JOURNAL OF ALGEBRA 185, 258᎐265Ž. 1996 ARTICLE NO. 0324
Representing the Sporadic Groups as Noncentral Automorphisms of p-Groups
Panagiotis C. Soules
Department of Mathematics, Uni¨ersity of Athens, Athens, Greece
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Department of Mathematical Sciences, Villano¨a Uni¨ersity, Villano¨a, Pennsyl¨ania 19085
Communicated by Gordon James
Received February 23, 1996
DEDICATED TO H. HEINEKEN ON THE OCCASION OF HIS 60TH BIRTHDAY
In Heineken and Liebeck w Arch. Math. 25 Ž.1974 , 8᎐16x it is proved that for every finite group K and odd prime p, there exists a p-group G of nilpotence class 2 and exponent p2 on which K acts as the full group of noncentral automorphisms. The authors investigate this problem under the additional requirement that GrZGŽ.be representation space for the p-modular regular representation of K. In particular, they prove that every sporadic simple group K can be so represented for any odd prime p. ᮊ 1996 Academic Press, Inc.
1. INTRODUCTION
Inwx 6 Heineken and Liebeck ask which finite groups can occur as the full automorphism group AutŽ.G of some nilpotent group G of class 2. Their motivation stems from the well known fact that not every finite group can be so realized when G is abelian.Ž For example, when G is abelian of order G 3 an obvious requirement is that AutŽ.G have even order..Ž. As they next observe, the structural requirements on Aut G when
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0021-8693r96 $18.00 Copyright ᮊ 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. REPRESENTING SPORADIC GROUPS 259
G is nilpotent of class 2 are not much less stringent than those correspond- ing to the abelian case. For example, in this case AutŽ.G must contain a noncyclic abelian normal subgroup consisting of all inner automorphisms of G. The situation changes dramatically if one passes to the quotient group Ž. Ž. Aug G rAut c G , i.e., the group of so-called ‘‘noncentral’’ automorphisms ŽŽ. Ž. of G. Here Aut c G denotes the kernel of the homomorphism Aut G ª AutŽŽ..GrZG , thus consisting of all automorphisms of G which induce the trivial automorphism on GrZGŽ... Indeed, Heineken and Liebeck were able to prove that every finite group could be so represented. In their proof they use graph theory in a novel way. Namely, starting from an arbitrary finite group K they construct a directed graph DKŽ., the full automorphism group of which is K. They next define, in terms of genera- tors and relationships, a p-group G of nilpotence class 2. Moreover, the manner in which G is defined allows one to readily verify that Ž. Ž. Ž. Aut G rAut c G is the full automorphism group of DK as well, thus Ž. Ž. providing the desired isomorphism K ( Aut G rAut c G . If there is a weakness to their solution it is the sheer enormity of the representing group G. Indeed, in order to select an appropriate digraph
DKŽ.they start from a Cayley color graph ⌫C Ž.K, S and then introduce so-called simple points as a device for distinguishing between color classes once these classes have been removed. As the rank of GrZGŽ.coincides with the number of vertices of DKŽ.Žconsisting both of simple points and the extant group points. , this rank can be quite large. As an example, their construction of digraph DSŽ.33, where S is the symmetric group on three letters, requires the addition of 18 simple points, so a total of 24 points in all. This leads to a p-group G for which the rank of GrZGŽ.equals 24ᎏfar greater than the desired rank of 6. As previously mentioned, our aim is to investigate the case where the rank of GrZGŽ.equals < 2. THE DIGRAPH DKŽ. Given a group K and generating set S of K Ž²:.so that K s S , we form the corresponding Cayley color graph ⌫C Ž.K, S as follows. The vertices of 260 SOULES AND WOLDAR ⌫C Ž.K,Sare the elements of K, and the arcs Ž directed edges . of ⌫C Ž.K, S are the ordered pairs Ž.x, y for which y s xs for some s g S. Moreover, we color all arcs of the form Ž.x, xs with the color s. Ž Arcs of the same color form what we refer to as a color class of ⌫C Ž..K, S . If color is suppressed in ⌫C Ž.K, S , we obtain what is called the Cayley digraph ⌫DŽ.K,S, and if orientation of arcs is suppressed in addition to color, the result is the Cayley graph ⌫Ž.K, S . It is well known that AutŽŽ..⌫C K, S is isomorphic to K and that the respective automorphism groups of these graphs embed as follows: AutŽ.Ž.⌫CDŽ.K , S F Aut ⌫ Ž.K , S F AutŽ.⌫ Ž.K , S . The disparity between these groups can be quite large; for example, if K is any group of order n and S consists of all nonidentity elements of K, then Ž Ž .. Ž Ž .. Ž . Ž . the index of Aut ⌫C K, S in Aut ⌫ K, S is n y 1 ! since ⌫ K, S is here the complete graph on n vertices, which has automorphism group Sn. As alluded to in the Introduction, what we would like to avoid is the necessity to add vertices to a Cayley digraph in order to ‘‘codify’’ its absentee color classes. It turns out this can be done if the choice of generating set is rather special. Our first result, which is a strengthened version of part of Lemma 3.1.1 fromwx 8 , illustrates a particular case of this. LEMMA 2.1. Let K be a finite group generated by two elements x, y with orders satisfying oŽ. x s 5 / oy Ž..Then K is isomorphic to the full automor- Ž Ä4. phism group of its Cayley digraph ⌫D K, x, y pro¨idedwx x, y / 1. In particular, the result obtains for any nonabelian group so generated. Proof. As K is clearly isomorphic to the automorphism group of its Cayley color graph ⌫C Ž K, Ä4x, y ., it suffices to show that arcs of color x can be distinguished from those of color y in ⌫DŽ K, Ä4x, y .. We do this by establishing that the only arcs that occur in directed 5-cycles of this graph are those of color x. Clearly, arcs of color x occur in directed 5-cycles by virtue of oxŽ.s5. Any other such cycle would correspond to a length-5 word in the genera- tors, distinct from x 5, which represents the identity element 1 of K. Thus we look at all possible length-5 words. It is easily seen that these fall into 5 ii 2 seven conjugate classes, with representatives xyy Ž.1FiF5,xyxy , and 2 yxyx .AsoyŽ./5 we can eliminate the case i s 5, while for 1 F i F 4, 5 ii 5i xyy s1 clearly implies wx y, yxwxs 1, but then x, y s 1 since 5 y i and 5 are relatively primeᎏa contradiction. Our final two cases are symmetric 2 1 so we treat only the first of these. Suppose xyxy s 1. Then xy yx s 12 211 Ž.Žxyxy xy x .Žs yyy .yysyᎏour final contradiction. REPRESENTING SPORADIC GROUPS 261 As will become apparent in future sections, our choice of digraph will Ž. Ž Ä4. always be of the form DK s⌫D K, x,y for appropriate x and y. 3. PASSAGE TO p-GROUPS In this section we briefly describe the construction introduced inwx 6 of the p-group G s GDŽ.associated with a digraph D. Under certain restrictions on Dᎏmost notably that it be strongly connected, have minimum degree at least 2, and have no undirected cycles of size 4 or less Ž. Ž. Ž. ᎏit is established that Aut D ( Aut G rAut c G . We shall not present details of the proofs here. The interested reader is referred to the articles wx6 and wx 8 ; see also w 10 x for a modified construction that includes the case p s 2. The Construction of G() D Let D be a digraph with vertices P12, P ,...,Pnand let p be an odd prime. We define G s GDŽ.to be the group with canonical generators p X X12, X ,..., Xn, with G : G such that 1. GX is the elementary abelian p-group freely generated by Ä4Xij, X ,1Fi-jFn. 2. For every i,1 i n, given that Ä4P , P ,...,P is the complete F F ii12 ik set of terminal vertices of arcs having initial vertex Pi, then the following defining relations holds in G: X p X , XX иии X . iiiiis 12 k Ž. Ž. Remark. That such G can be constructed for which Aut G rAut c G Ž. Ž (Aut D depends on the observation that the map Pii¬ P a permutation ofÄ4 1, 2, . . . , n .is an automorphism of D if and only if the map Xii¬ X extends to an automorphism of G. A determination of conditions on D under which this is ensured comprises a major part of the work done inwx 6 . Our next result describes conditions under which D can be chosen as a Cayley digraph. A proof can be found inwx 8 . LEMMA 3.1. Let K be a finite group generated by elements x, y with Ž. Ž. Ž Ä4. oxs5and o y ) 5 and let ⌫D K, x, y be its corresponding Cayley Ž. Ž. Ž Ž Ä4.. digraph. Then, with G as abo¨e, Aut G rAut cDG ( Aut ⌫ K, x, y 262 SOULES AND WOLDAR pro¨ided the following hold: 1. w x, y 2 x / 1, 2. Ž.xy 2/ 1, 3. Ž xyy1 .2 / 1. LEMMA 3.2. Let K be a centerless groupŽŽ.. i.e., ZK s1 generated by elements x and y with oŽ. y ) 2. Thenw x, y 2x/ 1. In particular, if group K of 2 Lemma 3.1 is centerless, the result of the lemma obtains pro¨idedŽ. xy / 1 andŽ xyy1 .2 / 1. 22 Proof. If wxx, y s 1, then y g ZKŽ.s1ᎏa contradiction. 4. MAIN RESULT LEMMA 4.1. Let K be a group generated by elements x, y with oŽ. x odd, oŽ. x not di¨isible by 3, and o Ž. y ) 2. Then there exists an element t g K for 1 which the following hold:iŽ.ot Ž .sox Ž .,ii Ž .oty Ž .)2, Ž iii .oty Ž y .)2, and Ž.iv K s ²t, y :. Proof. We show that t can be chosen as one of x, x 2,or x3. Obviously, 1 if oxyŽ.)2 and oxy Žy.)2, simply choose t s x. So we assume that 2 12 either Ž.xy s 1orŽxyy.s 1. 2 2 Case 1. Ž.xy s 1. We first show that oxyŽ .)2. Suppose not, so 2 2 1 1 2 1 1 2 1 1 1 Ž xy. s1. Then yy s xyx s xxyxy Ž . sxyxxy Ž y y . sxyxy y y 2 syᎏa contradiction to oyŽ.)2. Thus we may choose t s x provided 2 1 oxyŽ y .Ž)2. Indeed as ox.is odd, one immediately has otŽ.soxŽ.and ²:t,ysKfor this choice of t. 2 1 23 So assume Ž xyy . s1. We claim that in this case t s x works. For this purpose it suffices to establish oxyŽ 3 .Ž)2 and oxy3y1.)2, since 3 3 both oxŽ .Ž.²sox and x , y:s k follow from our assumption that 3 does not divide oxŽ.. 3 2 1 33 2 Assume first that Žxy. s1. Then yxy y sxysxxyŽ.s 2 1 1 2 1 1 2 1 3 1 xyxŽy y.Žsxyy. xysŽyxy. xys yxy whence y s yy, contra- 3 1 2 3 diction. Next assume that Ž xyy . s1. Then, similar to the above, yxy 3 1 2 1 2 2 1 1 2 1 3 sxyy sxxyŽ y .Žsxyxy.Ž.s xy xys Žyxyy. xysyxyy 1 whence the contradiction y s yy is again reached. We conclude that one 2 3 of t s x , t s x works in this case. 1 2 1 1 Case 2. Ž xyy . s 1. Apply Case 1, replacing y by yy . Since oyŽ y . 1 soyŽ.and K s ²x, yy :the argument remains valid. Remark. Consideration of the dihedral group D2 n, n odd, shows that the assumption oyŽ.)2 cannot be removed from the lemma hypotheses. REPRESENTING SPORADIC GROUPS 263 Similarly, the group S3 illustrates that the assumption that oxŽ.not be divisible by 3 cannot be completely eliminated, though it is unclear to what extent the full strength of this assumption is needed. THEOREM 4.2. Let K be a sporadic simple group, n s < 5. EXPLICIT REPRESENTATIONS OF THE MATHIEU GROUPS The results ofwx 9Ž generation of sporadic groups by pairs of elements Ä4x,ywith x / 1 and y of maximum prime order. are nonconstructive, though evidence strongly supports that such pairs occur in great abun- dance for most values of x. Below we derive for each Mathieu group M, explicit generators Ä4x, y of the type required to ensure representation of M as the full group of noncentral automorphisms of an appropriate p-group. That is, we show Ms²:x,yfor certain elements x, y g M of respective orders 5 and p, where p is the maximum prime divisor of < The Group M11 ²: Fromwx 3 we have the explicit generation M11 s a, b, c , where a s Ž.0,1,2,...,10 , bsŽ.Ž.1,4,5,9,3 2,8,10,7,6 , csŽ.Ž.1,5,4,3 2,6,10,7 . 264 SOULES AND WOLDAR 1 1 1 Set x s Žaby c.y bŽ aby c.s Ž0, 6, 7, 9, 4 .Ž 2, 8, 5, 10, 3 . and y s a. The or- der of group ²:x, y must certainly be divisible by 5 and 11, but also 8 and 3 2 since xy s Ž.Ž.0, 7, 10, 4, 1, 2, 9, 5 6, 8 and xy s Ž.Ž0, 8, 7 1, 3, 4, 2, 10, 5 .Ž. 6, 9 . ²: ²: This proves < M11: x, y The Group M12 A presentation for M12 was obtained by Leechwx 7 in terms of the two generators a s Ž.2,3,4,...,12 , bsŽ.Ž.Ž.Ž.Ž.Ž.12, 3 11, 6 10, 9 8, 7 5, 4 2, 1 . 42 Set x s Žab.sŽ.1, 11, 10, 4, 6Ž. 2, 5, 12, 7, 9 and y s a. The only maximal subgroups of M12 of order divisible by 55 are isomorphic to either M11 or L2Ž.11 , and neither of these groups has an element of order 10. As y1 Ž.Ž.²: xy s 1, 10, 3, 2, 4, 5, 11, 9, 12, 6 7, 8 we conclude that x, y s M12 ,as desired. The Group M22 We use generators from Janko’s presentationŽ seewx 5. : a s Ž.1,2,3,...,11Ž. 12,13,14,...,22 , bsŽ.1, 4, 5, 9, 3Ž. 2, 8, 10, 7, 6Ž. 12, 15, 16, 20, 14Ž. 13, 19, 21, 18, 17 , csŽ.Ž.Ž.Ž11,22 8,14 4,5,3,9 13,18,17,19 .Ž.Ž. 2,16,10,15 7,20,6,12 . a Set x s b s Ž.Ž.Ž.Ž.2, 5, 6, 10, 4 3, 9, 11, 8, 7 13, 16, 17, 21, 15 14, 20, 22, 19, 18 c and y s a s Ž.1, 16, 9, 5, 3, 12, 20, 14, 4, 15, 22Ž 7, 18, 8, 2, 10, 19, 17, 13, 6, 21, 11. . The only maximal subgroup of M22 with order divisible by 55 is Ž. Ž. Ž. isomorphic to L2211 . As oxys7 and L 11 has no elements of that ²: order, we conclude that x, y s M22 . The Group M23 We use two of three generators that occur in a presentation due to Fryerwx 4 : a s Ž.1,2,3,...,23 , csŽ.3, 17, 10, 7, 9Ž. 5, 4, 13, 14, 19Ž. 8, 18, 11, 12, 23Ž. 15, 20, 22, 21, 16 . Here simply choose x s c and y s a. As the only maximal subgroup of Ž² :. M23 that contains y is its normalizer Ny(23:11, we immediately see ²: that M23 s x, y . REPRESENTING SPORADIC GROUPS 265 The Group M24 Inwx 1 Conway describes M24 as a four-generator group, three generators of which we list for use below: a s Ž.1,2,3,...,23 , csŽ.Ž.Ž.Ž.Ž.Ž.1, 24 4, 16 8, 14 15, 19 6, 10 11, 17 Ž.Ž.Ž.Ž.Ž.Ž.9, 21 5, 18 3, 12 2, 23 13, 22 7, 20 , dsŽ.15, 18, 12, 20, 23Ž. 21, 11, 8, 6, 22Ž. 19, 5, 3, 7, 2Ž. 9, 17, 14, 10, 13 . ac Set x s d s Ž.2, 13, 3, 21, 20Ž. 4, 15, 22, 9, 24Ž. 5, 19, 17, 8, 6Ž. 7, 10, 16, 14, 12 and y s a. The only maximal subgroups of M24 of order divisible by 5 и 23 are isomorphic to M23 , but all such groups are intransitive in their natural representation on 24 letters. As ²:x, y is clearly transitive, we conclude ²: that x, y s M24 . ACKNOWLEDGMENTS This research was conducted at the University of Athens. The author A. Woldar wishes to express his gratitude for the support and hospitality extended him during his stay at the University. 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