5^ cannot be present, as it would have to be intransi- 
 tive with the systems of intransitivity 5, 5 or 5, 5, 5. In the one case, it would 
 have to be contained self-conjugately in a group of order 5l 8m, in the other, in 
 a group of order 5^. 3 . 772 . In either case, a circular substitution of order 5 would 
 be present, which is impossible. 
 
 The factor 5 may be contained in the order, as it is possible to establish a 
 (5, 1) isomorphism between the cyclical group of degree 15 and the cyclical 
 group in the remaining three letters. 
 
 The order must, therefore, be a factor of 2^^. 3^ 5 . 7 . 1 7. 
 
 Simply Transitive Groups. 
 
 The maximal subgroup G^ that leaves a^ unaffected is intransitive (Theorem 
 XII), and its order is, therefore, a factor of 2^^ 3l 5.7. Moreover, its class can- 
 not be less than 6, for if it were 2, 3 or 5, G^ would necessarily contain a transi- 
 tive subgroup of too low a degree, and it cannot be of class 4, if G is to be 
 primitive. (Netto, 1. c, p. 138.) 
 
 By Theorem XIV, Cor. I, it is evident that Gi cannot contain a transitive 
 constituent of degree 13 or 11; by Theorem XIV it cannot contain a transitive
 
 20 Martin : On the Imprimitive Substitution Groups of Degree 
 
 constituent of degree 15 or 14, and by Theorem XV, Cor., it cannot contain a 
 constituent of degree 2 . 
 
 If one of the transitive constituents of G^ is of degree 1 2, the other must be 
 of degree 5. The isomorphism between the transitive groups of degrees 12 and 
 5 must be an (a, 1) isomorphism, where a itself may be equal to 1. By Theorem 
 XIV, the order of the group of degree 5 must contain the factors 5, 3, 2; there- 
 fore this group must be either the alternating or the symmetric group of 
 degree 5. If the isomorphism is more than simple, then the group of degree 12 
 must be an imprimitive group with 6 systems of imprimitivity. The head for 
 such an imprimitive group as we require is the intransitive group of order 2 
 and degree 12 given by [l, aiotg • ^3^*4 • %<^6 • ^t'^s • o^9<^io • ^u^d • The group 
 (rti3ai4«i5«i6'^i7) pos contains 
 
 24 conjugate substitutions of order 5 and degree 5, 
 20 conjugate substitutions of order 3 and degree 3, 
 15 conjugate substitutions of order 2 and degree 4. 
 
 Corresponding to these in the group of degree 17, we have 1 substitution of 
 degree 12 and order 2, 24 of degree 15 -f- 2a and order 5 or 10, 24 others of 
 degree 15+ 2a' and order 5 or 10, 20 of degree 15 + 2(5 and order 3 or 6, 
 ogether with 20 of degree 15+ 2/3' and order 3 or 6, 15 of degree 12 + ly 
 and order 2 or 4, and 15 of degree 12 + 2^' and order 2 or 4, where 
 
 a = 0, 1 ; a' = 0, 1 ; /^ = 0, 1 ; ;^' = 0, 1 ; 7 = 0, 1, 2; / = 0, 1, 2. 
 
 By Theorem IX, the following equation must be satisfied : 
 
 12 + 24(15 + 2a) + 24(15 + 2a') + 20(15 + 2/3)+ 20(15 + 2/?') 
 
 + 15 (12 + 2y) + 15 (12 + 2/) = 120 . 15. 
 
 The only type of solution is given by a = /? = ^ = |5' = , a' = 1 , }^' = 2 . 6^^, 
 therefore, contains both a self-conjugate subgroup of degree 12 and order 2, and 
 15 conjugate subgroups of the same type. But, by Theorem XIII, only 5 such 
 conjugate subgroups should exist if this group is to be the G^ of a simply transi- 
 tive primitive group. This intransitive group gives us, therefore, no such group 
 as we require. 
 
 For precisely the same reason the intransitive group formed by establishing 
 a (2, 1) isomorphism between an imprimitive group of degree 12 and order 240,
 
 Fifteen and the Primitive Sahstitution Groups of Degree Eighteen. 21 
 
 and the symmetric group of degree 5 cannot be employed in the formation of a 
 simply transitive primitive group of degree 18. 
 
 If the isomorphism is simple, the group G^, including the alternating group 
 of degree 5, must contain 24 substitutions of degree 10 or 15 and order 5, 20 of 
 degree 6, 9, 12 or 15 and order 3, 15 of degree 6, 8, 10, 12, 14 or 16 and order 2. 
 The following equation must, therefore, be satisfied : 24 (10 + a) + 20 (6 + /^) 
 + 15 (6 + ^) = 60 X 15, where a = 0, 5 ; ^ = 0, 3, 6, 9 ; ^ = 0, 2, 4, 6, 8, 10. 
 The only solution is a = 5, /i? = 9, )/ = 10. 6^^ is, therefore, of class 15. 
 
 A group G of degree 18, formed with the help of this 6^^ and, therefore, of 
 order 60. 18, contains 36 conjugate subgroups of order 5, each of which is con- 
 tained self-conjugately in a group of order 30. As each of these subgroups of 
 order 5 is already self-conjugate in a group of order 10, the construction of the 
 generating substitutions of such a group is an easy matter. G^ is generated by 
 
 and by f = a^a^ . a^a^ . a^cti^ . o^a^ . a^ai^ . a^a-^^ . ai3«i5 . aua^^ , 
 
 and contains, as one of the above-mentioned groups of order 10, the group gene- 
 rated by 
 
 V = s~Us = a^a^ . a^a^ . a^a^ . a^a^ . agajg . aio«ii . ctisdi^ • «i4«i6. 
 
 The group \u, v\ is a subgroup of a group of degree 18 and order 30 formed 
 by establishing a (5, 1) isomorphism between an imprimitive group of degree 15 
 and order 30 with u and its powers as head and the symmetric group in the three 
 elements a^a^a^^. The question then reduces to that of the determination of a 
 substitution of degree 18 and order 3 that will transform the head \u\ into itself, 
 interchange cyclically the three systems of \u\, and be in its turn transformed 
 into its square by v. An examination of all substitutions fulfilling these condi- 
 tions results in finding none that do not give, when combined with other substi- 
 tutions of Gi, substitutions that cannot possibly belong to a simply transitive 
 primitive group containing Gi as a maximal subgroup. 
 
 There is no primitive or imprimitive group of degree 12 simply isomorphic 
 to the group [ahcdef) ^20', consequently, no isomorphism can be established 
 between the symmetric group of degree 5 and a transitive group of degree 12. 
 
 There remains the question whether the symmetric group of degree 5 can 
 be put in a simply isomorphic relation to one of the imprimitive groups of degree
 
 22 Martin : Oti the Imprimitive Substitution Groups of Degree 
 
 12 and order 120 that have both six and two systems of imprivitivity. Such 
 o^roups of degree 12, however, contain two self-conjugate subgroups of orders 2 
 and 60 respectively, and, therefore, are not in a simply isomorphic relation to 
 the symmetric group of degree 5. 
 
 If one of the transitive constituents is of degree 10, the other can only be of 
 degree 7. By Theorem XIV, the group of degree 10 must contain 7 as a factor 
 of its order, and, therefore, must be either the alternating or the symmetric 
 group. It is impossible to establish an isomorphic relation between either of 
 these groups and one of degree 7 without introducing substitutions of too low a 
 degree. 
 
 If one of the transitive constituents is of degree 9, the remaining constituent 
 may be either intransitive in two systems of four elements each or intransitive in 
 eight elements. The isomorphism can in neither case be simple, as an examina- 
 tion of all groups of degree 8 and orders equal to those of transitive groups of 
 degree 9 shows that in each case a system of intransitivity of degree 2 enters, 
 with the single exception of a group of order 144. Here, however, the group of 
 degree 9 contains a substitution of order 8, while an inspection of the corres- 
 ponding groups of degree 8 shows no substitution of that order. 
 
 The isomorphism is, therefore, an (a, p) isomorphism, where a and /3 are 
 not simultaneously equal to one. 
 
 When neither a nor p is equal to one, G^ must be formed from an imprimi- 
 tive group of degree 9 and an intransitive group of degree 8. The order of each 
 transitive constituent must contain 3 as a factor, and, therefore, the group ot 
 degree 8 must be some combination of the alternating and symmetric groups of 
 degree 4 in two systems of elements. The only combinations possible, consistent 
 with the requirements of class, are got by establishing a simple isomorphism 
 between the two symmetric groups of degree 4 or between the two alternating 
 groups of the same degree. Every relation of isomorphism established between 
 these groups of degree 8 and any impriraitive groups of degree 9 consistent with 
 the requirements of class, results in a Gi that contains a self-conjugate subgroup 
 of order 4 and degree 8, and no other subgroups of the same order. This case, 
 therefore, gives no simply transitive primitive group. 
 
 When a becomes 1, the group of degree 8 must, as before, be composed of 
 either the symmetric or the alternating groups of degree 4 in two sets of 
 elements put into the relation of simple isomorphism. The order of such a group 
 does not contain 9 as a factor; therefore this case gives no possible G^.
 
 Fifteen and the Primitive Substitution Groups of Degree Eighteen. 23 
 
 When /? becomes 1, the group of degree 9 must be impriraitive. No tran- 
 sitive group of degree 8 stands, however, in the given relation of isomorphism 
 towards an imprimitive group of degree 9. The only permissible intransitive 
 groups of degree 8 are combinations of the symmetric and alternating groups of 
 degree 4 in two sets of elements, and none of these are isomorphic in the given way 
 to any imprimitive group of degree 9. 
 
 If one of the transitive constituents is of degree 8, we may have the systems 8, 
 6, 3 or 8, 3, 3, 3. In both cases we have an (a, 1) isomorphism between an 
 intransitive group of degree 14 and the symmetric group of degree 3. The group 
 of degree 8 is not primitive, as no suitable isomorphic relation can be established 
 between a primitive group of degree 8 and an imprimitive or an intransitive group 
 of degree 6. The only imprimitive groups of degree 8 that can be used are those 
 with the head (1, aia^-dia^ . OrfitQ •a.^cig) that are isomorphic to a group of degree 
 4 and order 1 2 or 24. Such groups, however, cannot be combined with the groups 
 in the remaining 9 elements in such a way as to generate a group capable of being 
 the Gi of one of the required primitive groups. 
 
 The case in which Gi contains a transitive constituentof degree 7 has already 
 been discussed, as according to Theorem XIV, Cor. I, the remaining constituents 
 must be of larger degree. 
 
 If Gi contains a transitive constituent of degree 6, the systems may be either 
 6, 6, 5 or 6, 4, 4, 3. For the former system the only possible arrangement is to 
 establish a simple isomorphism between the three groups {aia./t^a^ar^aQJQQ , {a^agaya^Q 
 «n«i3)6o» (^I3f«i4%5«^i6^j7) pos, or between the groups {axa.^a.ja^a^aQ\c,Q, («7«8«/'io«ii«i2) leo- 
 (ajgai^aisaieai,) all. An examination of the two groups (rj formed from these iso- 
 morphisms shows that these are not the maximal subgroups of simply transitive 
 primitive groups of degree 18. 
 
 If the systems are 6, 4, 4, 3, only the imprimitive groups of degree 6, the 
 alternating and symmetric groups of degree 4, and the symmetric groups of degree 
 3 are involved. The group formed by the system 6, 4, 4 has an (a, 1) isomorphism 
 to the group of degree 3, and this isomorphism cannot be simple. No combination 
 of these groups can be found fulfilling all the necessary conditions. 
 
 The case in which Gi contains a transitive system of degree 5 has already 
 been discussed, as the remaining systems must be of degree greater than 5. 
 
 If (?! contains a transitive system of degree 4, the only arrangement possible 
 is 4, 4, 3, 3, 3. The groups involved are therefore the symmetric groups of degree 
 3 and 4, and the alternating group of degree 4. One group consistent with the
 
 24 Martin : On the Imprimitive Suhstitution Groups of Degree 
 
 requirements of class is got by establishing a (1,4) isomorphism between the 
 group, 
 
 This group contains, however, one and only one subgroup of degree 8 and order 2. 
 
 A second group is got by first establishing a (4, 1) isomorphism between {a^a^a 
 a^.a^a^a^a^ all and (orgajoaji) all; and then establishing a (12, 3) isomorphism between 
 the group of order 24 so formed and the group {a^^a^^a^^ . ais^ie^n) all. This group 
 of degree 17 contains only one subgroup of order 4 and degree 8; therefore it 
 cannot become a G^ . 
 
 Gi cannot contain onl}^ systems of degree less than four, as in such a case 
 a system of degree 2 would have to enter. 
 
 There is, therefore, no simply transitive primitive group of degree 18. This 
 result when joined to all other determinations of similar groups shows that there 
 is no simply transitive primitive group of degree p +1, i> a prime number and 
 <17. 
 
 Multiply transitive groups. 
 
 Among the transitive groups of degree 17 five contain a self-conjugate sub- 
 group of order 17. These are of order 17, 2.17, 4.17, 8.17, 16.17 respectively, 
 while all excepting the first are of class 16. 
 
 If a primitive group of degree 18 and order 18.17 existed, such agroup would 
 contain 18 conjugate subgroups of degree 17. It would therefore contain 17 sub- 
 stitutions of degree 18 and 18.16 of degree 17. By Sylow's theorem since 18.17 
 =: 2.3-. 17, such a group contains either 1 or 34 subgroups of order 3~. A sub- 
 group of this order must be intransitive, therefore cannot be self conjugate, and 
 it is impossible to form 34 subgroups of order 9 from 17 substitutions of degree 18. 
 No such group of degree 18 exists. 
 
 A primitive group of degree 18 and order 18. 17.2 = 2l3^ 17 would contain 
 among its substitutions 153 of class 16 and order 2, 288 of class 17and order 17, 
 170 of class 18. This group must contain either 1, 4, or 34 conjugate subgroups 
 of order 3'^ As before, a subgroup of this order cannot be self-conjugate, as 
 it is intransitive. If there were 4 conjugate subgroups, each would be self- 
 conjugate in a group of order 3l 17 involving all 18 letters and necessarily tran- 
 sitive. Such a group is non-existent. If there were 34 conjugate subgroups 
 they must be of degree 18, and there are not enough substitutions of class 18 to 
 form all these subgroups.
 
 Fifteen and the Primitive Substitution Groups of Degree Eigldeen. 25 
 
 A primitive group of degree 18 and order 18.17.4 = 2^31 17 contains among 
 its substitutions 476 of degree 18, 459 of degree 16, 288 of degree 17. According 
 to Sylow's theorem it contains either 1, 3, 9, 17, 51, or 153 conjugate subgroups 
 of order 2^. Now the group leaving one element unchanged contains 17 conju- 
 gate subgroups of degree 16 and order 4; therefore the group of degree 18 contains 
 153 distinct conjugate subgroups of order 4; therefore it contains 153 conjugate sub- 
 groups of order 2^. Each of these is contained self-conjugately in no larger group. 
 
 The number of systems of intransitivity in any one is got from the following 
 equation, where x denotes the number of substitutions of degree 18 and a the 
 number of systems: 
 
 18cc + 16 (7 — x)=: 8(18 — a), where a :^ 1, x-< 8. 
 
 There are two sets of solutions, either x- = 0, a = 4, or a; = 4, a = 3. 
 The group of degree 17 is generated by, 
 
 where t and its powers form a self- conjugate subgroup of the group of order 2^ and 
 degree 18 that is now under discussion. It is impossible to so connect the systems 
 and introduce the remaining elements that the first solution may give the group 
 of order 2^ Making use of the second solution we have only to combine with 
 the group generated by ^ a substitution of degree 18 that connects the two remain- 
 ing elements by a transposition, and unites the cycles of ^ in pairs. The 153 
 groups of order 2^ give in this way 153.4= 612 distinct substitutions of degree 
 18, while there are only 476 in the group. This group of degree 18 does not exist. 
 If there is a primitive group of order 18 . 17. 2^ = 2^ 3l 17, it contains 288 
 substitutions of degree 17, 1071 of degree 16, 1088 of degree 18. The group of 
 degree 17 which is generated by 
 
 s -=i a^a^a-ia^a^a^a^a^aQaiifi-^ya^Mi2,ciiiai^a^^a^^ 
 and t = a^iCt^Qai/^ai^ayia^arflz . ajjUi^a^aycft^^a^ai^a^ , 
 
 contains 17 conjugate subgroups of degree 16 and order 8; therefore in the 
 group of degree 18 there are 153 such conjugate subgroups, and each of these 
 is self-conjugate in a group of order 2^ and degree 18. Denoting by a the 
 number of systems of intransitivity of this group of order 2*, and letting x denote the 
 number of substitutions of degree 18 contained in the group, we have the equation 
 18a: -h 16 (15 — ic)= 16 (18 — a), where a ^l. There are two solutions, x= 8, 
 a=:2;a;=0, a=3. The first solution would involve a larger number of substi-
 
 26 Martin: On the Imprimitive Substitution Groups of Degree 
 
 tutions of degree 18 than are actually present in the group under consideration. 
 The second solution shows that the group must contain 153 conjugate subgroups of 
 order 2^ and degree 18 consisting of substitutions of class 16 only, and involving 
 three systems of intransitivity. A substitution must therefore be combined with< 
 that transforms t into one of its powers, and has its head in the group generated 
 by t] moreover, this substitution must have as one of its cycles the transposition 
 (ajajg), and must have systems of intransitivity apart from this cycle consistent 
 with the systems of t. Such a substitution is a'=-a.^aiQ . a^ «i4 . ci^cLy, . a^ni^ ■ Oo^is • «7«ii. 
 a^ai^.ttiaiQ. The required group is therefore \s, t, a\. 
 
 It is not necessary to prove that these three substitutions give a group of 
 the required order, as such a group would be necessarily doubly transitive, and 
 it is known that there is a doubly transitive group of degree 18 and of the 
 required order. B}' the mode of construction of the substitutions, it is evident 
 that there is only the one type of group of this degree and order. 
 
 Any primitive group of degree 18 and order 18 . 17. 16 contains 2312 sub- 
 stitutions of degree 18, 288 of degree 17, 2295 of degree 16. The group of 
 degree 17 contains 17 conjugate cyclical subgroups of degree 16 and order 16, 
 therefore, the group of degree 18 contains 153 subgroups of order 16, each of 
 which is self-conjugate in one of 153 conjugate subgroups of order 32. Giving 
 a andx the usual meanings, we find that the group of order 32 involves the equa- 
 tion 18a: + 16 (31 — x)= 32(18 — a), where a =f= 1. The only solution is a = 2, 
 x = 8; therefore, the group of degree 12 and order 32 must be intransitive 
 with two systems of intransitivity, and must contain 8 substitutions of degree 
 18, 23 of degree 16. We have to add, therefore, to the cyclical group of degree 
 16, 8 substitutions of degree 16 and 8 of degree 18, all of them containing as one 
 cycle the transposition of the remaining two letters. 
 
 The group of degree 17 and order 17.16 has as generators 
 
 and u = a.M^a^^a^^a^^aQa^^ayMi^al^a^a^a^a^^a^a^ . 
 
 The substitution t = (i\U\% • n^a^ . a.^aio • ^s'^ii • ^'e^'s • «i/^i6 • «n«i3 • «i2<^*i5 generates with 
 s and u the required group of degree 18 and order 17.16.18. A triply transi- 
 tive group of such an order is known to exist (Burnside, 1. c, p. 158); so no 
 further proof that \s, u, t:\ is a group is necessary. It is easy to see that the 
 even substitutions of the group just found form the simple group of order 
 18 .17.8.
 
 Fifteen and the Primitive Substitution Groups of Degree Eigliteen. 27 
 
 The three remaining transitive groups of degree 17 each contains 120 con- 
 jugate subgroups of order 17. They are of orders 15.16.17, 15. 16. 17. 2, 
 15.16.17.4 respectively. 
 
 The group of degree 18 and order 15 . 16 . 17 . 18 would necessarily contain 
 816 conjugate subgroups of order 5. Each is self- conjugate in a group of order 
 90 connecting the remaining three elements transitively. This group is intran- 
 sitive with two transitive constituents, one of degree 15 and order 90, the other 
 of degree 3. The first, however, is non-existent, therefore, the group of degree 
 18 is non-existent. 
 
 The two remaining groups also, if they can generate primitive groups of 
 degree 18, would generate groups that each contain 816 conjugate subgroups of 
 order 5. In the one case, we should have to make use of an intransitive group 
 containing as a transitive constituent a group of degree 15 and order 180, in the 
 other, the transitive constituent would enter as a group of degree 15 and order 
 360. Both of these groups are non-existent; therefore, the three groups of 
 degree 17, at present under discussion, furnish us with no new groups of 
 degree 18. 
 
 As the case now stands, the conclusion arrived at may be summed up as 
 follows : 
 
 There are no simply transitive primitive groups of degree 18, and in addi- 
 tion to the symmetric and alternating groups, there are only two multiply transi- 
 tive groups of this degree, viz., the two given by 
 
 ^ («2«io«i4«i6<^i7«9«6«3 • «4«n<^6«i2«i5«8«i3«7) . I of ordcr 2448, 
 
 I (ag^io • otgaii . a^ay, . a^a^^ . a^a^^ . a^a^^ . a^a^^ . a^a^^) , J 
 
 r (aia2«3«4«5«6«7«8^9«10«n«13«13«fl4«15«16«17) » "j 
 
 \ («3«4«io«n^i4<^6«i6«i2<^i7«i5«9«8«5«i3<^3«7)» }- of Order 48 96 . 
 
 The second of these is triply transitive, and contains the first, which is 
 doubly transitive and simple, as a self-conjugate subgroup. 
 
 The works consulted in the preparation of this paper have included, in addi- 
 tion to the standard works on the subject by Jordan, Serret, Netto, and Burn- 
 side, the following papers : 
 
 Askwith, " On Possible Groups of Substitutions that can be formed with 
 3, 4, 5, 6, 7 Letters Respectively." Quar. Jour. Math., v. XXIV (1890), pp. 
 111-167.
 
 28 Martin : On the Imprimitive Substitution Groups of Degree^ etc. 
 
 " On Groups of Substitutions that can be formed with Eight Letters." Quar. 
 Jour. Math., v. XXIV (1890), pp. 263-331. 
 
 "On Groups of Substitutions that can be formed with Nine Letters," Quar. 
 Jour. Math., v. XXVI (1892), pp. 79-128. 
 
 Cayley, " On Substitution Groups for Two, Three, Four, Five, Six, Seven, 
 and Eight Letters." Quar. Jour. Math., v. XXV (1891), pp. 71-88, 137-155. 
 
 Cole, " List of the Substitution Groups of Nine Letters." Quar. Jour. Math., 
 V. XXVI (1892), pp. 372-388. 
 
 "The Transitive Substitution Groups of Nine Letters." I3ull. New York 
 Math. Soc, V. II (1893), pp. 250-258. 
 
 "List of the Transitive Substitution Groups of Ten and of Eleven Letters." 
 Quar. Jour. Math., v. XXVII (1894), pp. 39-50. 
 
 "Note on the Substitution Groups of Six, Seven, and Eight Letters." Bull. 
 New York Math. Soc, v. II (1893), pp. 184-190. 
 
 Miller, " Intransitive Substitution Groups of Ten Letters." Quar. Jour. 
 Math., V. XXVII (1894), pp. 99-118. 
 
 "List of Transitive Substitution Groups of Degree Twelve." Quar. Jour. 
 Math., V. XXVIII (1896), pp. 193-231. 
 
 "Note on the Transitive Substitution Groups of Degree Twelve." Bull. 
 Amer. Math. Soc, v. I (1894-1895), pp. 255-258. 
 
 "On the Transitive Substitution Groups of Degrees Thirteen and Fourteen." 
 Quar. Jour. Math., v. XXIX (1898), pp. 224-249. 
 
 " On the Primitive Substitution Groups of Degree Fifteen." Proc Lon. 
 Math. Soc, V. XXVIII (1896-1897), pp. 533-544. 
 
 "On the Primitive Substitution Groups of Degree Sixteen." Amer. Jour. 
 Math., V. XX (1898), pp. 229-241. 
 
 "On the Transitive Substitution Groups of Degree Seventeen." Quar. 
 Jour. Math., v. XXXI (1899), pp. 49-57. 
 
 "A Simple Proof a Fundamental Theorem of Substitution Groups, and 
 Several Applications of the Theorem." Bull. Amer. Math. Soc, v. II (1895- 
 1896), pp. 75-77. 
 
 " On the Limit of Transitivity of the Multiply Transitive Substitution Groups 
 that do not Contain the Alternating Group." Bull. Amer. Math. Soc, v. IV 
 (1897-1898), pp. 140-143. 
 
 Jordan, " Sur la classification des groupes primitifs," Comptes Rendus, t. 
 LXXIII (1871 j, pp. 853-857. 
 
 " Sur Venumeration des groupes primitifs pour les dix-sept premiers degres." 
 Comptes Rendus, t. LXXV (1872), pp. 1754-1757. 
 
 Philadelphia, January 1901.
 
 LIFE. 
 
 I was born in Elizabeth, New Jersey, December 30, 1869. In 1890, I 
 entered Bryn Mawr College, selecting, as my major studies, Mathematics and 
 Latin. In 1894, I received the degree of A. B. from this college. The first 
 semester of the following year, 1894-1895, was spent at Bryn Mawr College as 
 a graduate student in Mathematics and Physics, the second semester, in teaching 
 at a preparatory school. During the year 1895-1896 I held the Fellowship in 
 Mathematics in Bryn Mawr College, remaining there the following year, 1896- 
 1897, as a graduate student. During the year 1897-1898 I held the Mary E. 
 Garrett European Fellowship from Bryn Mawr College, and spent the entire 
 year at the University of Gottingen, where I attended the lectures of Professors 
 Klein and Hilbert. I then returned to Bryn Mawr College, where I was Fellow 
 by Courtesy in Mathematics during the year 1898-1899. In the spring of 1899 I 
 passed the examination at Bryn Mawr College for the degree of Doctor of 
 Philosophy. My major subject was Mathematics, pursued under the direction 
 of Professors Scott and Harkness, while my double minor was Physics, pursued 
 under the direction of Professor Mackenzie. 
 
 My gratitude is due to all the Professors under whom I have studied, and 
 especially to Professor Harkness, under whose direction this paper was prepared. 
 
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