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Report No. 199 
 
 MATHEMATICS 
 
 THE AUTOMORPHISM GROUP OF A STRONGLY CONNECTED 
 AUTOMATON AND ITS QUOTIENT AUTOMATA 
 
 by 
 Rudolf Bayer 
 
 February lU, 1966 
 
Report No. 199 
 
 THE AUTOMORPHISM GROUP OF A STRONGLY CONNECTED 
 AUTOMATON AND ITS QUOTIENT AUTOMATA 
 
 *>y 
 
 Rudolf Bayer 
 
 February 1^, 1966 
 
 Department of Computer Science 
 
 University of Illinois 
 
 Urbana, Illinois 
 
ACKNOWLEDGEMENT 
 
 The research carried out in this paper was initiated by a suggestion 
 of Professor M. Paul. 
 
 The author wants to thank Professor F. E. Hohn and Professor M. Paul 
 for many valuable discussions and for the encouragement they gave during the 
 course of this investigation. 
 
 Professor J. Nievergelt remarked that the theory developed in this 
 paper might lead to interesting applications in the design of highly parallel 
 computers, since it seems reasonable to expect that certain subcomponents of 
 such computers like ILLIAC IV should have a relatively large automorphism 
 group . 
 
 ■Hi' 
 
TABLE OF CONTENTS 
 
 Page 
 
 I. INTRODUCTION . 1 
 
 II. THE AUTOMORPHISM GROUP OF A QUOTIENT AUTOMATON 5 
 
 III. THE SUBGROUP LATTICE OF G(A) AND THE POSET OF QUOTIENT 
 
 AUTOMATA OF A 11 
 
 IV. JORDAN-HOLDER -SCHREIER THEORY AND QUOTIENT AUTOMATA 18 
 
 V. AN APPLICATION TO COSET ALGEBRAS 22 
 
 PRINCIPAL NOTATIONS 31 
 
 BIBLIOGRAPHY 33 
 
 -IV- 
 
I . INTRODUCTION 
 
 In this section we give the basic definitions and the general 
 background of the following theory. Also the major results obtained by 
 various authors in this research field will be stated as far as they 
 are needed later in this paper. 
 
 Definition 1 : An automaton A is a triple 
 
 A = (S, 1, 6) where 
 
 S is a finite set (set of states) 
 I is a finite set (input alphabet) 
 5 is a function from S x I into S 
 
 Definition 2: I* is the free semigroup of strings over the alphabet I 
 with the operation of string concatenation. We make the 
 convention that I* contains the empty string. 
 
 We extend 5 to a function from S x I* into S by defining inductively: 
 
 5 (s,z) = s for all s e S where z is the empty string 
 
 6 (s,xy) =5(5 (s,x),y) for x s I* and y e I. 
 
 Our definition of an automaton differs slightly, but not essentially 
 from the definitions given in [l] , [ 2 ] and [3], the reason being that 
 we want to keep the mathematical theory of automata as close to 
 physical representations as possible. 
 
 -1- 
 
Definition 3 : A = (S, I, 5) is strongly connected 
 
 < > (Vs £ S) (Vt e S) (3x e I*) such 
 
 that 5 (s, x) = t 
 
 Definition ki By a function h from A = (S, I, 5) to 
 
 B = ( T, I, \ ) we mean a function from S into T. h 
 
 is called operation preserving if (Vx e S) (Vx e I*) it is 
 
 true that h (b(s,x)) = (\(h(s),x). An operation preserving 
 
 function is called a 
 
 homomorphism if it is from A to B 
 endomorphism if it is from A to A 
 isomorphism if it is from A to B and 1-1 
 automorphism if it is from A to A and 1 - 1 . 
 
 Clearly the set of automorphisms of A is a group under function 
 composition. We denote this group by G(A). 
 
 For the rest of this paper we assume that all the automata are 
 strongly connected. This seems like a severe restriction for the 
 mathematical theory, one should realize, however, that all automata 
 that are interesting for applications, e.g. as computing devices, 
 must be strongly connected. 
 
 Theorem 1 : (Weeg [2]): If A is strongly connected, then G(A) is a 
 a group of regular permutations on S . 
 
 Lemma 1 ; (Fleck [l] ) : Let h , h e G(A). 
 
 If h (s) = h (s) for some s £ S then h = h * 
 
 -2- 
 

 Definition 5 ; Define the equivalence relation = on I* by: 
 
 x s y iff (Vs 6 S) (6(s,x) = 6(s,y)) 
 
 Note that = actually is a congruence relation and therefore we can define 
 the factor semigroup I* i . 
 
 Definition 6 : The semigroup of the automaton A is defined to be 
 I*/_ and is denoted by S(A). 
 
 Definition 7 : We call the strongly connected automaton A total if the 
 order of G(A) is equal to the number of states of A. 
 
 Notation : Denote isomorphism between groups or automata by 'V". 
 
 Theorem 2 : ( [5] ) : G(A) ~ S(A) if and only if A is total. 
 
 Definition 8 : Let H be a subgroup of G(A). Then we define the 
 equivalence relation = on the state set S of A by: 
 
 a 
 
 S l "H S 2 
 
 <=> (a h e H) such that h(s ) = s 
 
 We denote the equivalence class of s with respect to this relation by 
 
 H— H— 
 s. Call s a transitivity class of H and denote the set 
 
 I s ! s € s| by ^H and 7 5(t,x) |t e s > by 5( s,x). 
 
 Then we get the following useful lemma: 
 
 -3- 
 
■ H- 
 
 H- 
 
 Lemma 2: 6( s,x) = 6(s,x) 
 
 H— H— 
 
 Proof: 6("s,x) = <6(t,x) t e s 
 
 H- 
 
 h (h (s),x)| h e HS 
 
 h (&(s,x)) h £ H( = "5(s,x) . This means that an 
 
 arbitrary input x maps a transitivity class of H 1 - 1 onto some 
 
 other transitivity class of H. 
 
 Notation; "H < G" means: "H is a subgroup of the group G" . 
 "H <l G" means: "H is a normal subgroup of G" . 
 
 Definition 9: Let H < G(A). Then we define the quotient automaton of 
 A modulo H , denoted by A/ to be: 
 
 A/ = (S , 1, o) where 5 is defined by 
 
 5 ( s,x) = 6(s,x) 
 
 Using lemma 2 it is easy to see that 5 is well defined and that A/ is 
 strongly connected if A is. 
 
 The following two theorems are obtained in [5] . 
 
 Theorem 3 ' Let A be strongly connected and let H be a normal subgroup 
 of G(A). Then the factor group G(A) , is isomorphic to a subgroup 
 of G(A^ H ). 
 
 Fheorem k : If A is total and H < G(A), then G(A) / ^ G(A, ). 
 
 Also in [5] the following open question was asked: "What can be 
 said about G(A/ ) if H is not normal in G(A)?" The rest of this 
 paper is concerned with that question and some consequences of the 
 partial answer which is obtained. 
 
 -k- 
 
..../•'■ 
 
 II. THE AUTOMORPHISM GROUP OF A QUOTIENT AUTOMATON 
 
 In the rest of this paper almost all the results which are obtained 
 for strongly connected automata can be strengthened for total automata, see 
 for example theorems 3 an( i ^-. Partial characterizations for strongly connected 
 automata turn out to be total characterizations for total automata, homomorphisms 
 become isomorphisms, etc. To show this parallelism where it exists we will call 
 the strengthened version of Theorem k, Lemma m, Definition n,... in the strongly 
 connected case Theorem Tk, Lemma Tm, Definition Tn, . . . resp. in case we have a 
 total automaton. 
 
 § 
 
 Definition 10 ; Let H < G. Then H is the unique maximal subgroup of G such 
 
 that H <l ET. 
 
 £ 
 Of course H depends on G, but it will always be clear from the 
 
 context 1 what G is and therefore we do not indicate this dependency in the 
 
 & 
 notation for H. The following is a generalization of Theorem 3: 
 
 Theorem 5 : Let H < G(A) and H < U<G(A). Then U / is isomorphic to a subgroup 
 of G(A^) . 
 
 Proof : The following proof is a generalization of the proof for theorem 3 
 
 given in [_5J . Let |Hu | u e Uv be the collection of cosets of H which 
 
 constitute U, . For each coset Hu define the map 
 / H 
 
 f u : A / H > A /H 
 
 by f u ( H s) = H uTTT 
 
 We use the normality of H in U to show that f is well defined. 
 
 -5- 
 
TT IT 
 
 Therefore let s = t and Hu = Hv, i.e., there exists h e H such that 
 s = h (t) and an h e H such that u = h g v . Thus we get: 
 
 ,H-x H 
 
 f ( s) = u(s) = h v(s) = v(s) = 
 
 H- 
 
 H 
 
 H-x 
 
 = V^Ct)) = n iy(tT = n 7TtT = f v ( t) 
 
 since vh = h v for some h e H by normality of H and f is well defined. 
 Next we want to show that f is 1 - 1: 
 
 •H- N 
 
 -Hr-> 
 
 H- 
 
 H- 
 
 sorae 
 
 So let f ( s") = f ( t), i.e., u(s) = u(t). Then there exists 
 h e H such that u(s) = h u(t) = uh (t) for some h e H again by the 
 normality of H But u is 1 - 1 as an automorphism of A and therefore 
 
 TT XT 
 
 s = h (t). But this means s = t and f is 1 - 1. Also since u is 
 
 onto clearly f is onto, 
 u 
 
 It is left to show that f is a homomorphism: 
 
 H 5(f u ( H F),x) = H 5( H uTFJ» = H 5(u(s),x) = 
 
 = H u(o(s,x)) = f u ( H oT^T) = f u ( H 5( H s,x)) 
 
 Thus f is an automorphism of A/„ and f e G(A/„). Next we have to 
 show that the correspondence it : Hu > f is 1 - 1. 
 
 TT TT 
 
 Suppose that u and v belong to distinct cosets and that f(s)=f(s) 
 
 TT TJ T T 
 
 for some s e S , then u(s) = v(s) and there exists an h e H such that 
 
 H 
 
 hu(s) = v(s). Then by lemma 1 we have: 
 
 hu = v and u and v are in the same coset of H which is a contradiction. 
 Thus it is 1 - 1. To show that it is a homomorphism: 
 
 ^(Hu Hv) = it(Huv) = f where uv = w 
 
 w 
 
 «(Hu)«(Hv) = f f . 
 
 ' U V 
 
 But f f ( s) = f ( vTsT) = uv[s7 = wTsT = f ( s) and f f = f and 
 
 u V x ' U V V ' ' V ' K ' W V ' U V w 
 
 is a homomorphism. Since it is also 1 - 1 it is an isomorphosm. 
 
 QED 
 
 ■6- 
 
Theorem T5 : Let A be total and H < G(A). Let J < G(A/ ). Then there exists 
 a subgroup U < G(A) such that H < U and U/ =, , J. 
 rhe proof makes use of the following lemma: 
 
 IT IT 
 
 Lemma 3 '• Under the conditions of theorem T5 let j e J and j ( s) = t. Let 
 g € G(A) and g(s) = t. Then Hg = gH. 
 
 H- 
 
 Proof of Lemma : Note that g exists since A is total. Clearly Hg(s) = t. 
 
 IT 
 
 To show: gH(s) = t 
 
 H H— H— 
 Let h e H and a e I* such that h(s) = 6(s,a), then clearly 6( s,a) = s and 
 
 5( t,a) = 5(j( s),_a) = j 6( s,a) = j( s) = t 
 
 H— 
 
 Then gh(s) = g(S(s,a)) = 5(g(s),a) = 5(t,a) € t and Hg = gH by lemma 1 which 
 completes the proof of the lemma. 
 
 Proof of T5 ; Define a map it from J into the cosets of H as follows: Choose 
 
 TT _ 
 
 a fixed s in S 
 
 H* 
 
 it: J ~ > J Hg|g e G(A) 
 
 it (j) = Hg if J( H s) = H t and g(s) ■. , t 
 
 TT TJ IT IT 
 
 To show: Jt is well defined. Let s = s, h (s) = s and t = t; 
 
 h £ (t) = t and g ][ (s 1 ) = t . Where h , h g e H and g e G(A). 
 
 Then g(s ) = gh (s) = h g(s) = h (t) for some h € H by lemma 3. h x g(s ) = 
 
 -1 
 
 t = h 
 
 ; x (t x ) 
 
 -7- 
 
Therefore h h^gCs.^ = t ± and h 2 h~ g = g 1 by lemma 1. 
 
 Therefore Hg = Hg and « is independent of the representatives we choose in 
 
 s and t. 
 To show that it is 1 - 1 let 
 
 *(j) = Hg and *(,],) = Hg 1 and Hg = Hg , i.e., g = hg for some h e H. 
 
 TJ TT TT II 
 
 Then let j ( s) = t and j ( s) = t^. 
 
 H- 
 
 Then g(s) = t and g (s) = t , therefore t = g(s) = hg (s) = h(t ) and t = 
 
 H- 
 
 t and j = j by lemma 1. 
 To show that " is a homomorphism let j-.jp - jo and ^(j- ) = H §- > * = 1> 2 ,3- 
 «(j 1 )«(j 2 ) = (Hg 1 )(Hg 2 ) = H( g;L H)g 2 = H(H gl )g 2 = (HH) gl g 2 = Hg ] _g 2 by lemma 3 
 and we need to show that Hg g = Hg . So let g.(s) = t. , then 
 
 g 1 g 2 (s) = g ] _( t 2 ) = g 1 (6(s,a)) = 6(g 1 (s),a) = 
 
 IT TT TT JJ 
 
 = 5(t ,3.) for some a e I*; but 5( t , ,a) - 5(j-.( s),a) = 
 
 H„,H- 
 
 -H-x H-— 
 
 = j ± B( s,a) = j 1 ( t 2 ) = j i j 2 ( s) = t and g^g^s) = 
 
 8(t 1 ,a) € t 
 
 i.e., g,g p (s) = h(t ) = hg (s) for some h e H and therefore g g = hg and 
 
 Hg n g p = Hhg = Hg which shows that it is a homomorphism. Therefore it is an 
 
 isomorphism between J and a group of cosets of H under coset multiplication. 
 Clearly the union of these cosets is a subgroup U < G(A) such that H O U and 
 
 u/ H =J. 
 
 QED. 
 
 As simple corollaries to Theorems 5 and T5 we get the following: 
 
 JL 
 
 sorem 6 : Let H < G(a). Then H"/ is isomorphic to a subgroup of G(A/ ) 
 
 -8- 
 

 Theorem T6: Let A be total. Then 
 
 H # / H ^G(A/ H ) 
 
 Note that theorems 3 and ^ are trivial corollaries of Theorems 6 and T6 resp. 
 for the case that IT = G(A). 
 
 Clearly Theorem 6 gives us the maximum amount of information about G(A/ ) that 
 can be extracted from G(A) using Theorem 5 > while Theorem T6 gives a total 
 characterization of G(A/ ) in terms of the lattice of subgroups of G(A) if A 
 is total. 
 
 Theorem "J : Let H < U < G(A). Then A/ is a homomorphic image of k/ 
 
 H 
 
 Proof : note that s = t => s = t 
 
 define f : A/ — > A/ by f( H s) = U i" 
 
 Then f is well defined and onto. 
 To show that f is a homomorphism: 
 
 f( H 5( H s,x)) = f( H 5li77T) = U oT^T = 
 
 U 6( U s,x) = U 5(f(%,x) 
 
 QED. 
 
 Definition 11 ; We call an automaton B a natural homomorphic image of A if 
 B is isomorphic to a quotient automaton of A. 
 
 With that much information available it is natural to consider the following 
 question: Let H < U < G(A). Then we know by theorem 7 that A/ is a homo- 
 morphic image of A/ . Under what conditions is this homormorphism a natural 
 one? The following theorem which resembles closely a basic theorem in group 
 theory gives a partial answer to this question. 
 
 -9- 
 
Theorem 8: (First Isomorphism Theorem): 
 
 Let H < U < G(A). Then 
 
 {k '*p/ n) = A /u 
 
 Here U/ stands for the isomorphic copy of U/ in G(A/ ) under the isomorphism 
 
 H 
 it defined in the proof of Theorem 5. 
 
 H' 
 
 Proof: Let card (A) be the number of states of A and let card (U) be the 
 
 order of U. Then obviously card (A) = card (U) • card (A/ ). Define the 
 
 U/ _ U 
 
 map fs A/ D — > (A/ H )/ (u/ } by f( D ?) = H ( H s). 
 
 / H u- U- 
 To show that f is veil defined let s = t, then s = u(t) for some u e U and 
 
 s = u(t) = f ( t) where f is as defined in the proof of Theorem 5. 
 
 "A 
 
 "/, 
 
 H ( H s") = U/ H ( H I) = U/ H f u ( H t) = U/ H ( H t) = "' H ( H t) since f u e o/ H and f 
 
 is well defined. Clearly f is onto; to show that f is 1 - 1 notice that 
 card (A/ TJ ) = card (U/ E )L 1 ))• 
 
 To show that f is a homomorphis.m: 
 
 U / W .n . . U A 
 
 H ( H 5) (f( U s),x) = H ( H 5) ( H ( H s),x) = 
 
 u/ 
 
 uA 
 
 H 5 ( H F,x) = " H ( H oT^7^D= f( u 5T^T = 
 
 f( U 5( U s,x)) 
 
 QED, 
 
 ■10- 
 

 III. THE SUBGROUP LATTICE OF G(A) AND THE POSET OF 
 QUOTIENT AUTOMATA OF A 
 
 With the results of section II we are now ready to study the 
 structure of the set of quotient automata of A under certain relations. 
 We denote the lattice of subgroups of G(A) by L P / A \ and the set of 
 quotient automata of A by L . Obviously if H < G(A) and K < G(A) 
 and H ^ K then also A/ ^ A/ since at least S ^ S . Thus we get a 
 
 natural 1-1 correspondence between L 
 
 G(A) 
 
 and L . Furthermore, we can 
 
 impose the lattice structure of L / \ onto L simply by transporting it 
 
 J G(A) 
 
 A 
 
 from L 
 
 G(A) 
 
 to L. using our 1-1 correspondence. It is also possible, 
 
 however, to define the same lattice structure on L. in a more natural 
 way in terms of certain properties of the quotient automata of A. This 
 was carried out by the author, but it was found neither conceptually 
 satisfying nor very useful. 
 
 A more reasonable approach seems to be to identify isomorphic 
 quotient automata and to drop down to equivalence classes of isomorphic 
 automata in the study of L . . If this is being done then the question 
 arises what happens to the lattice L p /.N and its structure, or in other 
 words, what are necessary and sufficient conditions on subgroups 
 H < G(A), K < G(A) that the corresponding quotient automata A/ and 
 A/ K be isomorphic. Again partial answers are obtained for strongly 
 connected automata and a complete characterization is obtained for 
 total automata. 
 
 It turns out that group isomorphism is not the proper concept 
 on Lp/.N to correspond to isomorphism on L . Too much collapses and too 
 
 -1.1- 
 
I 
 
 much structure is lost in Iw A \ if one considers classes of isomorphic 
 subgroups of G(A) to study the equivalence classes of L . The author 
 found the concept of conjugacy between subgroups of G(A) to be more 
 fruitful to the approach of this study. 
 
 For completeness sake we state here the standard definition of 
 conjugacy in group theory. 
 
 Definition 12: The subgroups H and K of G are called conjugate 
 if there exists some element g € G such that gHg " = K. Conjugacy 
 clearly is an equivalence relation and the equivalence class of H is 
 called the conjugacy class of H. Denote conjugacy between groups by "~". 
 
 Notation: Denote the set of conjugacy classes of subgroups of G(A) by 
 P r /.\ and the set of isomorphism classes of L. by P.. Denote the 
 conjugacy class of H < G(A) by H and the isomorphism class of A/ g L. 
 
 :> 
 
 Definition IS: We define the relation < on P /.\ by: H < U <= 
 H is conjugate to a subgroup of U. It is clear that < is a partial order 
 on PpfAN and if in the sequent we speak of the "poset P / s" we mean the 
 set P r fA\ with the partial order <. 
 
 Definition 1^-: We define the relation < on P. by: A/ < A/ < > 
 ./ is a homomorphic image of A/ . It is clear that < is a partial order 
 on P. and if in the sequent we speak of the "poset P " we mean the set 
 P. with the partial order <• 
 
 A 
 
 -12- 
 
We do not distinguish in notation between the two different partial 
 orderings in definitions 13 and 1^ but it will always be clear from 
 the context which one is meant. 
 
 
 Th 
 
 eorem 9: Let H < G(A), K < G(A) and H ~ K. Then A/ ~ A / 
 
 K 
 
 Proof: Fix g e G(A) such that gH = Kg. 
 
 r > A/ by 
 
 Vh 
 
 -H- 
 
 K- 
 
 f ( s) = gR 
 
 TT TT 
 
 To show that f is well defined let s = t and t = h(s); h e H. 
 
 Then f ( H t ) = K gTtT = K ihTsT = K kgT^T = K g(iy 
 
 for some k e K since H ~ K and f is well defined. Clearly f is 
 onto since g is. Also f must be 1 - 1 since card (A/„) = card (A/ ). 
 To show that f is a homomorphism: 
 
 f( H 5( H s,x)) = f( H oTI^Ty = K g(5(s,x)) = \(g(s),x) - 
 
 ViTiXx) = fyf (*£)>*) 
 
 QED 
 
 Our first isomorphism theorem and theorem 9 together give us the 
 following slightly stronger version of the first isomorphism theorem: 
 
 Theorem 8.1: Let N <J H < G(A) and let M be a conjugate of N. Then 
 A/-rj is a natural homorphic image of A/ . 
 
 -13- 
 
Proof: Let Ng = gM for some g e G(A). Then M is normal in g""Tlg. 
 Then by the first isomorphism theorem we have 
 
 A 
 
 ^Hg . ( A /m } 
 
 (g"^g/ M ) 
 
 and by theorem 9 
 
 we have A/ ~ A / _1 u • So A /u is isomorphic to a quotient 
 
 automaton of A/„. 
 As a simple corollary of Theorems 9 a nd 7 we get the following 
 generalization of Theorem 9* 
 
 Theorem 10: Let H < G(A); K < U < G(A) and H ~ K. Then A/ is a 
 homomorphic image of ki . 
 
 Proof: Combine the homomorphism in theorem 7 and the isomorphism 
 in theorem 9* 
 
 Theorem 11: Let A be total, H < G(A)_, U < G(A) and A/ a homomorphic 
 image of A/ . Then U contains a subgroup K such that H ~ K. 
 
 Proof: Let f: A/ ^ A/ be the homomorphism. 
 
 -HK U- U-7— y 
 
 ;(s) 
 
 Let for fixed s: f( lx s) = U t = U £ 
 
 for some now fixed g e G(A) since A is total. 
 Then for arbitrary t e S we have: 
 
 -lk- 
 
f( H ") = f( H 5li^77 = f( H 5( H s,x)) = U 5(f( H s),x) = 
 
 
 U c /U- 
 
 U; 
 
 u- 
 
 u- 
 
 8 ( U iT7yTx) = U 5(g(s),x) = U g(o(s,x)) = U i(t) 
 
 /here x is some input in I* such that 5(s,x) = t. But for f to be well 
 
 H)( u iTt7 
 
 U- 
 
 defined we must have ( Vh e H) [ g(t) = gh(t) 1 ; i.e. Ug(t) = Ugh(t) 
 
 _1 
 for all h e H. Therefore Ug = UgH or e qui vale ntly U = UgHg . But 
 
 then gHg is a subgroup of U. Let K = gHg and K ~ H. 
 
 As a corollary we obtain easily: 
 
 Theorem T 9: Let A be total and H < G(A); K < G(A). 
 Then H ~ K < > A/ ~ A / K 
 
 Proof: c=^> This is Theorem 9 
 
 < by Theorem 11 K contains a subgroup 1C conjugate to H, 
 But K = K since card (K) = card (h) = card (K_). 
 
 Theorem T 10: Let A be total, H < G(A), U < G(A). Then A/ is a 
 homorphic image of A/ 
 
 < > U contains a subgroup K such that H ~ K. 
 
 Proof: 
 
 <- 
 
 :> This is Theorem 11 
 This is Theorem 10. 
 
 -15- 
 

 Definition 15: Call an order preserving map of a poset onto another 
 poset a -poset homomorphism, an order reversing map a poset antihomomorphism . 
 If the maps are 1-1 call them poset isomorphism and poset anti- 
 isomorphism respectively. 
 
 Note that Theorems 9, 10, T9, T10 characterize the poset 
 p of isomorphism classes of quotient automata in terms of the automorphism 
 group G(A) or better in terms of the poset of subgroup conjugacy classes 
 of G-(A). The relation between these two poset s is made precise in the 
 following two theorems. Again we obtain a partial knowledge for the 
 case of the strongly connected automaton and perfect knowledge for the 
 case of the total automaton. 
 
 Theorem 12; The map f : P 
 
 G(A) 
 
 -> P 
 
 A 
 
 defined by: f (H) = A/ is an antihomomorphism from the 
 poset P^/aN onto the poset P.. 
 
 Proof: f is well defined by Theorem 9* f is clearly onto, f is 
 order reversing by theorem 10. 
 
 Theorem T 12 : If A is total, then the map of Theorem 12 is an 
 antiisomorphism between the posets P«/ a n a nd P . 
 
 Proof: The homorphism f is 1-1 by Theorem T9» Therefore f is 
 
 defined. The fact that f " is also order reversing follows 
 
 from Theorem T10. QED 
 
 -16- 
 
Now we can strengthen our first isomorphism theorem for quotient 
 automata in case A is total. 
 
 Theorem T 8 : (Second Isomorphism Theorem): Let A be total. Then 
 A/ is a natural homorphic image of A/ < > 
 N is a normal subgroup of a conjugate group of H. 
 
 Proof: 
 
 <= 
 
 this is Theorem 8 and Theorem 9« 
 
 > Let J < G(A/ N ) and A/ ~ (A/ )/j. Then by theorem T 5 
 there exists a subgroup U of G(A) such that N <\ U and U/ ~ J* 
 
 Then A 
 
 /U = (A V/(U /H ) : ( A /V 7j = A /H ^ the first 
 
 isomorphism theorem and U ~ H by theorem T9« 
 
 -17- 
 
IV. JORDAN-HOLDER-SCHREIER THEORY 
 AMD QUOTIENT AUTOMATA 
 
 The Jordan -Holder -Schreier Theory about normal series and 
 composition series of groups can now be used conveniently to investigate 
 series of natural homomorphisms for quotient automata. We need some 
 definitions first: 
 
 Definition l6: Let 
 
 12 n 
 
 (1) 
 
 and N. < G(A); i = 1, 2, ..., n. 
 
 Then this series is called a partial normal series. It is called lower 
 
 sem 
 
 inormal if I, = i^S where 1 is the identity of G(A), upper 
 
 seminormal if N = G(A) and normal if it is both lower and upper semi- 
 
 normal. If all the factor groups N 
 
 1+1 /N. 
 l 
 
 for i = 1, 2, . .', n - 1 are simple and non trivial then we call the 
 series (l) partial composition, lower semicomposition, upper semi- 
 impositio n and compos i tio n series respectively. 
 
 Definition 17: We call a series of quotient automata 
 
 A, 
 
 -> A, 
 
 -> 
 
 -> A 
 
 (2) 
 
 a partial natural series if A. is a natural homomorphic image of 
 
 A. } i = 1, 2, . . . , n - 1. We call (2) lower seminatural if A ~ A/ ,.^, 
 
 v n = /G(A)' 
 
 upper seminatural if A = A and natural if it is both upper and lower 
 
 seminatural. We call the series (2) partial tight , lower semitight, 
 
 -18- 
 
upper semitight and tight respectively, if all the automorphism groups 
 
 used to obtain the natural homomorphisms are simple and if in addition 
 
 7 A - i 
 t i +1 
 
 A. 7 k. ,., for i = 1, 2, . .., n - 1. 
 
 Theorem 13: Let N < G(A), let N_ < N_ <J . . . < N 
 — n 12 n 
 
 and N. ~ M. for i = 1, 2, .*., n. If the series (3) is partial 
 normal, lower seminormal, upper seminormal, normal then the series 
 
 (3) 
 
 A / 
 
 NL, 
 
 -> A 
 
 '/k 
 
 -> 
 
 -> A 
 
 .1 is partial natural, 
 n 
 
 1 2 
 
 upper seminatural, lower seminatural, natural respectively 
 
 Note that ] , -, is isomorphic to a subgroup of G(A/ ) 
 N. 1 
 
 i+l/j 
 
 by theorem 5 (i = 1; 2, . . . , n - l). Also (A/ 
 
 N. / (N. , ) ~ A/^ T 
 
 1 / 1+1 /n. = /N i + i 
 
 by our first isomorphism theorem and A/ is a natural homomorphic 
 
 7 i+1 
 
 image of A 
 
 /N.* 
 
 x 
 
 QED 
 
 Theorem ik: Let N < G(A); let ff. <1 N_ <J .. . <» N 
 
 n 12 n 
 
 and N. ~ M. for i = 1, 2, . .., n. If the series (h) is partial 
 composition, lower semicomposition, upper semicomposition, 
 composition then the series 
 
 w 
 
 A 
 
 '/k, 
 
 -> A 
 
 'At 
 
 "> 
 
 -> A 
 
 /m 
 
 1 ' 2 
 
 is partial tight, upper semitight, lower semitight, tight respectively. 
 
 -19- 
 
Proof: Note that the factor groups N in the proof of 
 
 /N. 
 
 theorem 13 are now simple groups, 
 
 QED 
 
 Theorem T 13 : Let A be total. Then there exists a partial normal, 
 
 lower seminormal, upper seminormal, normal series 
 
 N <q N^ <J . . . <J N such that N < G(A) and N. ~ M. for i = 1, 2, .... n 
 12 n n li 
 
 <- 
 
 :> A/ 
 
 M 
 
 -> A 
 
 1 
 
 '/K- 
 
 •> 
 
 -> A/ 
 
 M 
 
 -> A 
 
 n-1 
 
 /m 
 
 is a partial natural, upper seminatural, lower seminatural, natural 
 series respectively. 
 
 Proof: 
 
 > this is theorem 13 
 
 <- 
 
 = By theorems T5 and T9 M. is a normal subgroup of a conjugate 
 
 K. , of M. _ for i = 1, 2, ..., n - 1. Then define N = M and if 
 
 l+l l+l ' ' ' n n 
 
 the conjugate N. , of M. n is defined, then N. , ~ K. n by the 
 l+l l+l l+l i+l J 
 
 transitivity of conjugacy of groups. So let N = g K & . 
 
 and define N. = g. , _ M. g. " . Then N. <\ N. . and N. ~ M. 
 
 l i+l l i+l l i+l i i 
 
 -1 
 
 QED 
 
 Theorem T lU : Let A be total. Then there exists a partial composition, 
 
 lower semicomposition, upper semicomposition, composition series 
 
 N n O N^. O ... <J N such that N < G(A) and N. - M. for 
 12 n nil 
 
 i = 1, 2, ..., n < > 
 
 -20- 
 
•'•4r«V 
 
 A 
 
 /m, 
 
 -> A 
 
 Ik 
 
 -> 
 
 -> A 
 
 /M ." 
 ' n-1 
 
 -> A 
 
 /M 
 
 1 '2 ■ n-± ' n 
 
 is a partial tight, upper semitight, lower semitight, tight series 
 respectively. 
 
 Proof: 
 
 > This is theorem 1^ 
 
 <: 
 
 Observe that in this case the factor groups arising in the 
 
 construction in the proof of theorem 1^+ are all simple and non trivial 
 by definition 17 • 
 
 Theorem 13: Let A 
 and A > A/ M 
 
 -> A/ 
 
 N, 
 
 -> ... 
 
 -> A 
 
 /N 
 
 -> ... 
 
 -> A 
 
 /m 
 
 (1) 
 (2) 
 
 1 n 
 
 be two tight series of the total automaton A. Then n = m and every 
 amtomorphism group used to define a natural homomorphism in series 
 (l) is isomorphic to an automorphism group used to define a natural 
 homomorphism in series (2) and vice versa. 
 
 Proof: This is an immediate corollary to theorem T ik and the 
 Jordan -Holder Schreier theorem on composition series of groups. QED 
 
 -21- 
 
V. AN APPLICATION TO COSET ALGEBRAS 
 
 In our investigation we have put some rather strong 
 restrictions on our automata. So far, however, we have not yet 
 answered the question whether automata with the required properties 
 exist. The following theorem gives a positive answer to the existence 
 question: 
 
 Theorem l6: (Stated without proof in Bavel [6]): Let H be 
 
 an 
 
 arbitrary finite group. Then there exists a total automaton A such 
 that G(A) ~ H. 
 
 Proof: Let A = (H, H, &) where 
 
 5 (h , h ) = h h (group multiplication in H). 
 
 To show that G(A) ~ H: 
 
 For every h e H define the map: 
 
 A 
 
 -> 
 
 A 
 
 by 
 
 f (h ) = h h (group multiplication in H) Vh e H 
 
 Clearly f. is 1-1 and onto. 
 J h 
 
 To show f, is an automorphism: 
 h 
 
 f h (5(h 1 , h 2 )) = f h (h 1 h 2 ) = h(h 1 h 2 ) = (h h 1 ) h 2 = 
 
 f. (h. )h = &(f, (h.).h^) and f. is a homomorphism. Hence f. is an 
 h 1 2 h 1 2 h h 
 
 automorphism and we have f e G(A). 
 
 -22- 
 
HHHW 
 
 Clearly h n 4 ^ > f, 4 f, and the map 
 1 ' 2 h, h„ 
 
 it : H 
 
 1 2 
 -> G(A) defined by « (h) = f 
 
 h 
 
 is 1 - 1. Since card H = card A and card G(A) < 
 
 card A we have that card G(A) < card H. Therefore n is onto 
 
 and we must have card H = card A = card G(A). 
 
 To show that n is a homomorphism: 
 
 it (h ± h 2 ) = f 
 
 = f^f = * (h^K (h 2 ) and H : G(A) 
 
 QED 
 
 Notation: We denote the automaton A defined in the proof of 
 
 theorem l6 by (?„ and call it the canonical automaton of H. 
 
 n 
 
 Remark : It is easy to see that it would be enough to use a set of 
 generators of the group H as inputs in the construction of the automaton 
 A in the proof of theorem l6. So we get the following corollary. 
 
 Corollary l6.1: Let H be a finite group and I„ a set of generators 
 h. 
 
 for H. Then the automaton 
 
 A = (H, I H , 5) 
 where 5 is group multiplication in H is total and G(A) ~ H. 
 
 These two results characterize a whole class of total automata for which 
 the special results obtained in sections II, III, IV for the case of 
 total automata are valid. Our considerations suggest that the theory of 
 
 -23- 
 
total automata might be used to study coset systems of arbitrary (not 
 necessarily normal) subgroups of a group. In the rest of this section 
 our intention is not to present new results but to demonstrate a new 
 approach to a classical field of mathematics using the theory of total 
 automata and their quotient automata. However some of the proofs given 
 here should be new. To make these ideas more specific we need a few 
 definitions first. 
 
 Definition 18 : A monadic algebra 'ffirC is a pair 
 
 -^Y, = (E ; 0) where 
 
 E is a set, the elements of "?9~d , and 
 
 is a set of unary operators on E, the operators of 'ffl'C* 
 Let o, , o , ..., o e 0, e e E. Then by o |e we denote the element of 
 E which we get if we apply the operator o. to the element e and by 
 
 o, • • • o^ o., e we mean o. ( o, n • > . (o_ e ) • • • ) ■ 
 
 k 2 l 1 k 1 k-1 l 1 
 
 Let G be a group and H a subgroup of G. Then define the monadic algebra 
 
 ^^■^ - the coset algebra of H in G- by 
 G, n 
 
 ^ G , H = ( I H g I g e G I , G) 
 where > Hg | g € G I is the collection of right cosets of H in 
 G and the operators g, e G are defined by 
 
 gjHg = Hg g ± 
 
 -2k- 
 
Mit»«M 
 
 This coset algebra of H in G can in a certain sense , which will "be made 
 precise in a moment, be considered as a generalization of the concept 
 "factor group of a normal subgroup of G" for the case that H is not 
 normal. 
 
 Definition 19 j We define the relation = on G by 
 
 <- 
 
 ■> g-jHg = g 2 |H§ 
 
 Vg € G. 
 
 °1 ~ & 2 
 
 We do not indicate the dependence of = on H in the notation. Denote the 
 collection of equivalence classes by G and the equivalence class of 
 g by g. 
 
 If in our coset algebra y 2'?L„ „ we want to have a non 
 redundant set of operators, i.e. we identify operators with the same 
 effect on the coset s Hg dropping from G down to G, then we get the 
 following non redundant monadic algebra of H in G: 
 
 1H 
 
 Gr,H 
 
 with g 1 j Hg = Hg 
 by definition 19. 
 
 = ( H 
 
 " f ^ 
 
 6 G i, G) 
 
 ; . It is clear that the operator g is well defined 
 
 Hotation: Let H be a subgroup of G. Then by En we denote the unique 
 maximal subgroup of H which is normal in G. 
 
 Theorem 17 '• = is a congruence relation on G and the factor group 
 G/_ is isomorphic to G/ 
 
 -25- 
 
Proof: Let g ± = g 2 , then (g g 1 )|Hg = Hg(g 3 g^ = 
 
 (Hg g ) g^_ = (Hg g 3 ) g 2 = Hg (g 3 g 2 ) = (g 3 g 2 )|Hg 
 
 and (g ± g )|Hg = Hg (^ g^ = (Hg g^ g 3 = (Hg g 2 ) g 3 = 
 
 Hg (g g ) = (g g )|Hg. Thus = is a congruence relation. Also 
 = is a refinement of the equivalence relation defined by the cosets 
 of H as equivalence classes since g. = g > 
 Hg = g JH = g |H = Hg and g and g are in the same coset of H. 
 
 Also if NOG and N < H then g = n g^^ (Vn e N) since 
 
 n g |Hg = Hg n g = Hn g g- L = Hg g 1 = g^^lHg for some n e N. 
 
 Therefore = is a refinement of the equivalence relation 
 defined by the cosets of H and the cosets of any subgroup of H which 
 is normal in G is a refinement of the congruence =. Consequently 
 = must be the same congruence relation as the one defined by En. 
 
 QED 
 
 Now if N < G, then N u = N and the non redundant coset algebra 
 
 ^T- , T of N in G is "isomorphic" to the factor group G/_ T in 
 
 Vn 
 
 the following sense : 
 
 1//t 
 
 G,N = (G /V 7 N } and 
 
 ■26. 
 
.-.IrU'- ' 
 
 Ng |Ng = Ng g = Ng Ng and the application of the operator Ng 
 
 corresponds to multiplication with the coset Ng from the right in 
 the factor group G/ . In this sense it is justified to consider the 
 coset algebra of an arbitrary subgroup H of G of the non redundant coset 
 algebra of H in G as a generalization of the factor group of a normal 
 subgroup. Since normal subgroups and the corresponding factor groups play- 
 such a central role in many branches of mathematics it should be interest- 
 ing to study the generalization to coset algebras. 
 
 Our theory of total automata and their quotient automata will 
 turn out to be useful for such an investigation since under a certain 
 interpretation all the results obtained for total automata carry over 
 immediately to coset algebras. 
 
 Let G be a finite group and CL the canonical automaton of G. 
 
 G 
 
 Let H be a subgroup of G. Remember that C n is total and that G(C„) ~ G. 
 
 G G = 
 
 So we can consider the quotient automaton (C r ) . . Then there is a very 
 
 /H 
 
 close relationship between the automaton (C )/ and the coset algebra 
 
 G / n 
 
 W'-- n „ in the following sense: 
 
 i. The states of (CL). are the right cosets of H in G and so are the 
 
 G /h 
 
 elements of the monadic algebra sty'^ 
 
 G,H' 
 
 ii. The inputs of (C ) are the group elements of G and so are the 
 
 G /h 
 
 operators of 1?j£ 
 
 G,H* 
 
 -27- 
 
111. 
 
 The transition function D of C & / H is multiplication of the coset 
 
 from the right by group elements and so is the application of an 
 operator - an element of G - to an element of / 7tTi^ c „ - a right 
 
 coset of H in G. 
 
 Formally this is: 
 
 S 
 
 G,H 
 
 = ( \ Hg | g € G i, G) 
 
 (O 
 
 /H 
 
 (S„, G, 5) 
 
 H 
 
 where \ Hg | g e G V = S 
 
 H 
 
 operation in '/<}£ 
 
 G,H' 
 
 5 I Hg = Hg g 1 
 
 transition in C 
 
 G/H' 
 
 o (Hg, g 1 ) = ( g, g ] _) = b(g, gj_) 
 
 H- 
 
 5-l = H S g-L 
 
 Therefore the quotient automaton (CL) / is just a different way of viewing 
 
 the monadic algebra ^^„ „. CL , could be considered as a machine 
 
 representation of 'c/J 6. n „ at work displaying what "happens" in $/£„ „• 
 
 Lr,n U-jii 
 
 Now it is routine work to carry over the concepts like isomorphism, 
 automorphism, automorphism group, natural homomorphism etc. from automata 
 to monadic coset algebras. Clearly all the theorems obtained for total 
 automata will carry over immediately to coset algebras, too. We will state 
 some of the more interesting ones here, numbering the theorem corresponding 
 to Tn now by Mni 
 
 -28- 
 
.<,tr<V 
 
 The 
 
 # 
 
 orem M 6: Let H < G. Then Ey ~ Q('7jt ^ R ) 
 
 Theorem M 7: Let H < U < G. Then ^2f„ TT is a homorphic image of 
 
 ■ u> u 
 
 ■OW 
 
 G,H* 
 
 Theorem M 8: (isomorphism theorem for coset algebras). 
 
 i. Let N <J H < G. Then ( 'TTtL- w )v 
 
 Lr,lM / 
 
 ( h /n } 
 
 G,H 
 
 ii. )j /■ is a natural homomorphic image of 
 
 GjH \kfxk 
 
 < > I is a normal subgroup of a conjugate group of H. 
 
 Theorem M 9: Let H < G; K < G. Then J 71 r t n „ ~ 7/T n v 
 
 \ijti = • • ' k} } is. 
 
 < ■ - > H, K are conjugate. 
 
 We give a proof of the following theorem of group theory using automata 
 
 theory techniques. 
 
 Theorem 18 : Let G be a finite group and let H,K be conjugate subgroups 
 of G. Then 
 
 *• H# / H : ^/ K 
 
 ii. W 
 
 is conjugate t 
 
 oi 
 
 -29- 
 
Proof: ?/ c~ G H ~ ■ '■• Q K by theorem M9. Thus G( 9/£ Q R ) ~ G( .y/ Q y ) , 
 but G(7y^ G H ) ~ H # / H and G( •V^' g r ) ~ K"/ K by theorem M6 and we get i). 
 
 Now using theorem M8 we get 
 
 'i- # ~ ( 
 
 G,IT = v 
 
 £ 
 
 1 G^/tJ 
 
 /<*/ H ) 
 
 
 g ' k 7(k*V) 
 
 Therefore H is conjugate to K. again by theorem M9 
 
 QED 
 
 We will not carry over the results of section IV. to monadic coset 
 algebras since it is entirely routine. But it should be noted that 
 all the results of section IV are valid for coset algebras. Also it 
 should be noted that analogously to "^^ T r we can define a monadic 
 algebra of left cosets instead of right cosets and all the techniques 
 and results obtained here carry over trivially. 
 
 -30- 
 
PRINCIPAL NOTATIONS 
 
 A,B/., 
 
 G(A) 
 G,H, . , 
 H < G 
 N < G 
 H# 
 
 A/ H 
 
 I 
 I* 
 
 b,\, . < 
 S,T,.. 
 
 V 
 
 a 
 
 A 
 
 H- 
 s 
 
 H 
 
 Finite state automata 
 
 The automorphism group of the automaton A 
 
 Groups and subgroups 
 
 H is a subgroup of G 
 
 N is a normal subgroup of G 
 
 The maximal subgroup such that H <l ET 
 
 The factor group of G modulo the normal subgroup N 
 
 Quotient automaton of A modulo H where H < G(A) 
 
 Set of input letters (input alphabet) of A 
 
 Free semigroup generated by I including the empty string 
 
 Transition functions of A,B, . . . 
 
 State set of A,B, . . . 
 
 Universal quantifier 
 
 Existential quantifier 
 
 Isomorphism between groups or automata or monadic algebras 
 
 Conjugacy between groups 
 
 Lattice of subgroups of the group G 
 
 Poset of conjugacy classes of subgroups of G 
 
 Poset of isomorphism classes of quotient automata of A 
 
 Partial order on the posets P or P. 
 
 State set of the automaton A/„ where H < G(A) 
 
 n 
 
 Transitivity class of the state s with respect to the subgroup 
 
 H of G(A) 
 
 Transition function of the automaton A/ 
 
 The canonical automaton of the finite group H 
 
 -31- 
 
ole 
 
 H 
 
 L # 
 
 Monadic algebra 
 
 Monadic algebra of cosets of the subgroup H in G 
 
 Non -redundant monadic algebra of cosets of the subgroup H in G 
 
 Application of the operator o to the element e in a monadic algebra 
 
 Maximal subgroup of the subgroup H of G such that H '.» <\ G 
 
 t 
 
 -32- 
 
REFERENCES 
 
 [l] Fleck, A.C.: "Isomorphism Groups of Automata" JACM 9,k (October 1962). 
 [2] Weeg, G.P. : "The Structure of an Automaton and its Operation Preserving 
 
 Transformation Group." JACM 9,3 (July 1962). 
 Pjj Barnes, B. : "Groups of Automorphisms and Sets of Equivalence Classes of 
 
 Input for Automata." JACM 12, k (October 1965). 
 \h~] Weeg, G.P.: "The Automorphism Group of the Direct Product of Strongly 
 
 Related Automata." JACM 12,2 (April 1965). 
 [5] Fleck, A.C.: "On the Automorphism Group of an Automaton." JACM 12,^4- 
 
 (October 1965). 
 \6] Bavel, Z. : "On the Structure and Automorphisms of Automata." Ph.D. 
 
 Thesis, Dept. of Math., University of Illinois, Urbana (October 1965); 
 
 or Report 192, Dept. of Computer Science, University of Illinois, Urbana, 
 
 Illinois (October 1965). 
 \Jj Bayer, R.: "Series -Parallel Decomposition of Sequential Machines." 
 
 University of Illinois, Dept. of Computer Science, File No. 677 
 
 (August 1965). 
 \Q] Weeg, G.P.: "The Group and Semigroup Associated with Automata." 
 
 Proceedings of the Symposium on Mathematical Theory of Automata. 
 
 Polytechnic Press, New York, 1962. 
 [9] Buchi, J.R.: "Mathematical Theory of Automata." University of Michigan, 
 
 Communication Sciences, (i960) . 
 
 ■33- 
 
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^"S^OFILLINOIS-UBBANA 
 
 30112103707086