ON THE POINT STABILIZERS OF TRANSITIVE GROUPS WITH NON-SELF-PAIRED SUBORBITS OF LENGTH 2

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1 University of Ljubljana Institute of Mathematics, Physics and Mechanics Department of Mathematics Jadranska 19, 1000 Ljubljana, Slovenia Preprint series, Vol. 37 (1999), 644 ON THE POINT STABILIZERS OF TRANSITIVE GROUPS WITH NON-SELF-PAIRED SUBORBITS OF LENGTH Dragan Marušič Roman Nedela ISSN March 8, 1999 Ljubljana, March 8, 1999

2 ON THE POINT STABILIZERS OF TRANSITIVE GROUPS WITH NON-SELF-PAIRED SUBORBITS OF LENGTH Dragan Marusic 1 Roman Nedela IMFM, Oddelek za matematiko Univerza v Ljubljani Jadranska 19, 1000 Ljubljana Slovenija dragan.marusic@uni-lj.si Katedra Matematiky Univerzita Mateja Bela Banska Bystrica Slovakia nedela@bb.sanet.sk Suggested running head: POINT STABILIZERS OF CERTAIN TRANSITIVE GROUPS 1 Supported in part by \Ministrstvo za znanost in tehnologijo Slovenije", project no. J Supported in part by the Ministry of Education of the Slovak republic, grant no. 1/331/96. 1

3 Abstract In [7] a characterization of transitive permutation groups having a non-self-paired suborbit of length (with respect to which the corresponding orbital graph is connected) was obtained in terms of their point stabilizers. As a consequence, elementary abelian groups were proved to be the only possible abelian point stabilizers arising from such actions, and D 8 was shown to be the only nonabelian group of order 8 with the same property. Constructions of such group actions with point stabilizers isomorphic to D 8 or to Z h, h 1, were also given there. These results are extended here to include a more in depth analysis of the structure of point stabilizers of such group actions, resulting in a set of necessary conditions allowing us to obtain a restricted list of 19 possible candidates for point stabilizers of such group actions when the point stabilizers have order h, h 8. (For h 5, this list gives a complete classication of such point stabilizers.) Furthermore, a construction of a transitive permutation group action with a non-self-paired suborbit of length and point stabilizer isomorphic to D 8 Z h;3 is given for each h MSC: 05C5, 0B5. 1 Introduction Throughout this paper groups are assumed to be nite, unless specied otherwise. By a graph we mean an ordered pair (V E), where V is a nite nonempty set and E is a symmetric irreexive relation on V whose transitive closure is the universal relation on V. (Graphs are thus assumed to be nite and connected.) For graph-theoretic and group-theoretic terms not dened here we refer the reader to [1,, 3, 13]. If a graph X admits a vertex- and edge- but not arc-transitive action of a subgroup G of its automorphism group Aut X, in short, a 1 -arc-transitive action of G. We saythatx is (G 1 )-arctransitive in this case. If G =AutX we say that X is 1 -arc-transitive. The work in this paper is motivated by results on point stabilizers of transitive permutation groups having suborbits of small length [8, 9, 11, 14] and by recent research in vertex- and edge- but not arc-transitive graphs

4 { see [5, 10, 1]. It is easily seen that the class of transitive permutation group having a connected self-paired suborbit of length consists of dihedral groups alone. (A suborbit will be referred to as connected if the corresponding orbital graph is connected.) On the other hand, we have the classical results on point stabilizers by Tutte [11] and Wong [14] in the case of self-paired suborbits of length 3, and by Sims [9] in the case of primitive permutation groups and self-paired suborbits of length 4. Some partial results for other lengths are also known (see [8]). The aim of this paper is to continue the study of the structure of transitive permutation groups having a connected non-self-paired suborbit of length (with the emphasis on their point stabilizers) initiated in [7]. From a graphtheoretic point of view this is equivalent to the study of the structure of connected graphs of valency 4 admitting a vertex- and edge- but not arctransitive group action, in particular, the study of the corresponding vertex stabilizers. A lack of general machinery for these sort of problems makes the study of the structure of transitive permutation groups having a (connected) nonself-paired suborbit of length aratherdicult task. As a starting point one would need to have a set of group-theoretic conditions fullled by a pair of groups (G H), where G has a (connected) non-self-paired suborbit of length in its action on the set of left cosets of H. Such a characterization was obtained in [7]. Such pairs are called ~-creators (see Section for the deniton). Now when investigating ~-creative pairs of groups, two or three steps are taken, depending on whether the context is group- or graph-theoretic. Step 1 would consist in the study of \local" structural properties of the group H (in the ~ -creative pair (G H)), that is, those properties of the point stabilizer H that are implicitly subsumed in the denition of a ~-creative pair of groups and can be, at least in theory, extracted from there. Such groups H are called concentric (for reasons that will become apparent in Section where the denition is given). As for Step, one needs to \imbed" a concentric group H into a supergroup G in such a way that the pair (G H) is a ~ -creator. Note that the diculties involved in each of these two steps are quite dierent in nature. The rst step is purely group-theoretic, whereas the second presupposes, at least implicitly, a graphical realization of some sort. Finally, Step 3 concerns the relative orbital graph corresponding to a (connected) non-self-paired suborbit in the action of G on (left) cosets of H, 3

5 and is of course, of interest only from a graph-theoretic point of view: is the group G the full automorphism group of this graph, or more precisely, is this graph 1 -arc-transitive of is it arc-transitive? It is the rst two steps that we are interested in here. As mentioned above, a characterization of transitive permutation groups having a (connected) non-self-paired suborbit of length in terms of their point stabilizers was obtained in [7]. (Note that the point stabilizer of such an action is necessarily a -group.) As a consequence, elementary abelian groups were proved to be the only possible abelian point stabilizers arising from such actions, and D 8 was proved to be the only nonabelian group of order 8 with the same property. Moreover, constructions of such group actions with point stabilizers isomorphic to D 8 or to Z h, h 1, were also given there, that is, an innite family of 4-valent graphs admitting a 1 -arc-transitive group action was constructed when the corresponding vertex stabilizer is either elementary abelian or the dihedral group of order 8. For other constructions of such actions with abelian point stabilizers see [4, 6]. In this paper we go a step further by presenting a more detailed analysis of the structure of concentric groups (Step 1), and hence, of the structure of point stabilizers of transitive groups with a (connected) non-self-paired suborbit of length, resulting in a restricted list of possible point stabilizers of order h, h 8, for such group actions. Namely, in Theorem 3.1 we prove that the point stabilizer is isomorphic to one of the following 19 groups: (i) Z h for 1 h 8 (ii) D 8 Z h;3 for 3 h 8 (iii) D 8 for 6 h 8 and (iv) H 7 Z h;7 for h f7 8g where D 8 denotes the dihedral group of order 8 and H 7 denotes a certain - extension of the direct product D 8 Z 3. (For h 5, this list gives a complete classication of such stabilizers.) The proof of this result uses three general lemmas about concentric groups (proved in Section ), Another byproduct of these lemmas is Corollary.4 where it is shown that a certain class of concentric groups of order h (notably the one consisting of groups of complexity 1 see Section for the denition) coincides with the direct products of the form 4

6 D h;d 8 Z 3d;h, where d. Moreover, Theorem 3. gives possible dening 3 relations for a group G containing a subgroup H belonging to one of the above four families of groups { for arbitrarily large h { where the pair (G H) is a ~-creator. (but see also Table 1). Finally, in Section 4 a construction of a transitive permutation group action with a (connected) non-self-paired suborbit of length and point stabilizer isomorphic to D 8 Z h;3 is given for each h 3 (Step ). As a consequence, we have for every h 3 a construction of a connected 4-valent graph admitting a 1 -arc-transitive group action with the respective stabilizer isomorphic to the group D 8 Z h;3, giving us a family of such graphs with arbitrary large nonabelian vertex stabilizers. For the rest of the paper by a suborbit we shall always mean a connected suborbit. The structure of point stabilizers An ordered pair of groups (G H) where H is a subgroup of G is said to be a ~ -creator (or ~ -creative) provided the following four conditions are satised. (A1) there are 1 H and a b G n H such that G = ha bi = ha 1 i = hb 1 i (A) H = h 1 ::: h i for some positive integer h, where i = a i b ;1 a ;i+1 is an involution for each i =1 ::: h (A3) H i j = h i ::: j i,1 i j h, has cardinality j;i+1 (A4) H is not normal in G. The following characterization theorem showing the correspondence between ~-creative pairs of groups and transitive permutation groups having a non-self-paired suborbit of length is proved in [7, Theorem 3.1]. Theorem.1 A pair of abstract groups (G H), where H G and G is not dihedral, is a ~-creator if and only if the action of G on the set of (left) cosets of H has a connected non-self-paired suborbit of length. 5

7 We say that a group H = h 1 ::: h i is concentric if it satises the following two conditions. (C1) the cardinality ofh i j = h i ::: j i is j;i+1 for j >i (C) there exists a rotary group isomorphism ' : H 1 h;1! H h such that '( i )= i+1 for i =1 ::: h; 1. (The name concentric is supposed to reect the fact that, in view of Lemma. below, concentric groups have a \large" center, generated by the middle section of the generators 1 ::: h.) Clearly, by (C1) each i is an involution. Moreover, it follows from (C1) that every element x H can be expressed uniquelly as a product x = x 1 1 x x h, where h x i Z for i =1 ::: h. Such an expression of x will be called the canonical form of x. Note that if (G H) is a ~ -creative pair of groups then H is concentric. (In general we do not know if the converse holds.) A positive integer d is a noncommuting distance H = h 1 ::: h i, if for each i = 1 ::: h, the elements i and i+d do not commute. Clearly, because of the \rotary" property (C) all possible noncommuting distances d appear if we consider just pairs of generators 1 and d+1. Let Dist(H) = (d 1 d ::: d k ) denote the sequence of all of noncommuting distances in H listed in the increasing order. The shortest noncommuting distance d 1 will be called the diameter of H, denoted by diam(h). If all the generators commute (that is, if the group H is elementary abelian) we set diam(h) = h. In other words, d = diam(h) isthemaximum subscript such that 1 d is an involution, that is, d ; 1 is the maximum commuting distance. Furthermore, let Z(H) denote the center of H. Lemma. (The Swinging Lemma). Let H = h 1 ::: h i be a nonabelian concentric group of order h and diameter d = diam(h). Then ( i i+d ) H h;d+i d;h+i Z(H), for i f1 ::: h; dg. In particular, d 3 h. Proof. In view of the \rotary"property (C) it suces to prove the statement for i =1. Let x =( 1 d+1 ) and m = h ; d. We use induction on m. Let m =1. Since, by (C1), both H 1 h;1 and H h are index subgroups of H, we have x =( 1 d+1 ) =( 1 h ) H 1 h;1 \ H h = H h;1 = H d. Suppose the statement holds for concentric groups with h;d = m;1. Let H be a concentric group satisfying h ; d = m. By the induction hypothesis 6

8 Figure 1: The Swinging Lemma. the statement holds for the subgroup H 1 h;1. Consequently, x H m d;m+. We need to see that x H m+1 d;m+1. Let z = ( 1 h ). Then z H h;1 H 1 h;1,by (C1). By induction hypothesis x is a central element ofh 1 h;1 and consequently, x and z commute. But z commutes with d+1, too. Namely, by induction hypothesis we have d (h;1) and so h;1 < d+1 < d Hence any j H h;1 commutes with d+1 for j d in view of the deniton of d, forcing z H h;1 to commute with d+1. Using the above facts about z, x and d+1, we have, 1 zx = 1 xz = d+1 1 d+1 z = d+1 1 z d+1 = d+1 h 1 h d+1 = h d+1 1 d+1 h = h 1 x h = 1 z h x h forcing x = h x h. Thus x centralizes h. On the other hand, since d is the diameter of H it follows that x centralizes 1. Thus x Z(H). Recall that, by induction hypothesis, x H m d;m+. But m h is a noncommuting pair of involutions. Thus x = m H m+1 d;m+ and so x H m+1 d;m+. Similarly, since 1 d;m+ is a noncommuting pair, we deduce that x H 1 d;m+1. It folows that x H m+1 d;m+1 = H h;d+1 d;h+1 Z(H). Furthermore, since x 6= 1 the subgroup H h;d+1 d;h+1 is nontrivial. Therefore h ; d +1 d ; h +1,implying that d h. 3 Lemma.3 Let H be a concentric group of order h and diameter d = diam(h). Then H h;d+1 d = Z(H) is the center of H. 7

9 Proof. By the Swinging Lemma we already know that H h;d+1 d Z(H). Let us assume, for the contrary, that there exist a central element x = H h;d+1 d. Let us consider the canonical form x = x 1 1 x ::: x h h of x, where x i f0 1g. There exists i h ; d or j d + 1 such that either x i =1 or x j =1. If x i =1wetake i to be the minimum subscript with this property. By the Swinging Lemma i+d commutes with all k for k fh ; d +1 :::hg. By the choice of i it commutes with all k for k fi +1 ::: h; dg. Thus i+d x 6= x i+d. If x j =1(for some j d +1)we take j to be the maximum subscipt with that property. Using a similar argument as in the previous case we deduce that j;d x 6= x j;d. We have thus derived a contradiction in both cases, and consequently Z(H) =H h;d+1 d. The number of dierent noncommuting distances in a concentric group H will be called the complexity of the group H with respect to the set of generators f 1 ::: h g. Corollary.4 A concentric nonabelian group H = h 1 ::: h i of order h, diameter d and complexity 1 is isomorphic to the direct product D h;d 8 Z 3d;h. Conversely, if H = D h;d 8 Z 3d;h, where d h, then there exists 3 a set of involutory generators f 1 ::: h g of H with respect to which H is concentric with diameter d and complexity 1. Proof. To prove the rst part note that K i = h i i+d i = D 8 for i =1 ::: h; d. Besides K i \ K j = 1 and x K i commutes with y K j for i 6= j. The result follows. As for the second part, it follows from the assumptions that H contains h;d disjoint copies K i of D 8 generated by pairs of involutions, say i and i. Furthermore let i be the generators of the 3d;h copies of Z in H. Dene i = i for i =1 ::: h; d, h;d+i = i for i =1 ::: 3d ; h, and d+i = i for i =1 ::: h;d. Since all the noncommuting pairs of the generators i are at distance d, the group H is concentric having diameter d and complexity 1 with respect to the set f 1 ::: h g. Let us denote by H 7 the group with 18 elements generated by 7 involutions! i, i f1 ::: 7g such that the following irreducible relations hold: (i) (! i! j ) =1ifji ; jj 4 8

10 (ii) (! 1! 6 ) =! 3 (iii) (!! 7 ) =! 4 (iv) (! 1! 7 ) =! 5. It is not dicult to see that H 7 is uniquelly determined by the above relations (up to group isomorphism). Lemma.5 Let H = h 1 ::: h i be a concentric group of order h, where h 6 and with the distance vector Dist(H) =(h ; h; 1). Then ( i j ) H 3 h;3 = Z(H) for all 1 i j h. Moreover, H is isomorphic to D 8 D 8 if h =6, and to one of the groups D 8 or H 7 Z h;7 if h 7. Proof. First, we note that by Lemma.3, Z(H) = H 3 h;. Let z 1 = ( 1 h;1 ), z = ( h ) and z 3 = ( 1 h ). The Swinging Lemma implies that z 1 H 3 h;3 Z(H), z H 4 h; Z(H) and z = '(z 1 ), where ' : H 1 h;1! H h is the rotary isomorphism existing by (C). Moreover, by (C1), z 3 H h;1 which is an elementary abelian group. Thus z 3 centralizes any j for j h ; 1. Since, z 3 is the central element of the dihedral group generated by 1 and h, it follows that z 3 Z(H) =H 3 h;. There are two possibilities for the group Z 0 = hz 1 z z 3 iz(h). Either Z 0 = hz 1 z i or Z 0 6= hz 1 z i. Thus either Z 0 = Z or Z 0 = Z 3. In the rst case z 3 = z 1 1 z where 1 f0 1g. Replacing 1 by 0 1 = 1 z 1 1 and h by h 0 = h z we get a new generating set h 0 1 ::: h;1 hi 0 where the only noncommuting pairs are ( 0 1 h;1) = z 1 and ( h) 0 = z. This canbeeasilyveried using the fact that z 1, z and z 3 are central elements of the group. Since, z 1 6= z it follows that H = D 8. If Z 0 6= hz 1 z i, and consequently Z 0 = Z 3,wehave the isomorphism H = h 1 h;1 h i Z h;7. Now the assignment 1 7!! 1, 7!!, h;1 7!! 6, h 7!! 7 extends to a group isomorphism h 1 h;1 h i!h 7. Let us remark that the groups D 8, H 7 Z h;7 are nonisomorphic for any h 7. Both have centers isomorphic to Z h;4, however the subgroup Z 0 (H) of the center of minimum order and such that the factor group H=Z 0 (H) is elementary abelian, has order 4 for H = D 8, while Z 0 (H) has order 8 for H = H 7 Z h;7. Using the same argument we can even deduce that H 7 cannot be a subgroup of the direct product D 8 9.

11 3 Point stabilizers of order h h 8 Using the lemmas of the previous section we are now able to give a restricted list of possible point stabilizers of order h 8 for transitive permutation groups having a non-self-paired suborbit of length. Theorem 3.1 Let G be a transitive permutation group having a (connected) non-self-paired suborbit of length and let H be a point stabilizer of order h h 8. Then H is isomorphic to one of the following groups: (i) Z h for 1 h 8 (ii) D 8 Z h;3 for 3 h 8 (iii) D 8 for 6 h 8 (iv) H 7 Z h;7 for h f7 8g. Proof. By Theorem.1, H is a concentric group. If H is abelian it must be elementary abelian (see Section 1 and also [7, Theorem 4.1]). Assume H is nonabelian. By the Swinging Lemma we have that diam(h) 3 h, and consequently, the complexity of H is either 1 or and if it is then the distance vector Dist(H) = (h ; h; 1). Now the statement follows combining together Corollary.4 and Lemma.5. It follows from Theorem 3.1 that concentric groups of order h h 8 split into four families characterized by the nonabelian factors in their direct product decomposition. Constructions of ~-creators of the form (G Z h ) are known for every h 1 (see [7]). In fact an even stronger result holds: for every h 1 there are innitely many 1 -arc-transitive graphs whose vertex stabilizer is isomorphic to Z h see [6]. Point stabilizers isomorphic to the direct products of one copy ofd 8 with an elementary abelian group are investigated in the subsequent section where ~-creators of the form (G D 8 Z h ) are given. As concerns the other two families, let us remark that we know no example of a transitive permutation group having a non-self-paired suborbit of length whose point stabilizer is neither elementary abelian nor a product of D 8 with an elementary abelian group. However, checking all possible relations between the involutory generators of H, a list of h relations in terms of the generators a, b of the group G can be produced (see part (A1) in the 10

12 deniton of a ~-creative pair of groups for the deniton of a and b). To nd an imbedding of a particular concentric group H into a (nite) group G such that the pair is ~ -creative, it amounts to nding a nite quotient of the group determined by the above mentioned list of relations in which the relations are irreducible. To be more precise we have the following characterization theorem. Theorem 3. Let G be agroup. Given elements a, b of G let i = a i b ;1 a ;i+1 for each i 1. The following statements hold: (i) A pair of groups (G H), where H = Z h, is a ~ -creator if and only if there are elements a b G such that G = ha bi, (a i b ;i ) = 1 for i = 1 ::: h are irreducible relations and a h a ;1 = H = hab ;1 ::: a h b ;h i (ii) Apair of groups (G H), where H = D 8 Z h;3, h 3, isa ~ -creator if and only if there are elements a b G and h 3 such that G = ha bi, and (a i b ;i ) = 1 for i = 1 ::: h ; 1, ( 1 h ) = z for some nontrivial z h ::: h;1 i are irreducible relations, and a h a ;1 = H = h 1 ::: h i (iii) Apair of groups (G H), where H = D 8, h 6, isa~-creator if and only if there are elements a b G and h 6 such that G = ha bi, and (a i b ;i ) = 1 for i = 1 ::: h;, ( 1 h;1 ) = z 1 for some nontrivial z 1 h 3 ::: h;3 i, ( 1 h ) = z for some z hz 1 az 1 a ;1 i are irreducible relations, and a h a ;1 = H = h 1 ::: h i (iv) A pair of groups (G H), where H = H 7 Z h;7, h 7, is a ~-creator if and only if there are elements a b G and h 7 such that G = ha bi, and (a i b ;i ) = 1 for i = 1 ::: h;, ( 1 h;1 ) = z 1 for some non-trivial z 1 h 3 ::: h;3 i, ( 1 h ) = z for some non-trivial z h 3 ::: h; inhz 1 az 1 a ;1 i are irreducible relations, and a h a ;1 = H = h 1 ::: h i. Proof. Let us note that part (i) of the theorem was proved in [7, Theorem 4.1]. The arguments used in the proof of each of (ii) -(iv) are similar. We show here only the proof of part (iii). Assume that the pair (G H) isa~-creator and that H = D 8. Let d = diam(h). We rst prove that d = h ;. If d h ; 1 then either H 11

13 is elementary abelian or, by Corollary.4, H is isomorphic to D 8 Z h;3. Assume d < h ;. The complexity of the subgroup H 1 d+ H is either 1 or. If it is 1 then H 1 d+ = D 8 Z d;4, by Corollary.4. Thus, in view of the assumption on H, we must have H = H 1 d+ H d+3 h. But d+3 does not commute with 3 H 1 d+, a contradiction. If the complexity of H 1 d+ is, then we have Dist(H 1 d+ ) = (d d +1) for the distance vector. Now Lemma.5 applies and we deduce that either H 1 d+ = D 8 Z d;4, or H 1 d+ = H7 Z d;5. The latter case is impossible since H 7 cannot be a subgroup of D 8 (as was remarked at the end of Section ). If H 1 d+ = D 8 Z d;4, then as above we derive a contradiction from the fact that d+3 and 3 H 1 d+ do not commute. Wehave therefore proved that the diameter of H is h ;. Now the relations (a i b ;i ) =1fori =1 ::: h; follow from part (i) and the fact that H 1 h; is an abelian group. The Swinging Lemma gives us ( 1 h;1 ) = z 1 H 3 h;3. As for ( 1 h ) = z the analysis goes as in the proof of Lemma.5. First, we deduce that z H 3 h; = Z(H). Second, either z hz 1 az 1 a ;1 i or z = hz 1 az 1 a ;1 i. But the second case yields H 7 H, a contradiction. We have thus established that G satises the required set of relations. Finally, assume for the contrary that a h a ;1 H. Since ah 1 h;1 a ;1 = H h this means that a normalizes H. But G = ha Hi forcing H to be a normal subgroup of G, acontradiction. Now let us assume that G satises the required conditions. We are going to prove that the pair (G H) is a ~-creator and that H = D 8. The relations together with part (i) imply that H = h 1 h;1 h i. The analysis of the relations gives h 1 h;1 h i = D 8 D 8. We conclude that H = D 8. Of course, the properties (A1) and (A) are satised by the pair of groups (G H). Combining the facts that H 1 j is elementary abelian for j h; and that H 1 h;1 is a -extension of H 1 h; (for ( 1 h;1 ) H 1 h; ), it follows that the order of H 1 j is j if j h;1. We already know that jhj = h. Since H i j = a i;1 H 1 j;i+1 a ;i+1,wegetjh i j j = j;i+1, and consequently, (A3) holds. Finally, the condition a h a ;1 = H implies that H is not a normal subgroup of G, thus (A4) holds too. The table below gives a set of possible relations of the group G = ha bi for each of the four families of concentric groups H G in Theorem 3.. Note that they arise from a particular choice of the elements z, z 1 and z. 1

14 TABLE 1: Possible relations for G = ha bi with (G H) a ~ -creator and jhj 8. stabilizer H relations for G = ha bi Z h (a i b ;i ) = 1 for i =1 ::: h D 8 Z h;3 (a i b ;i ) = 1 for i =1 ::: h; 1, a h b ;h a h b ;1 a ;1 b ;h+ =1 D 8 Zh;6 (a i b ;i ) = 1 for i =1 :::h; h, a h;1 b ;h+1 a h;1 b ;1 a ;1 b ;h+3 =1 (a i b ;i ) = 1 for i =1 ::: h;, H 7 Z h;7 a h;1 b ;h+1 a h;1 b ; a ;1 b ;h+4 =1, (ab ;1 a h b ;1 a ;h+1 ) a 4 ba ;5 =1 If h is not too large it is not dicult to nd all possible sets of relations depending on the choices of the elements z, z 1 and z in Theorem 3.. For instance, if H = D 8 D 8 then part (iii) of Theorem 3. applies. We have no choice for z 1 = 3, while we have four choices for z = h 3 4 i. Thus we have the following corollary characterizing ~-creative pairs of the form (G D 8 D 8 ). (Note that D 8 D 8 is the smallest concentric group H for which a ~ ; (G H)-creator is not known.) Corollary 3.3 Let (G D 8 D 8 ) be a ~-creative pair of groups. Then G is a nite quotient of one of the following four groups: where G j = h a b R 1 R R 3 R 4 R 5 Rel j i j =1 3 4 (i) R i (a i b ;i ) = 1, for i = 1 3 4, and R 5 a 5 b ;1 a 5 b ; a ;1 b ; = 1, while 13

15 (ii) Rel 1 a 6 b ;6 a 6 b ; a ; b ; = 1, Rel a 6 b ;6 a 6 b ;3 a ;1 b ; = 1, Rel 3 a 6 b ;6 a 6 b ; a ;1 b ;3 =1, and Rel 4 a 6 b ;6 a 6 b ;6 =1. Moreover, any nite quotient G of one of the groups G 1 ::: G 4 for which a 7 b ;1 a ;6 = hab ;1 a b ;1 a ;1 ::: a 6 b ;1 a ;5 i and the respective relations above are irreducible, determines a ~-creative pair of groups (G D 8 D 8 ). Since we know no ~-creators of the form (G D 8 D 8 ), the groups G j, j =1 3 4, may not exist. However, if they exist one can prove that they are necessarily innite. 4 Constructing actions with point stabilizers D 8 Z h;3 Let h 3 and H = D 8 Z h;3. By Corollary.4 there exists a set of involutory generators f 1 ::: h g of H with respect to which H is concentric, of complexity 1 and diameter d = h ; 1. Note that in any representation of H as a concentric group the complexity is 1 and diameter is h ; 1. Basically, all such representations dier only in the choice of the central element z = ( 1 h ) H h;1. We are now going to prove that for any concentric representation of H there exists a supergroup G such that the pair (G H) is a ~-creator. Theorem 4.1 Let h 3 be an integer and let H = D 8 Z h;3. Then for any concentric representation of H there exists a supergroup G such that the pair (G H) is a ~-creator. Proof. In view of Theorem.1 it is sucient to nd a supergroup G H for H such that(g H) isa ~ -creative pair of groups. In what follows we assume that H = h 1 ::: h i = D 8 Z h;3, where all i are involutions, and i j = j i unless jj;ij = h;1. Since 1 and h are involutory generators of a dihedral subgroup of order 8, there is an involution z H h;1 such that ( 1 h ) = z. Since every element x of the concentric group H = h 1 ::: h i can be expressed in a unique way in the canonical form x = x 1 1 x x h h, where x i Z we may think of the group H as a "sqewed" vector space V = Z h (so to speak) with the unit vectors e i as generators. Note, however that the binary operation induced by H is not commutative inthiscase. 14

16 We now dene a regular permutation representation H of H on V by rst taking the right regular representation of H (on itself) and second, by making an appropriate identication of H with V given via the above canonical form of its elements. In view of the dening relations for H we have that i (x) = x x h z if i = 1 and i (x) = x + i for i = ::: h, where i H denotes the permutation acting on V, corresponding to i H. Now we nd an appropriate permutation ' of V such that the pair of groups (G H) is a ~ -creator, where G = h' Hi. To this end let ' act by the following rule '(x) = R(x + x 1 x h z), where R is the linear transformation cyclically permuting the unit vectors, that is, R(e i ) = e i+1, and x =(x 1 x ::: x h ), x i Z. It is easy to verify that ' is injective. To see that G = h 1 'i gives rise to a ~-creative pair (G H) of groups, it is sucient to prove that '( i )' ;1 = i+1 for i f1 ::: h; 1g and that '( h )' ;1 = H. We distinguish two cases. Case 1: i h ; 1. We have by computation, ' i (x) ='(x + e i )=Rx + x 1 x h Rz + e i+1 = i+1 (R(x + x 1 x h z)) = i+1 '(x): Case : i =1. In this case the denition of 1 implies ' 1 (x) ='(x + e 1 + x h z)=rx + e + x h Rz +(x 1 +1)x h Rz = = Rx + e + x 1 x h Rz = (Rx + x 1 x h Rz) = '(x): Thus the conjugation by ' sends H1 h;1 onto H h. We complete the proof by checking that '( h )' ;1 = H. Assume for the contrary that '(h )' ;1 H. Since ' h ' ;1 (0) = e 1 = 1 (0), the regularity of the action of H on V forces '( h )' ;1 = 1, which is equivalent to the identity ' h = 1 '. However, ' h (e h;1 )=e 1 + e h and 1 '(e h;1 )=e 1 + e h + z, acontradiction. By restating the above theorem in graph-theoretic terms we have, in view of Theorem.1, the following corollary. Corollary 4. Let h 3 be an integer. Then there exist a connected 4- valent graph X and a group G Aut X such that X is (G 1 )-arc-transitive with vertex stabilizer isomorphic to D 8 Z h;3 ). 15

17 Remark 4.3 The binary operation on vectors considered above can be de- ned by means of h Boolean functions in variables x 1 ::: x h y 1 ::: y h, for each coordinate of x y we have one function. Thus the problem of characterization of concentric groups is related to the problem which sets of h Boolean functions of h variables dene a binary group operation on vectors over Z. For instance, for h =3the respective binary operation used in the above proof is (x 1 x x 3 ) (y 1 y y 3 )=(x 1 + y 1 x + y + x 3 y 1 x 3 + y 3 ). References [1] N. Biggs and A. T. White, Permutation groups and combinatorial structures, Cambridge University Press, [] A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier, New York, [3] J. D. Dixon and B. Mortimer, \Permutation groups", Springer-Verlag, New York, [4] A. Malnic, D. Marusic, Constructing 1 -transitive graphs of valency 4 and vertex stabilizer Z Z, submitted. [5] D. Marusic, Recent developments in half-transitive graphs, Discrete Math., 18 (1998), 19{31. [6] D. Marusic, Constructing half-arc-transitive graphs of valency 4 with large vertex stabilizers, submitted. [7] D. Marusic, R. Nedela, Transitive permutation groups with non-self-paired suborbits of length and their graphs, submitted. [8] P. M. Neumann, Finite permutation groups, edge-coloured graphs and matrices, \Topics in Group Theory and Computation", (Proc. Summer School, University Coll., Galway, 1973), 8{118, Academic Press, London, [9] C. C. Sims, Graphs and nite permutation groups, II, Math. Zeitschr. 103 (1968), 76{81. [10] D. E. Taylor, M. Y. Xu, Vertex-primitive 1 -transitive graphs, J. Austral. Math. Soc. Ser. A 57 (1994), 113{14. [11] W. T. Tutte, A family of cubical graphs, Proc. Camb. Phil. Soc. 43 (1948), 459{

18 [1] R. J. Wang, Half-transitive graphs of order a product of two distinct primes, Comm. Algebra (1994), 915{97. [13] H. Wielandt, \Finite Permutation Groups", Academic Press, New York, [14] W. J. Wong, Determination of a class of primitive permutation groups, Math. Zeitschr. 99 (1967), 35{46. 17