Probabilistic Generation of Finite Simple Groups 1

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1 Journal of Algebra 3, 7379 Ž 000 doi:101006jabr , available online at on Probabilistic eneration of Finite Siple roups 1 Robert M uralnic Departent of Matheatics, Uniersity of Southern California, Los Angeles, California E-ail: guralnic@athuscedu and Willia M Kantor Departent of Matheatics, Uniersity of Oregon, Eugene, Oregon E-ail: antor@athuoregonedu Counicated by Alexander Lubotzy Received January 10, 1997 FOR HELMUT WIELANDT ON HIS 90TH BIRTHDAY For each finite siple group there is a conjugacy class C such that each nontrivial eleent of generates together with any of ore than 110 of the ebers of C Precise asyptotic results are obtained for the probability iplicit in this assertion Siilar results are obtained for alost siple groups 000 Acadeic Press 1 INTRODUCTION It is well nown that any finite siple group can be generated by two eleents In fact, the probability that two eleents generate approaches 1 as Di,KaLu,LiSh In KaLu it was ased whether one could obtain an analogous result by using any given nontrivial eleent 1 This research was supported in part by the NSF $3500 Copyright 000 by Acadeic Press All rights of reproduction in any for reserved

2 7 URALNICK AND KANTOR of together with a rando eleent and still generate with probability 1 as ; however, this is not the case KS Žfor exaple, a 3-cycle in An does not generate An together with each eleent in a large proportion of the group This paper has the sae thee, but we restrict our rando eleents to a fixed conjugacy class C, chosen so that each eleent in C is contained in very few axial subgroups Using estiates for nubers of fixed points of eleents on the cosets of these axial subgroups Žwhich have turned out to be of independent interest, we deduce estiates, whenever 1 g, for the probability that a rando eleent of C together with g generate We obtain an absolute lower bound for this probability, as well as asyptotic bounds as the siple group gets large In particular, if is a group of Lie type over a field of q eleents and q, then this probability usually tends to 1 Ža siilar but slightly different result for classical groups is given in KS We also prove an analogous result for alost siple groups In order to state our results ore precisely, we introduce soe notation For any finite group, let O Ž denote the last ter in the derived series for Let PCŽ denote the following conditional probability: ² : PCŽ ax in Pr g, s O Ž s s 1s 1g Note that, for an alost siple group with socle S, PCŽ ax in Pr² g, s: S s s Ž 1 s 1 g A group is called alost siple if it lies between S and AutŽ S for soe nonabelian finite siple group S Thus, for at least one conjugacy class C s,pc is a lower bound for the proportion of ebers of C each of which generates a subgroup containing O Ž together with any given nontrivial g Using this notation we will prove the following two theores: THEOREM I If is a finite alost siple group then PCŽ 110 THEOREM II li infpcž for alost siple li infpcž for siple 1 We will obtain ore than just the stated li infs If Ž i is any sequence of pairwise nonisoorphic groups of Lie type then li PCŽ i 1 unless Ž contains a subsequence Ž with Ž 1, q i i i i for a fixed j j j prie power q, in which case li PCŽ 1 1q The value of PCŽ A ij n is related to arithetical properties of n; we find that li inf PCŽ A l and li inf PCŽ A 3 Moreover, li inf PCŽ A l n i 1 if and only if the sallest prie dividing ni tends to infinity More precise inforation concerning PCŽ A is obtained in Section 7 Žcf Rear at n

3 PROBABILISTIC ENERATION OF ROUPS 75 the end of the section, where it is also shown that 1, 1 is the set of liit points of ŽPCŽ S n One surprising fact is that, for infinitely any n, PCŽ S n is not attained when C consists of n-cycles; instead it is attained by eleents s having two cycles of length approxiately n Soewhat siilarly, for any of the classical groups we use eleents s having two irreducible constituents on the underlying vector space, of approxiately the sae degrees In soe of these cases there is in general no single optial choice for the set of conjugates of ² s : It is natural to as whether the requireent ² g, s: O Ž in the definition of PCŽ is too wea: perhaps we should have required that ² g, s: provided that is cyclic odulo its socle S Žif S is not cyclic, then taing g S shows that not every eleent has a ate such that the pair generate However, this stronger notion would not give the desired type of li inf even when is a syetric group Žcf Rear 1 at the end of Section 7 The above theores do not iply the results in Di,KaLu,LiSh On the other hand, Theore I iplies, in particular, the following property of 1 siple groups, called 1 generation Žcf DT, Wo The property was first conjectured by Steinberg Ste in 196, who observed that his conjecture, if true, would quite liely require ethods uch ore detailed than those used here COROLLARY Any nontriial eleent of a finite alost siple group belongs to a pair of eleents generating at least the socle of This can be proved using less effort than we eploy here for Theore I In particular, if x is an eleent contained in at ost axial subgroups ² yž g of a siple group, then g, x : for any nontrivial eleent g and soe yž g Žsee, In ost cases one can produce such an eleent x The proofs of the theores rest on the classification of finite siple groups However, when is alternating or classical, a ore eleentary proof of Theore I is possible Ž but with a poorer bound ; the case PSLŽ d, q is contained in Ka3, using very eleentary ethods, while the alternating groups are dealt with in Ž 71 below All proofs of this type of result follow siilar patterns: bounding the nuber of ways not to generate by using inforation concerning axial subgroups of For any a, b write P Ž b Pr² a, b: O Ž a a a Then ² : P b P a Pr a, b O a a, b b and PCŽ a b ax in Ž1 PŽ g Thus, we focus on estiating PŽ g 1 s 1 g s s for carefully chosen s as indicated above and for all g 1 This is discussed fro a general viewpoint in Section, relating our tas to estiating fixed point ratios of eleents of perutation groups Section 3 provides soe such estiates for the classical groups, leading to a proof of the theores

4 76 URALNICK AND KANTOR for these groups in Sections and 5 Exceptional and sporadic groups are dealt with in Section 6, while the alternating and syetric groups are considered in Section 7 We have already entioned that the latter cases lead to any of the situations we encounter in which a subsequence of a sequence in Theore II converges to a liit other than 1; hence they require ore effort than ight be expected Our results on siple groups were presented at the roups and eoetries Conference in Siena in Septeber 1996 shortly before the present paper was subitted Later soe of the results were stated in H and a proof of the corollary for siple groups was given in St We are grateful to T Breuer and Malle for providing us with AP coputations, and to J Buhler, Z Reichstein, and A Shalev for helpful coents Finally, we than the referee for his helpful coents, including the suggestion that we should extend our earlier results to alost siple groups FIXED POINT RATIOS; NOTATION All groups will be finite, as will the sets on which they act For any action of a group on a set X, and for any g, consider the set Fix Ž g X of fixed points of g, and the fixed point ratios Ž g, X Fix Ž g X and X Ž, X ax Ž g, X g, g 1onX These are related to PŽ g and hence to PCŽ s as follows Let s be such that the conjugacy class s O Ž generates a subgroup containing O If g, then P g P s Ý g M g s g M M Ž s, where MŽ s is the set of all subgroups M of containing s and axial with respect to not containing O Ž Žin particular, if is siple, then MŽ s is precisely the collection of axial oergroups of s in If M denotes the conjugacy class of M MŽ s, then g, M g M g M g 1 Žwe use the fact that M N Ž M, which follows fro the axiality of ² O Ž : ² O Ž M together with our hypothesis O s M : In particular, P Ž g g, M s Ý M MŽ s Thus, it suffices to estiate Ž, X for suitable choices of X Note, however, that is a crude estiate, since it ignores the overlaps of ebers of MŽ s For any group, let Ž ax Ž, X is nontrivial and priitive on X Also, let F* Ž denote the generalized Fitting subgroup of

5 PROBABILISTIC ENERATION OF ROUPS 77 we will not define this here, but note that if is alost siple then F* Ž is the socle of The proof of Theore II for classical groups over fields of size q uses together with a difficult general upper bound for Ž obtained by Liebec and Saxl: THEOREM 3 LiSa Suppose that S F* Ž is a siple group of Lie type oer q not isoorphic to any -diensional linear group, alternating group or PSp Ž, 3 Then Ž, M 3q 910 for any axial subgroup M of In particular, Ž, M 0 as q In order to use this for Theore II we erely need to choose s so that MŽ s stays bounded as q This theore also shows that Theore I holds for groups of Lie type provided that MŽ s 1 The next section provides soe ore precise bounds for Ž, X in the case of classical groups in natural perutation actions Exceptional groups of Lie type will be dealt with in Section 6 We will use in conjunction with another siple observation: LEMMA Let A B If all -conjugates of A lying in B are B-conjugate, then A lies in N Ž A :N Ž B N Ž A conjugates of B In particular, if also B is self-noralizing, then A lies in N Ž A :N Ž A B conjugates of B Another well-nown and eleentary result about fixed points is the following: LEMMA 5 If acts transitiely on a set X and g, then Ý Fix Ž g C Ž g :C Ž g, X i H i i where H is the stabilizer of a point in X and the g are a set of representaties i for the H-conjugacy classes that intersect the -conjugacy class of g 3 SOME FIXED POINT RATIOS FOR CLASSICAL ROUPS Recall that a group is called quasisiple if it is perfect and siple odulo its center Let S be a quasisiple classical Ž linear group defined on a d-diensional vector space V over Ž or in the unitary case q q In this section we will consider groups such that S N Ž S L ŽV : for each type of group we need to consider the nuber of fixed points of an r-eleent g whose order odulo scalars is the prie r More precisely, we will provide bounds on Ž g, M for various subgroups M of On occasion we will replace by ZŽ S, which will not effect any of

6 78 URALNICK AND KANTOR our estiates Žsince ZS always acts trivially in the perutation representations we consider Soe unpublished results are nown for the groups PSLŽ d, q Sh ; for the reaining classical groups Pucontains related estiates when q is sufficiently large 31 SLŽ d, q PROPOSITION 31 Let LŽ V LŽ d, q, and let S denote the set of all -spaces of V, where 1 d Then Ž i, S q, and d1 ii, S in 1, 1q 1q 1 Proof Let g act nontrivially on S We ay assue that g has prie order odulo Z and is seisiple, unipotent, or a field autoorphis There are four cases to consider: Case A g is seisiple and acts hoogeneously on V with each irreducible subodule having diension e, where 1 e and e d Case B g is seisiple and does not act hoogeneously on V Then V V1 V for nonzero g-invariant subspaces Vi having no coon ² g: -irreducible constituent, and di V e where 1 e d Case C Case D g is unipotent g is a field autoorphis of prie order 1 d d q As usual, write S a aussian coefficient, or just when the field is evident LEMMA 3 If Case A holds, then g, S 1q Proof In this case g preserves an q e-linear structure on V and indeed with respect to this structure acts as a scalar that generates qe q Note that if g has any -diensional invariant subspace, then e and Fix Ž g S is the set of e-spaces of V viewed as a de-diensional q e-space So de d e e de e Ž e Ž d Ž g, S Ž q q 1q e q e q LEMMA 33 Assue Case B holds Then Ž a g, S q ; d1 b if 1, then g, S 1q 1q ; and Ž c if q, then Ž g, S 1

7 PROBABILISTIC ENERATION OF ROUPS 79 Proof ŽŽ a, b Note that Fix Ž g S is contained in the set of -spaces of the for X X with X V, where 1 i i in e, e d e S e Ž d e ; Ž 3 Ý j j Thus, we ay apply Ž 37 below in order to conclude that Ž a and Ž b hold Ž c Suppose that q and If g has no fixed points on V, then the arguent of the previous lea yields the result Thus, we ay assue that g is trivial on V1 or V and has no invariant 1-spaces on the other subspace Let d di C Ž g and d dig, V Ž 1 V one of these is e If 1, the nuber of invariant -spaces is d 1 1 d 1 and so Ž c holds If, then the nuber of -diensional g-invariant subspaces of V is the nuber of -diensional g-invariant subspaces of C Ž g V plus the nuber of -diensional g-invariant subspaces of g, V It is easily seen that an upper bound for the latter is d 3 where d is the diension of the subspace generated by the -diensional ² g: -irreducible subspaces Thus, the total nuber of invariant -spaces is at ost Ž d 1 1Ž d 11 d This quantity is largest when d1 d and d, and the result follows If, the result follows fro Ž a LEMMA 35 j0 Assue Case C holds Then Ž a g, S q ; and Ž b if 1, then Ž g, S 1q Proof We ay replace g by a polynoial in g and hence assue that the inial polynoial of g is Ž T 1 Let W g, V and e di W Then W C Ž g V and hence e d iven a j-space J of W with j, we will count the nuber of g-invariant -spaces U of V such that U W J Here UJ is an g-invariant subspace of VJ such that 0 Ž UJ Ž WJ UJ, g,so that UJ is a Ž j -space of C Ž g Clearly di C Ž g Ž d j V J V J d e di VJ, g d j e j d e Thus, J produces at ost choices for U It follows that the nuber of g-invariant -spaces of V that e d e intersect W in a j-space is at ost Then Fix Ž g j j S is bounded above by the quantity Se Ž Ž d e ; in Ž 3, and Ž a follows fro Ž 37 below If H is any g-invariant hyperplane containing C Ž g V, then every g-invariant 1-space is contained in H Thus, if 1, then Ž g, S 1q, which proves Ž b q q j

8 750 URALNICK AND KANTOR Ž d LEMMA 36 Assue Case D holds Then g, S q q for d 3 and Ž g, S 1Ž q 1 1 Proof Let g be a field autoorphis of prie order r Let D PLŽ V There is a unique conjugacy class of eleents of order r in the coset gd Žcf L, 7, so we ay tae g to be the standard field autoorphis Indeed the sae arguent shows that there is also a unique such class in gk where K is the stabilizer of a point in S Fro 5 it follows that the nuber of fixed points of g is precisely C Ž g :C Ž g K, ie, the nuber of -subspaces over the fixed field of g A straightforward coputation now yields the result We now turn to a cobinatorial observation that is crucial for the above arguents Recall that Se Ž Ž d e ; was defined in Ž 3 for 1 d and 1 e d Ž Ž LEMMA 37 a Se d e ; S q Ž d1 b If 1, then S e d e ; S 1q 1q Proof Recall that Se Ž Ž d e ; counts a set of -spaces of a d-diensional q-space V Naely, write V V1 V where di V1 e Then Se Ž Ž d e ; is the size of the set of all pairs Ž X, X 1 of subspaces Xi of V such that each Xi Vi and di X1 di X We begin by disposing of two special cases of the lea If 1, then e de d S Ž q 1 q 1 q 1 q 1 q d1 1 q d 1, Ž Ž so that a and b are clear If e and d, then ½ 5 1 S q 1 q 1 q 1 q 1 q 1 Ž q 1 1 q Consequently, for the reainder of the proof, assue that ; if e then d Ž 38 Let i denote the projection onto Vi copatible with the decoposition V V V Define : S by Ž W ŽW V, Ž W 1 i 1 1 and Ž W Ž Ž W, W V 1 Let S 1 and S be disjoint copies of S and define : S 1 S by i W W whenever W S for i 1, We clai that i i i i 1 Ž X, X q for each Ž X, X, Ž fro which it follows that S q, as desired

9 PROBABILISTIC ENERATION OF ROUPS 751 Fix X, X with di X j e First note that Ž ejž j Ž X, X HoŽ X, V X q jž dej Ž X, X HoŽ X, V X q 1 1 Naely, if HoŽ X, V X 1 1 and eleents of V1X1 are viewed as 1 subsets of V, then x x x X lies in Ž X, X 1 1 ; and this construction easily reverses 1 Then Ž X, X i 1 1 for i 1,, and we will prove the following, which iplies Ž 39 : 1 Ž X, X q for i 1 or Ž 310 i 1 If j 0or then Ž e jž j e or jd Ž e j d Ž e, respectively, and Ž 310 holds We now assue that 0 j, and divide the reainder of the proof of Ž 310 into various cases: Case 1 e and j e Here Ž e jž j Ž ež e Ž ež provided that e Since 0 j e, the only reaining possibility is Ž e, j Ž 3, 1, in which case Ž e jž j Ž 1 by Ž 38 Case e and j e Here jd Ž e j Ž ežd e e Ž ež e for e This leaves the possibilities e j 3orŽ e, j Ž 3,, Ž, 1 If e j, then jd Ž e j jd Ž IfŽ e, j Ž 3,, then jd Ž e j Ž d 1 Ž 1 If Ž e, j Ž, 1, then jd Ž e j d 1 by Ž 38 Case 3 e and j Here Ž e jž j Ž e Ž Ž ež if e By Ž 38, the only reaining possibility is Ž e, j Ž 3, 1, and then Ž e jž j Ž 1 Case e and j Since d e, and j 1 by Ž 38, we have jd Ž e j je Ž j j The next lea deals with graph and field-graph autoorphiss of order This is needed for Theore 8 as well for the general alost siple case LEMMA 311 Let be an alost siple group with socle PSLŽ d, q PSLŽ V with d 3 Fix a positie integer d Let X1 be the -set consisting of copleentary pairs of subspaces one of which has diension Let X be the -set of flags of type Ž, d Ž here d If g is

10 75 URALNICK AND KANTOR an inolution inducing a graph or field-graph autoorphis, then, for i 1,, Ž a g, X i q ; d1 b if 1, then g, X i 1q 1q ; Ž c Ž g, X i 1; and d if d, then g, S q Proof We ay tae Aut PSLŽ V First suppose that g induces a field-graph autoorphis There is a unique such class in Žsee L, 7, so we ay assue that g corresponds to the standard heritian for on V Then g fixes U 1, U X i if and only if U U 1 Thus, the nuber of fixed points of g is either the nuber of totally singular -spaces Ž on X or the nuber of nondegenerate -spaces Ž on X 1 A straightforward coputation now yields uch better estiates than claied Siilarly, if d then g fixes an eleent of S if and only if the subspace is a axial totally singular subspace, and the desired estiates hold Now assue that g induces a graph autoorphis Since g is an involution, there is a nondegenerate bilinear for associated to g and the fixed points of g can be identified as above Again, a straightforward coputation yields uch better estiates than claied LEMMA 31 Let be an alost siple group with socle PSLŽ d, q PSLŽ V, d 3 Let 1 d, and let X i denote the -set in the preious lea If 1 g then, for i 1,, Ž a g, X i q ; d1 b if 1, then g, X i 1q 1q ; and Ž c Ž g, X i 1 Proof We ay assue that g has prie order If g is a graph or field-graph autoorphis, the previous lea applies Otherwise, we can siply use Ž 31 Ž since X aps onto the -set S i except in the case d and X i X 1 We now assue that d and that g arises fro a seilinear transforation If g has odd order, then fixing Ž U, U 1 is equivalent to fixing both U1 and U and again the result follows fro Ž 31 The reaining case is when g is an involution If g is a field autoorphis, then argue precisely as in Ž 36 If g cannot interchange two copleentary subspaces, then the estiate for the action on S again applies Thus, we ay assue that g arises odulo scalars fro 0 si ž I 0 /

11 PROBABILISTIC ENERATION OF ROUPS 753 for soe scalar s So we see that one of the following holds: Ž a g is diagonalizable with distinct eigenvalues, both of ultiplicity ; Ž b the natural odule is a direct su of isoorphic -diensional irreducible ² g: -odules; or Ž c q is even and the Jordan for for g has all of its blocs of size In general, the fixed points of g are either of pairs of fixed -subspaces or pairs of g-conjugate subspaces Suppose that U1 is a g-invariant -space and that g fixes soe copleent U Let Q be the radical of the axial parabolic subgroup leaving U invariant Then Q acts regularly on 1 the set of copleents to U 1 Thus, the nuber of g-invariant pairs of copleentary subspaces with first ter U is precisely C Ž g Consider- 1 Q ing each of the three cases above, we see that C g q Ž Q note that Q, considered as a ² g: -odule, is just U1 U, where U denotes the adjoint of U Let t denote the nuber of eleents of S fixed by g The nuber of pairs of copleentary -spaces interchanged by g is certainly at ost Ž 1ŽS t Thus, if g fixes s pairs of copleentary -spaces, then s tq Ž 1ŽS t The nuber of pairs of copleentary - spaces is S Q S q, whence the result follows 3 The Reaining Classical roups Let S be a classical group SpŽ V, Ž V or SUŽ V on a d-diensional vector space V of Witt index over the field Žor q q in the unitary case We will need to consider the action of S on totally singular subspaces and on nonsingular spaces ŽWe use the ter totally singular instead of separating into totally isotropic or totally singular subspaces according to the type of space V We use nonsingular to ean having 0 radical Let S N Ž S L ŽV Let TS and NS denote -orbits on the set of totally singular or nonsingular -spaces, respectively These are S-orbits except in soe orthogonal settings where they can be unions of two S-orbits: when S Ž V and the action is on TS, or when S Ž V d with both q and odd and the action is on NS In all of the latter cases any eleent either preserves the two orbits or else interchanges the and hence has no fixed points In our results on orthogonal groups, we also include the case S Ž q Žbecause inductively we need results about all such groups with Witt index at least The fact that S is not quasisiple will ae no difference in the coputations In the next section we will prove Theore I except in the case of very sall-diensional spaces In order to iniize the nuber of special

12 75 URALNICK AND KANTOR arguents needed in sall diensions, in this section we will be soewhat careful about boundsleading to relatively ugly-looing estiates In this direction, we introduce additional paraeters, * for S, as follows: S Sp, q, q, q 1, q SU, q SU 1, q * Note that 1 If x is any singular 1-space, then x x has exactly TS Ž 1 totally singular 1-spaces The eaning of * will appear within the proof of Ž 313 Both of these quantities will be carried along during various fixed point estiates We will require an iportant and useful subgroup Let Q denote the centralizer of a given totally singular -space W Then Q has the following structure, with each indicated odule a natural one for LŽ W, where we also use a LŽ W -invariant subgroup Z of Q: SpŽ, q Z Q can be viewed as the space S Ž W of syetric -tensors If q is even, this is an indecoposable odule with coposition factors W W and a Frobenius twist of W Ž, q Z Q W W Ž 1, q Z W W and QZ W, where Z ZQ if q is odd Ž, q Z ZQ W W and QZŽ Q W, where W W q q SUŽ, q Q can be viewed as the q-subspace W W of W W spanned by all w w w with, w W SUŽ 1, q Z ZQ W W and QZŽ Q W LEMMA 313 Let S be an orthogonal, unitary, or syplectic group acting on its natural odule If g is a nonscalar eleent of N Ž S LŽV acting nontriially on the g-inariant totally singular -space W, then C Q Ž g Q * Proof Let h denote the linear transforation of W induced by g, so h 1 We will obtain a lower bound for dih, Z Žand so an upper bound for di C Ž h Z Case A: S is not unitary Subcase: h is unipotent Decopose W W1 W where h acts nontrivially and indecoposably on W, so that h, W 1 1 is a hyperplane of W 1 Then Z contains the subodule W W IfSSpŽ, q 1, then Z even contains Ž W W S Ž W, where S Ž W is the syetric square of W 1

13 PROBABILISTIC ENERATION OF ROUPS 755 By considering a axial h-invariant flag on W, we obtain an h-invariant flag on W W all of whose quotients are ² h: 1 -isoorphic to W 1 Thus, dih, Z dih, W W Ž di W 1Ž di W 1 1 This shows that the diension of the centralizer of h in Z has codiension at least, which is * when S Ž, q In the syplectic case, h is also nontrivial on S Ž W 1 and so the codiension of the centralizer is at least 1 * This proves the desired estiate except when S is Ž, q or Ž 1, q, where * is or 1, respectively If S Ž, q, then QZ W q and hence di h, QZ, whence q q Ž * Q C h q q IfSŽ 1, q Q then QZ W and di h, QZ 1, which yields the desired result q Subcase: h is seisiple We will abuse notation and consider the odules W and Z over the algebraic closure: the diensions and codiensions of centralizers will be the sae whether we consider Z over q Ž its field of definition or over the algebraic closure First suppose that h induces a scalar on W If the scalar is not 1, then Z h, Z and the result follows easily If h acts as I on W, then Q Z Ž since h is not a scalar on V Then h, QZ QZ and the result follows in this case So we assue that h has at least distinct eigenvalues on W We will show that h, Z has diension at least 1 unless 3 and h has exactly eigenvalues whose product is 1 Let the eigenvalues of h Ž over the algebraic closure on W be a 1,,a Then di C Ž h is the nuber of ordered pairs Ž i, j such that Ž i Ž ii Z aa i j 1 for i j if S is orthogonal; or aa i j 1 for i j if S is syplectic Suppose that h has distinct eigenvalues a and b with ab 1 Žthere ay be further eigenvalues Decopose W W W for ² h: 1 -odules Wi such that di W1 e with 1 e and if ai is any eigenvalue on W i, then aa 1 Since W W is isoorphic to an ² h: 1 1 -subodule of Z and h has no fixed points on this subodule, it follows that dih, Z e Ž e 1 So assue that h has exactly eigenvalues a and a 1 with ultiplicities e and e Let W and W denote the eigenspaces of h Then Ž W 1 i Žor S Ž W in the syplectic case i are subspaces of Z, having 0 intersec- tion, such that h has no fixed points on either If is orthogonal then dih, Z ee Ž 1 Ž ež e 1 1 unless 3,

14 756 URALNICK AND KANTOR in which case dih, Z If S SpŽ, q then dih, Z ee Ž 1 Ž ež e 1 1 Now coplete the arguent as in the unipotent case Žconsidering h, QZ Case B: S is unitary We again abuse notation and consider the odules W and Z over the algebraic closure: the diensions and codiensions of centralizers will be the sae whether we consider Z over Ž its field of definition q or over the algebraic closure Since the result is stated over q rather than over, we ust divide our estiate of dih, W W by q 1 Subcase: h is unipotent We decopose W W1 W as above Then Ž Z is isoorphic to a direct su of the ters Wi Wj since Wi Wi as ² h: -odules for i 1, The arguent in Case A shows that di h, W 1 W and dih, W W are at least Siilarly, di 1 h, W1 W Thus, dih, Z 1 This yields the result when d Ž recall that we ust divide by When d 1, QZ W, QZ :C h q and so Q :C Ž h Q Z Q q, as required Subcase: h is seisiple If h has eigenvalues a 1,,a, then q di C Z h is the nuber of pairs i, j so that aa i j 1 Arguing as above, we see that the result is true if h induces a scalar on W or has eigenvalues a, b with ab q 1 So assue that h has eigenvalues a and b q with ab 1, and let e be the ultiplicity of a Then di h, Z e Ž e Ž 1 This yields the result when d, while for d 1 we proceed exactly as in the unipotent case LEMMA 31 Ž, TS * 1 5 * if 3 Proof It suffices to assue that g has prie order e and is not a scalar Let g act nontrivially on TS Write C C Ž g V We consider various cases which ay overlap but contain all possibilities In Cases A and B, we assue that g acts linearly on V Case A g is seisiple and has a 1-diensional inariant subspace on V Suppose first that V V1 V is a nontrivial orthogonal decoposition of V, that there are no nontrivial ² g: -hooorphiss fro V1 to V, and that di V1 di V Note that in particular this holds if C 0 Žwith V, V C, g, V 1 In particular, every axial totally singular g-invariant subspace of V is spanned by ones of V and V Then Fix Ž g 1 TS is bounded above by the product of the nuber of axial totally singular subspaces of a nonsingular subspace of diension and the nuber of

15 PROBABILISTIC ENERATION OF ROUPS 757 axial totally singular subspaces of a nonsingular space of diension d for soe 1 If d 1, then Ž g, TS Ž 1, and otherwise Ž g, TS is saller than this This handles all possibilities except when V V1 V for axial totally singular subspaces V1and V on each of which g induces a scalar Then V ust be a unitary space Any g-invariant axial totally singular subspace X ust have the for X Ž X V Ž X V 1 where X Ž X V 1 X V Thus, the nuber of such X is the total nuber of subspaces of V, so that Ž, TS Ž TS 1 * Case B g is unipotent, or g is seisiple and has no 1-diensional inariant subspace We ay assue that g does stabilize soe W TS If every g-invariant eleent of TS lies in C C Ž g V, then rad C 0, so that Fix Ž g TS is at ost the nuber of axial totally singular subspaces of x x for a singular 1-space x of rad C, and hence is at ost TS Ž 1 Hence, we ay assue that g acts nontrivially on our g-invariant subspace W TS Let S Ž i denote the set of ebers of TS whose intersection with W is an i-space Since Q acts regularly on S Ž 0, Fix Ž g is either 0 or C Ž g By Ž 313 S Ž0 Q, since g is nontrivial on * W we have Fix g Q S Ž 0 * S Ž0 Now consider S Ž i,0i Since S Ž W we will focus on the case i 1 If U Si Ž is g-invariant then I U W is a g-invariant i-space; the nuber of such U for a given I is the nuber of g-invariant totally singular copleents to WI in I I Conversely, if we start with a g-invariant i-space I W, we wish to count the nuber of g-invariant U S Ž i such that U W I First consider those I for which g is nontrivial on WI Then i di WI since we have assued that g has no eigenvalue in 1 In particular, the Witt index of I I is i, as required to apply Ž 313 : g fixes at ost S Ž i *i ebers of S Ž i By Ž 31, there are fewer than S Ž W i i choices for a g-invariant i-space I of W Thus, the nuber of g-invariant U TS such that either U W or x is nontrivial on WŽ U W is at ost S Ž W S Ž i S Ž 0 S Ž W S Ž i TS i i 1 Ý Ý i *i * * * 1 0 Next consider those I S Ž i for which g is trivial on WI, ie, such that g, W I The nuber of totally singular -spaces of V containing g, W is just the nuber of axial totally singular subspaces of Ž g, W g, W, which is at ost TS q 1 TS q

16 758 URALNICK AND KANTOR * Consequently, g fixes fewer than TS TS q ebers of TS, as required We now consider the cases where g acts seilinearly but not linearly on V These are eleents which involve field autoorphiss of the split group We note two instances If S is a unitary group, then there are involutory autoorphiss which are a product of an involutory field autoorphis and a graph autoorphis of the corresponding linear overgroup; these are handled below in Case D On the other hand, when S is of type D the involutory field autoorphis of the split group acts linearly on the natural orthogonal S-odule and so was dealt with previously Case C g is a field autoorphis of prie order e Žwith e odd if S is unitary or of type D, or g is a field-graph autoorphis of order for S of type D Argue as in the proof of Ž 36 to conclude that the nuber of fixed points is the nuber of totally singular subspaces of diension defined over the fixed field of the field autoorphis A straightforward coputation now yields a uch better estiate than claied Case D S is a unitary group and g is a nonlinear inolution Let h denote the standard field autoorphis of order on UŽ d, q, so gs hs The conjugacy classes of such involutions g are given in LiSa In particular, if d is odd, there is a unique class If d is even, then there are two classes of involutions if q is even and three if q is odd Note that by L, 7, g is conjugate to h in LŽd, q : ² h : In particular, W C Ž g V is a d-diensional q-subspace and each d-diensional g-invariant subspace of V is generated over by V W, a d-diensional subspace over q The description of the conjugacy classes entioned above shows that the restriction of the Heritian for on V to W is an nondegenerate bilinear for over Ž q For exaple, if d is even and q is odd, the three conjugacy classes correspond to the cases where this restriction is alternating or either of the two classes of syetric bilinear fors Thus, the nuber of fixed points of g is the nuber of totally singular -spaces of U A straightforward coputation now yields a uch better estiate than claied PROPOSITION 315 Ž, TS * 1q 1 wheneer 1 and 3 Proof Let g act nontrivially on TS By the preceding lea we ay assue that 1 The proof here, and in the reaining estiates in the section, can be viewed as eleentary conditional probability estiates Tae any totally singular -space U U g, and then choose

17 PROBABILISTIC ENERATION OF ROUPS Ž Ž one of at least of the g -spaces in U not in U By Ž 31, 1 Ž g, TS 1 Ž g, TS 1 1 Ž 1 ž / 1 * 1q * 1q 1 We now consider actions on nonsingular -spaces N, where 1 d 1 We do not restrict to be at ost d In fact, we will apply the following estiate either to N or to N, depending in part on the Witt index requireent in the proposition PROPOSITION 316 If 3 and if l 1 is the Witt index of the ebers of NS, then Ž, NS is bounded as follows: S, NS SpŽ, q, q odd d q 1q 1q 1q SpŽ, q, q even 1 d q 1q 1q 1q Ž, q 1 l d q 1q 1q 1q Ž 1, q, q odd 1 l d q 1q 1q 1q Ž n, q n n l d q 1q 1q 1q SUŽ, q q 1q 1q 1q SUŽ 1, q q 1q 1q 1q Ž 1 l Ž d Ž 1 1 l Ž d Proof Let g act nontrivially on NS As in the proof of Ž 315, tae any L L g in TS l; then g oves any N L in NS such that N L g, so that 1 Ž g, NS Ž 1 Ž g, TS 1 ax PrL N L N NS l ž L, LTS l / LL Ž 317 We only need to consider those distinct L, L TSl that lie in soe N NS Since l is the Witt index of N, if i di L L for such a pair L, L then ² L, L: Ž L L Z for a nonsingular Ž l i -space Z of Witt index l i Here, i l 1 since L L Moreover, the set-stabilizer of L is transitive on the set S Ž L l i of all L TSl such that di L L i and ² L, L: Ž L L is nonsingular

18 760 URALNICK AND KANTOR For any such given L, L TSl for which i di L L di rad² L, L: l 1, consider the probability Pi Pr L N L N NS on the right side of Ž 317 For given I L N NS such that i di I we also have PŽ i PrL N L TS, L L I rad² L, L : l Write l Table I lists S Ž l 0, as well as the size of the set S Ž 0 N of ebers of S Ž 0 lying in N Ž l l Here, p is the characteristic of and O Ž S is regular on S Ž i p L l Here PŽ 0 S Ž 0 N S Ž 0, and Pi Ž l l is obtained by replacing d,,, n, and l by d i, i, i, n i, and l i, respectively Žnaely, we pass to I I Then Pi is given in the last colun of Table I, including a bound that is achieved when l i 1 Now Ž 316 follows iediately fro the table, since 1 Ž, NS 1 * 1 ax Pi Ž 1 * 1 0 i l Pl Ž 1 by Ž 315 and Ž 317 Note that the previous result also handles cases of Witt index 0 For exaple, when is orthogonal and NS consists of anisotropic -spaces we can apply the proposition with d A siilar rear holds for nonsingular 1-spaces, but here it is convenient to prove slightly ore precise estiates than in the preceding result: LEMMA 318 Let S F* 1, q, and let NS denote the S-orbit of nonsingular hyperplanes of V of type Ž, q, where Let 1 g Ž i If q is een and g is a transection, then 1q 1q Ž Ž g, NS 1q 1q, g, NS 1q, and 1q 1q Ž g, TS 1 Ž ii If q is een and g is not a transection then g, NS 1q 1q 1 iii If q is odd and g is a reflection, then 1q 1q Ž g, X for X NS, TS 1 TABLE I Ž Ž Ž S O S S 0 S 0 N P i p L l l l l1 SpŽ, q Ž l q Ž qž ŽliŽ 1q 1q Ž 1, q Ž l Ž l Ž l l q q ŽliŽ 1 1 1q 1q Ž n, q Ž n nž nl Ž nlž l Ž l l q q ŽliŽ n n 1q 1q SUŽ d, q Ž d Ž d l Ž l q l l q ŽliŽ d Ž d 1q 1q

19 PROBABILISTIC ENERATION OF ROUPS Ž Ž 1 Proof i g, NS q q q 1 and g, TS Ž 1 Ž q 1 q 1 Ž ii We ay assue that g has prie order e and fixes soe eber of NS If e and g acts linearly on V, then V C Ž g V, g V, and the fixed hyperplanes not containing the radical V are V V, g 0 for the hyperplanes V of C S not containing V Here, div, g is even Ž 0 V as eigenvalues occur in inverse pairs If di C Ž g V 1, then Ž and hence g, NS q q 1 q q 1 1q 1q Ž note that the signs need not atch up here Suppose that e and g acts linearly on V Since V, g is the intersection of the fixed hyperplanes of g it does not contain V By considering VV we see that V, g has a nonzero radical Also, div, g since g is not a transvection Thus, V, g contains a totally ² : Ž ² : singular -space x, y Clearly, g, NS Pr H x, y H NS Then, for H NS, g, NS Pr ² x, y: H ² x, y: is a totally singular -space Ž Ž q 1Ž q 1 1Ž q 1 1Ž q 1 Ž q 1Ž q 1 1q 1q Suppose that g does not act linearly on V As usual, by L, we ay assue that g is the standard field autoorphis and hence has a fixed point; let J denote the stabilizer of this point If e is odd, then again by J L, g J g and so g has exactly q : JŽ q 0 0 fixed points where e q q The result follows easily If e, then using L 0 again we see that every g-invariant hyperplane has a basis over q 0 Thus, the total Ž 1 nuber of g-invariant hyperplanes is at ost q 1 Ž q The Ž total nuber of points is 1 q q 1, so the fixed point ratio is at Ž 1 Ž ost q 1 q 1 q q and the result follows Ž iii Ž g, NS Ž q Ž q q Ž q for, Ž Ž 1 Ž 1 and 0 or 1, and g, TS q 1 q 1 q 1 1 PROOF OF THEOREMS I AND II FOR CLASSICAL ROUPS WHOSE DIMENSION IS NOT SMALL Before starting the proof of Theores I and II for classical groups, we indicate our general approach using priitive prie divisors Let be a group such that S F* Ž is classical group with natural odule V of diension d We will assue that S is quasisiple and linear rather than

20 76 URALNICK AND KANTOR siple, since this aes no difference for our estiates We choose an eleent s of S and deterine the set MŽ s of overgroups of s axial with respect to not containing S The reducible axial subgroups containing s are obvious: they are just the stabilizers of the nonsingular or totally singular subspaces left invariant by s Thus, we need only classify the axial irreducible subgroups of S containing s Žor in the case of Ž 1, q with q even, those that act irreducibly odulo the radical V In ost cases, the noralizers of these axial subgroups will be the axial subgroups of containing s but not S In a few cases, an outer autoorphis will fuse eleents of MŽ s ; this only aes the arguents easier In all cases s acts irreducibly on a subspace of diension e with e d Moreover, by Zsigondy s Theore Zs, soe prie order subgroup of ² s: will act irreducibly on this space as well unless either Ž q, e Ž, 6 or e and q is a Mersenne prie If e, then d 3 and all axial subgroups are nown Whenever the case Ž q, e Ž, 6 coes up in our proof it is handled individually So we consider the case when soe prie order eleent of ² s: acts irreducibly on a subspace of diension e d and apply PPS, which classifies all subgroups H of LŽ d, q containing such an eleent of prie order The exaples fall into several failies The ost natural are other classical groups of the sae diension over subfields Žnot necessarily proper and saller classical groups over extension fields Žthis includes the iportant case of SUŽ d, q in orthogonal and syplectic groups of even diension d The reaining subgroups H are usually in sall diension or have soe other special properties which allow us easily to see that they do not contain our eleent s Indeed, in ost cases the eleent of H of prie order which acts irreducibly on the subspace of diension e has sall order Žusually coparable in agnitude to d or at worst d and its centralizer in H is quite sall Žin particular, saller than s With this in ind we will prove the following Žwhere always denotes the Witt index : PROPOSITION 1 Theores I and II hold for the classical groups in Table II Proof The action of s on V is given in Table II, including all irreducible constituents in the fourth colun On the first constituent s induces a linear transforation whose order is the first indicated factor Ž divided by a sall nuber so as to have deterinant and spinor nor 1 Siilar stateents hold for the reaining constituents The set MŽ s is obtained using PPS When S SLŽ d, q the irreducible constituents have relatively prie diensions, thereby eliinating Ž 1 the possibility SL de, q e M MŽ s for soe e 1 Moreover, if

21 PROBABILISTIC ENERATION OF ROUPS 763 TABLE II S s divides M Ž s and decoposition of V SL Ž, q excluding SL Ž,, where satisfies: Ž Ž odd q 1 q 1 Ž Ž Ž 1 Ž 1 even q 1 q SL 1, q excluding SL 11,, where Ž 1 Ž q 1 q 1 Ž 1 Sp Ž, q and also, 5, 6, 8, 10 if q is even, and also satisfies: SU Ž, q SU Ž 1, q 1 1 odd Ž q Ž 1 1 Ž q Ž 1 1 Ž 1 Ž 1 &O 1 0 od Ž q 1 1 Ž Ž Ž q 1 Ž q 1 Ž Ž &O od Ž q 1 Ž 1 Ž q 1 Ž 1 Ž q 1 Ž 6 1 Ž 1 Ž 1 Ž 6 &O Ž Ž 1 1 odd q 1 q 1 even Ž q 1 Ž q 1 Ž 1 Ž 1 Ž Ž odd q 1 q even Ž q 1 Ž q 1 Ž 1

22 76 URALNICK AND KANTOR Ž, q, 3,, 6, 8, 10 and Ž, q Ž 5,, and also satisfies: 1 1 odd Ž q Ž 1 Ž 1 q Ž od 8 Ž q 1 1 Ž Ž q 1 Ž q 1 Ž Ž Ž 1 od 8 Ž q 1 1 Ž Ž Ž 1 q 1 q 1 1 od 8 Ž q 1 1 Ž Ž Ž Ž q 1 Ž q 1 Ž Ž Ž 6 Ž od 8 Ž q 1 1 Ž Ž Ž Ž Ž 1 q 1 q 1 6 Ž n, q Ž and n 1 where n 7 also satisfies: 1 1 od Ž q Ž n3 1 1 Ž n1 Ž n5 1 1 Ž Ž 1 q 1 q 1 n 3 n 1 n od Ž q 1 Ž n 1 Ž n Ž q 1 Ž n 1 Ž n 1 Ž n od Ž q 1 1 Ž n n Ž n 1 1 Ž Ž 1 q 1 q 1 n n n od Ž q Ž n Ž 1 q Ž n Ž 1 q Ž n6 1 n 1 n 1 n 6 Ž 1, q excluding Ž 5, 3 and Ž 7, q, where q is odd and q 1 1

23 PROBABILISTIC ENERATION OF ROUPS 765 g induces a graph autoorphis, we see that MŽ s 1 Then the result follows by 3 In all other cases, by PPS the only possible irreducible axial overgroups M of s are the noralizers of naturally ebedded subgroups of the following sorts: Ž, q SpŽ, q when q is even Ždenoted O in Table II, SUŽ, q SpŽ, q,su, Ž q Ž, q,spt, Ž q t SpŽ, q, and Žt, q t Ž, q, where t However, except for the cases O none of these can occur because of the following siple conditions we have iposed on the nonsingular constituents: in all cases two of their diensions differ by, so SpŽt, q t SpŽ, q and Žt, q t Ž, q are ruled out; and unitary subgroups are ruled out because for one of the diensions in Table II the correspond ing factor in colun 3 is of the for q Ž 1 Following 3 we noted that, if q, then Theore II holds In particular, to prove Theore II we ay assue that q is fixed and d is large Case 1 S is not Ž 1, q for q of any parity The last colun of Table II gives diensions to use in the estiates obtained in Section 3; there is soe choice here, so we will choose to have the Witt index of a nonsingular subspace as large as possible By Ž 31, 315, 316, Ž, M Ž d18 q for any M MŽ s Thus, by, 1 PCŽ Ý Ž, M M 0q Ž d18 0as d This proves Theore II for these classical groups Slightly ore care with these sae estiates shows that 1 PCŽ 910 in every case within Table II, as required in Theore I We will give an exaple of this verification in Case below In any of the situations excluded in Table II only the cases q and possibly q 3 still need to be considered, but in the next section we will not bother to ae this restriction Case S Ž 1, q for q of any parity Here MŽ s contains the Ž stabilizer of a nonsingular hyperplane U, and g, U 1q 1q by 318 Then 1 PC 1q 1q when q is odd When q is even S SpŽ, q and MŽ s contains further subgroups, but proceeding as Ž d18 above we see that 1 PC 1q 13q 0as q, provided that d 13 ŽWe have used very different s for q odd and even so as to avoid dealing in the latter case with syplectic groups over extension fields On the other hand, for any choice of s, there is soe s-invariant hyperplane Žsince d 1 is odd and eigenvalues other than 1 ust coe in inverse pairs, and hence 1 PC Ps g 1q 1q when g is a reflection or a transvection, by Ž318Ž i, Ž iii Now for any q we Ž d18 have 1q 13q 1 PC Ps g 1q 1q, and hence li PCŽ 1 1q for fixed q d

24 766 URALNICK AND KANTOR It follows that, in order to deterine li inf PCŽ for classical, it suffices to consider only the present odd-diensional case and sequences Ž i for which q is bounded By passing to a subsequence we ay assue that q is fixed, and then li PCŽ d 1 1q is sallest when q, in which case this liit is 1 This copletes the proof of Theore II for the groups considered in this section Once again, ore care with these sae estiates yields 1 PCŽ 910 in every case within Table II For exaple, if Ž od with 10 and if q is even, then MŽ s consists of the stabilizers of the three indicated subspaces, together with a subgroup O Ž, q Use Ž 318, and the g-invariant nonsingular subspaces of diensions, Ž 3, Ž 1 and 3 6 in 316, in order to obtain P g 3q 1q s Ž Ž Ž Ž Ž3 Ž Ž 6 1q 1q 1q 1q 1q Ž36 Ž 1q 1q 1q 910 Rear We decoposed V as the orthogonal direct su of nonsingular subspaces whose diensions were approxiately Ž di V or Ž di V Other choices would have produced the sae results This flexibility eans that, at least asyptotically, the class ² s: is not at all uniquely deterined On the other hand, if we ignore asyptotic results and wish for a precise optial PCŽ, presuably an irreducible s will produce the best possible bound However, there are groups where no irreducible eleent s exists, such as Ž, q ; and in that case we could not use s of order q 1, since a list of all axial overgroups of such an eleent is not presently nown 5 CLASSICAL ROUPS: ADDITIONAL CASES We exclude those groups that are already Ž central extensions of alternating groups There are a nuber of cases oitted in the preceding section, all in diension at ost 0 Here we will settle ost of those, postponing until Ž 63 the following groups: Ž 5, 3, PSUŽ,, PSUŽ 5,, PSUŽ 6,, PSUŽ 3, 3, PSUŽ, 3, Ž 8,, P Ž 8, 3, SpŽ 6,, SpŽ 8,, P Ž 10,, SpŽ 10,, and PSLŽ 11, In each case we will see that 1 PCŽ 910, and that 1 PCŽ 0as q The latter fact follows again by noting that MŽ s is bounded independently of q We will list groups, bound the order of a torus in which s lies, and list MŽ s The arguents used to show that Theore I holds are essentially identical for the alost siple case By the rear after 3, no further justification is needed for the cases for which MŽ s 1

25 PROBABILISTIC ENERATION OF ROUPS 767 All assertions concerning MŽ s follow fro PPS PSLŽ, q, q 7orq9; s Ž q 1 Ž, q 1; MŽ s N Ž² s :, and 3 applies Ž PSL 3, q, q Let s q q 1 Ž 3, q 1; MŽ s is N Ž² s: if q and SLŽ 3, if q, and 3 applies PSUŽ, q, q 3, and PSUŽ 6, q, q ; s preserves a decoposition 1 1 as in Section Žbelow we will use abbreviations such as s :11; MŽ s consists of the stabilizer of a 1-space, and 3 applies PSUŽ 3, q, 5 q 3, PSUŽ 5, q, q, and PSUŽ 7, q ; Ž d s q 1 Ž q 1Ž d, q 1; MŽ s N Ž² s :; and 3 applies P 8, q ; s q 1; MŽ s N Ž Ž, q ; and 3 applies 5 P 10, q ; s q 1; MŽ s N ŽSUŽ 5, q ; and 3 applies 6 P 1, q ; s q 1; MŽ s N Ž Ž, q 3,N ŽP Ž6, q ; and 3 3 gives 1 PC 3q 3q P 8, q, q ; s : 6 ; MŽ s consists of the stabilizer of a nonsingular -space, together with two subgroups obtained fro it by applying triality; and if g does not induce a triality, then Ž 316 gives 1 PCŽ 33q Ž 1q 3 1q 6 If g does induce a triality outer autoorphis, then MŽ s 1 and 3 applies P Ž, q, 1, 16, 0; s : Ž ; MŽ s consists of N ŽP Ž, q and the stabilizer of a nonsingular -space; Ž 316 and 3 give the desired bounds Ž 3 7, q, q 5 odd; s q 1 Žso s :16 ; MŽ s consists of the stabilizer of a nonsingular 1-space; and 3 applies ŽNote that Ž q does not occur because s q q 1 PSp, q, q ; s q 1; if q is even, then MŽ s consists of O Ž, q and PSpŽ, q Žnote that these subgroups are isoorphic and are interchanged by a graph autoorphis ; the results follow by 3 If q is odd, then only the group PSpŽ, q occurs and 3 yields the result PSpŽ 6, q, q ; s :; MŽ s consists of the stabilizer of a nonsingular -space and, if q is even, also a subgroup O Ž 6, q ;1PCŽ is at 3 Ž ost q 1q 1q 1q 910 for odd q 5 and q 3 1q 1q 1q Ž1q 1q for even q, by Ž 316 and Ž 318 Sp 8, q, q even; s q 1; MŽ s SpŽ, q, O Ž 8, q ;1 PCŽ 3q 1q 910 by 3 and Ž 318 SpŽ 10, q, q even; s :8; MŽ s consists of the stabilizer of a Ž 3 nonsingular -space and a subgroup O 10, q ; 1 PC q 5 1q 1q 1q Ž1q 1q for even q, by Ž 316 and Ž 318

26 768 URALNICK AND KANTOR Sp, q, q even, 3 or 5; s q 1; MŽ s SpŽ, q, Ž Sp, q, O, q ; 1 PC 3q 1 :Sp,q Ž 3q 1 :Sp, Ž q 1q 910 by 3 and Ž Sp 16, q, q even; s q 1; MŽ s SpŽ8, q, O Ž 16, q ; 1 PC 3q 1 :Sp8,q Ž 1q 910 by 3 and Ž EXCEPTIONAL ROUPS AND SPORADIC ROUPS We next consider the exceptional and the sporadic siple groups, as well as the few classical groups not dealt with in the preceding section PROPOSITION 61 Let be an alost siple exceptional group of Lie type other than Ž 3 PSLŽ, 8, Ž PSUŽ 3, 3, Ž 3, Ž, F Ž, F Ž q, q 3, E Ž q, q 3, and E Ž q, q 3 Let ² s: 6 7 be a cyclic axial torus of F* Ž whose order is gien in Table III Then P Ž g Ž s 3, where Ž is gien in the table Moreoer, for such groups, Theore I holds and li PCŽ 1 Proof By 3 we have Ž 3q It follows fro We that an upper bound for MŽ s is as given in Table III The result in We is only for the case siple If MŽ s 1 in the siple case, then clearly this is true as well for Žrecall that we are excluding axial subgroups which contain F* Ž In the reaining cases, it is easy to copute that the bound for MŽ s is still valid for The result follows by using Ž MŽ s Ž We have excluded Ž 3 PSLŽ, 8, Ž PSUŽ 3, 3, Ž 3, Ž, F Ž ; and also, for q 3, F Ž q, E Ž q, and E Ž q 6 7, because these groups are excluded in We We now consider these excluded groups as well as the sporadic groups PROPOSITION 6 Let F* Ž be a siple sporadic group or Ž 3, Ž, F Ž, F Ž q, q 3, E Ž q, q 3 or E Ž q, q 3 Let s F* Ž 6 7 be an eleent whose order is gien in Table IV Then P Ž g Ž s 910, where Ž is gien in the table Proof The values for Ž are given in Ma for the sporadic groups We use the wea bound 3q in 3 for the larger exceptional groups Žnoting that Ž 1 also holds M The bounds for the saller exceptional groups can be coputed fro the character tables So our entire proof aounts to defining s and T ² s: so that MŽ s is sall First consider the case that F* Ž is a sporadic siple group other than B and M Then the conjugacy classes of all axial subgroups H are nown Žcf CCNPW, JLPW If T H then, since T is a Sylow subgroup

27 PROBABILISTIC ENERATION OF ROUPS 769 TABLE III F* s M s 1 ' ' ' 3 ' Ž B q, q q, 1 q q 1 1 3q q, q 3 q, 1 q 3 q 1 1 3q F q, q q, 1 q q q q 1 1 3q q, q 5 q q 1, 1 if 3 q 83q D q q q 1 1 3q F q, q q q 1, 1 if q 83q E q, q q q 1 1 3q E q q q 1 1 3q E q, q q 1 q q 1 1 3q E q q q q q q q 1 1 3q of, by the nuber of ebers of H containing T is N Ž T :N Ž T This leads us to our coputation of MŽ s N In Table IV we have listed the axial subgroups in the siple case In the alost siple case, MŽ s either decreases or is unchanged We can alost always tae Ž Ž MŽ s There are two special cases If HS, then we note fro CCNPW that Ž x Ž 1 1 for every nontrivial x and any character 1of with ²,1: 1 In particular, Ž 1 Žthis is slightly better than the estiate in Ma : the eleent appearing there and requiring a larger estiate is an outer involution and hence does not concern us in the siple case If : HS, then it follows fro CCNPW that MŽ s 1, whence the result follows The second special case is F* Ž M If M, one coputes Ž 1 1 using the perutation characters of the three axial subgroups as given in CCNPW that we ay tae Ž 59 If Aut M, then by CCNPW, it follows that MŽ s 1 and so we ay tae Ž 3 If B or M, then enough is nown about the axial subgroups to show that the only possible axial overgroups of T are either N Ž T or alost siple groups Žcf CCNPW, JLPW The only possible alost siple groups containing an eleent of order 7 and whose order divides B have socle PSLŽ, 7 ; however, PSLŽ, 7 is not a subgroup of B since B contains no dihedral group of order 6 This shows that N Ž T is the unique axial overgroup of T when B, as we have claied in Table IV If M, the only possible siple subgroup of order dividing that of M and containing an eleent s of order 59 is H PSLŽ, 59 Let E be a subgroup of order 9 in N Ž T, so N Ž E Ž 9: 1 3 Then H contains all involutions in N Ž E Let U be the dihedral subgroup of order