JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 5, Number 1, January 1992

THE UNIQUENESS OF GROUPS OF LYONS TYPE

MICHAEL ASCHBACHER AND YOAV SEGEV

We give the first computer free proof of the uniqueness of groups of type Ly. We also show that certain simpicial complexes associated to the Lyons group Ly have the same homotopy type as the Quillen complex for Ly at the prime 3; we show this complex is simply connected, and we calculate its homology. Finally we supply simplified proofs of some properties of groups of type Ly, such as the group . A group of type Ly is a G possessing an involution t such that H = CG(t) is the of the AJ1 and t is not weakly closed in H with respect to G. We prove Theorem 1. Up to there exists at most one group of type Ly. Lyons was the first to consider groups of type Ly in [7], where he determined the local structure, the group order, and the character table of such groups. Lyons left open the existence and uniqueness of groups of type Ly. These ques- tions were settled by Sims in [9] using extensive machine computation. Our proof is based on the theory of uniqueness systems developed in [2]. We apply this theory to the 3-local geometry r of a group G of type Ly and its collinearity graph ~. We also use ~ to give a much more elementary derivation of the group order. Given the theory in [2], the most difficult step in the uniqueness proof is to show that the graph ~ is simply connected in the language of [2]. Associated to any graph 0 is its clique complex K (0) , which is the simplicial complex whose vertices are the vertices of 0 and whose simplices are the cliques of O. We define the homology of 0 to be the homology of the topological space of its clique complex, and we say two graphs have the same homotopy type if the homotopy type of the topological spaces of their clique complexes is the same. By Remark 5 in [2], a graph is simply connected if and only if the topological space of its clique complex is simply connected. Finally we recall that the Quillen complex of a finite group G at a prime p is the complex of the poset of elementary abelian p- of G (cf. [8]). We prove

Theorem 2. Let G be a group of type Ly, ~ the commuting graph on 3-central subgroups of G oforder 3, A the commuting graph on subgroups of G oforder

Received by the editors September 18, 1990. 1991 Mathematics Subject Classification. Primary 20DOS; Secondary SSUOS. This work was partially supported by BSF 88-00164. The first author was partially supported by NSF DMS-8721480 and NSA MDA90-88-H-2032.

© 1992 American Mathematical Society 0894-0347/92 $1.00 + $.25 per page

75

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3, and r the 3-local geometry of G. Then ( 1) ~,A, r, and the Quillen complex ..w; (G) of G at the prime 3 all have the same homotopy type. (2) Each of these graphs and complexes is simply connected. (3) iii(~) = 0 for i -12, while dim(H2(~)) = 531,228,318,624. We do not assume any of Lyons's work from [7], but [7] was invaluable in suggesting what to prove. Also our initial analysis in §4 is much like that of Lyons. It should be added that Lyons's paper is a pleasure to read. While we do not assume any facts about Ly, we do assume certain facts about McLaughlin's group that are recorded in the first four lemmas of §3.

1. SOME RESULTS ON GRAPHS In this section ~ and r are graphs. (The term graph will always mean undirected graph.) We adopt the terminology and notational conventions of §§2 and 3 of [2]. In particular, if 8 is a set of cycles of ~ then (8) denotes the closure of 8 in the path groupoid P(~) of ~, as defined in §2 of [2]. Recall Bas(~) is the smallest closed subset of P(~). We write ~(~) for the closure of all cycles of length at most n. Given a set X and a symmetric relation R on X, we can regard X as a graph with edge set R. For x EX, we write R(x) for the set of y distinct from x such that (x, y) E R. We say X is R-connected if the graph (X, R) is connected. Recall that for x E ~, x-L = {x} u~( x) and a morphism ¢: A -+ ~ of graphs is a map of vertices such that ¢(x-L) ~ ¢(x)-L for all x EA. Throughout this section ¢: A -+ ~ is assumed to be a morphism of graphs. We extend ¢ to a map ¢: P(A) -+ P(~) via ¢(xo··· x n) = ¢(xo)··· ¢(xn). Then the extended map ¢ is a morphism of groupoids; that is, Lemma 1.1. Let p, q be paths in A with end(p) = org(q). Then ( 1) ¢(p -I) = ¢(p) -I . (2) ¢(pq) = ¢(p)¢(q). Lemma 1.2. Let 8 be a set of cycles of A. Then (1) ¢(Bas(A)) ~ Bas(~). (2) ¢((8)) ~ (¢(S)). (3) If p, q are paths in A with p "'(S) q then ¢(p) "'(¢(S)) ¢(q).

Proof. Let '" be an invariant equivalence relation on ~ and define a relation :::- on A by p :::- q if p and q have the same origin and end and ¢(p) '" ¢( q) . Using Lemma 1.1 we check that :::- is an invariant equivalence relation on A. In particular ker(:::-) is closed and consists of the cycles in ¢ -I (ker( '" )). As ker(:::-) is closed it contains Bas(A), and applying this observation to the basic relation in the role of "', we get (1). Similarly applying it to '" (¢(S)) in the role of "', we get (2). Finally assume the hypotheses of (3). Then pq-I E (8), so by Lemma l.l, ¢(p)¢(q)-I = ¢(pq-I) E ¢((8)) ~ (¢(8)), so ¢(p) "'(¢(S)) ¢(q). That is (2) implies (3).

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Lemma 1.3.

pairs (x, y) such that y E 112(x) and 11(x, y) is R(connected, let Roo = Ui Ri and p = Xo ... x4 a square with x 2 E Roo (xo)' Then p E ~ (11) . Proof. By hypothesis x 2 E Rm(xO) for some m ~ I. We induct on m. If m = 1 the result is 3.4 in [2], so take m > 1 . Then by hypothesis there is an Rm -path Xl = Yo'" Yk = X3 in 11(xo' x 2)· As Yi+l E Rm(yJ, Pi = XOYiX2Yi+IXo E ~(11) by induction on m. Thus as p is in the closure of the paths Pi' 0::; i < k, also P E ~(11) .

Lemma 1.5. Let p = Xo ... Xs be a pentagon in 11 such that xt n xt n xt =f:. 0. Then p E ~(11).

Proof. Let X E xt n xt n xt . Then p is in the closure of the triangle XX2X3X and the squares xox l x 2xxo and XOXX3X4XO'

2. SOME GROUP THEORETIC PRELIMINARIES Lemma 2.1. Let G be a finite group, t an involution in G, H = CG(t), and T E Syl2 (H). Assume (1) H = (t G n H). (2) O(H) = 1. (3) (t) = Z(T). (4) There is a 4- E of G with EUS;;; tG. Then G is a nonabelian .

Proof. First O(G) = (CO(G)(e) : e E E U), so as O(H) = 1, we have O(G) = 1. Let M be a minimal of G. As O( G) = 1, IMI is even, so as (t) = Z(T), t EM. Therefore H = (tG n H) ::; M. In particular by (3), T E SyI2(M) , so by a Frattini argument, G = MNG(T). Then as (t) = Z(T), G=MH=M. Therefore G is simple, and as E exists, G is not of prime order. Lemma 2.2. Let n = {I, ... , II}, G = Alt(n), and 11 the commuting graph on subgroups of G of order 3 generated by a cycle. Then (1) 11 is of diameter 2 and simply connected. (2) Let Mil ~ M ::; G, e be the set ofsubgroups of M oforder 3, and A be the bipartite graph on Au e obtained by joining x E 11 to a E e if [x, a] = 1. Then A is connected. Proof. For S S;;; G let Fix(S) and Mov(S) be the set of points of n fixed by S and moved by S, respectively. Then the map x 1---+ Mov(x) is a bijection

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between Ll and 3-subsets of Q. Let Lli(X) = {y Ell: I Mov(y) n Fix(x)I = i}.

Then by 9-transitivity of G on Q, Lli(X) , 0::; i::; 3 are the orbits of Gx on Ll. Further Ll(X) = Ll3(X) and for all YEll, I Fix((x, y))1 ~ 5, so Ll(x, y) i- 125 and hence Ll has diameter 2. Now to prove (1), it remains to show Ll is simply connected; equivalently by 4.3 in [2], we must show each cycle in Ll is in ~(Ll). As Ll is of diameter 2 it follows from 3.3 in [2] that it suffices to show each square and pentagon in Ll is in ~(Ll). Next we observe that if YEll, (x) then Ll(X, y) is connected, so in the language of Lemma 1.4, Ll, (x) ~ R, (x). Similarly if z E Ll2(X) then Ll(x, z) is R,-connected, so Ll2(X) ~ R 2(x). Thus ~(Ll) = ~(Ll) by Lemma 1.4. Finally let p = xo' .. X5 be a pentagon in Ll. Then I Fix( (xo ' x 2' x 3)) I ~ 4, so Ll(Xo' x 2' x 3 ) i- 125. Hence Lemma 1.5 completes the proof. Now to the proof of (2). Recall M is 4-transitive on Q and if a = (g) E e then g has 3 cycles gi on Q of length 3. Thus Xi = (gi) , 1 ::; i ::; 3 are the 3 members of A( a) . In particular, M is a group of automorphisms of A, and as M is 3-transitive on Q, M is transitive on pairs (b, x) with bEe, x Ell, and x E A( b). Thus A is connected if and only if M = (Ma ' Mx ). But this I holds as M is a maximal subgroup of M that does not contain Ma' XI Lemma 2.3. Let F be a field, G a finite group and V a finite-dimensional faithful FG-module with Cv(G) = O. Then ( 1) There exists a largest F G-module V = V (G, V) such that V ::; V, C u (G) = 0, and [V, G] ::; V . (2) dim(H'(G, V)) = dim(VjV). (3) The dual v' of V is the largest FG-module W such that W = [W, G] and WjCw(G) ~ V· . ( 4) The representation of N GL(V) (G) on V extends to a representation on V. Proof. See 17.11 and 17.12 in [1] for parts (1)-(3). Part (4) holds as V may be regarded as the group of automorphisms of the GV cen- tralizing V and GVjV; hence NGL(V)(G) is faithfully represented on V via conjugation.

Lemma 2.4. Let G ~ A 6 , V a faithful GF(3)G-module of dimension 4, A = NGL(V)(G) , and V = V(G, V). Then (1) V is determined up to isomorphism as a GF(3)G-module, G is abso- lutely irreducible on V, and AjG ~ D8 . (2) dim(H'(G, V)) = 2. (3) A has two orbits on hyperplanes of V containing V with representatives V, and V2 such that NA(V,) ~ Z2 x S6 and NA(V2) ~ Z2 x M IO • Proof. The first two statements in part ( 1) are well known. For the third observe that the semidirect product S = GV is a local subgroup of H = V4 (3) and

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if B = Aut(H) then B I H ~ Ds ' B = H NB(S) ,and V is self-centralizing in NB(S) = D. Thus DIV ::; A with DIGV ~ Ds. Further as G is absolutely irreducible on V, ICA(G)I = 2, while I Out(G)1 = 4, so indeed DIV = A. Next G has two conjugacy classes of subgroups L G and KG with L ~ K ~ As, L = K a for some a E A, and V ~ [lj, G1/Cv. (G), where lj is the I permutation module for Gover F on the cosets of I = L or K. Notice, however, that UL = VdCv. (G) is not isomorphic to UK = VKICv. (G) as L L K fixes a point in the former but not the latter. Thus dim(HI (G, V)) ~ 2. Observe next that dim(HI(L, V)) = 1. This is because WICw(L) U(L, V), where W is the six-dimensional permutation module for L, since a DIO-subgroup of L fixes a point in U - V for any extension U of V by the trivial L-module. Thus if Z is an FG-module with Cz(G) = 0 and [Z, G] = V and dim(ZIV) > 2, then there exists points X, Y of Z fixed by L, K, respec- tively with V + X = V + Y. Thus V + X ~ UL ~ Uk' a contradiction. So (2) is established. Finally by Lemma 2.3, the action of A = NGL(V)(G) extends to U. Let e E A induce scalar action on V. Then (e) = CA(G) and as Cu(G) = 0, C u (e) = 0 , so AI G ~ Ds is faithful on U I V and hence has two orbits on the points of UIV with representatives UJV, i = 1,2. We may pick UI ~ UL , so NA(UI) ~ Z2 x S6. As U2 is the restriction to G of the five-dimensional irreducible for Mil discussed in the next lemma, NA(U2 ) ~ Z2 x M IO •

Lemma 2.5. Let G ~ Z2 X Mil' M = [G, G], V a faithful five-dimensional GF(3)G-module, and K = GV the semidirect product. Then (1) V is determined up to isomorphism as a GF(3)M-module, so K is determined up to isomorphism. (2) M has two orbits on the points of V of length 11 and 110. (3) If x is a point in the orbit of length 11 then Kx = Aut(Kx).

Proof. For (1) see James [5]. Then (2) is an easy calculation. Moreover Mx ~ MIO and E = E(MJ ~ A6 is indecomposable on V, so V is isomorphic to the dual of the module U2 of Lemma 2.4(3) as an E-module. Thus NGL(V)(E) = I Gx by Lemma 2.4(3). Further as V = [V, Z(G)], H (Gx ' V) = o. Then (3) follows from these two facts.

3. GROUPS OF TYPE M c A finite group G is of type M c if G possesses an involution t such that CG(t) is the covering group of As and t is not weakly closed in CG(t) with respect to G. In this section we assume G is of type M c and let G = Aut( G) and H = CG(t). In Lemmas 3.1-3.4 we record the facts we will be assuming about G. Most of them are well known (cf. [lO, 6, 4]). Lemma 3.1. (1) Up to isomorphism there is a unique group of type M c . (2) G has one class of involutions.

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(3) I Out(G)I = 2, G is transitive on involutions in G-G, and CG(s) ~ Mll for each such involution. ( 4) The of G has order 1 or 3. (5) IGI = 27 .36 .53.7.11. (6) Elements ofodd order inverted by involutions in G are of order 3 or 5. Let .£1 be the commuting graph on the 3-central subgroups of G of order 3; we also write .£1 for the set of vertices of this graph. Lemma 3.2. (1) G has two classes of elements of order 3. (2) G is transitive on .£1 andfor x E.£1, Gx is the split extension of 0 3(G x ) = R ~ 3'+4 by Z2/(Z4 * SL2(5)). (3) Let T E Sy13(Gx) and A = (T n.£1). Then A = J(T) ~ Es, and N(j(A) is the split extension of A by Z2 x MIO with Tn.£1 of order 10 and A inverted by an involution in G- G . (4) T = RA and all elements of order 3 in T are contained in R or A. (5) x is weakly closed in R with respect to G. (6) Gx hasfiveorbits.£1i(x), O:::;i:::;4,on.£1,with.£1o(x)={x}, .£1,(x) = .£1(x), and (x, y) ~ SL2(3), SL2(5), Z3/5'+2 for y E .£1i(x) and i = 2, 3, 4, respectively. (7) Gx is transitive on involutions s E Gx - Gx and Gx,s ~ M 9 . Lemma 3.3. (1) G has two classes ofsubgroups of order 5 with representatives X and X 2 • (2) NG(X) is the split extension of F*(NG(X» = 0s(NG(X» = P ~ 5'+2 by ZS/Z3. (3) N(j(X2 ):::; N(j(X) with CG(X2 ) = X X 2 • (4) X is weakly closed in NG(X) with respect to G. (5) There is an involution s E N(j(X) - G with C(s) n N(j(X) equal to X 2 extended by an element of order 4.

Lemma 3.4. (1) G has a class ru of subgroups isomorphic to V4 (3). (2) For V E ru, V has 3 orbits on ru with representatives V, V, ' and V2 , where Vn V, is L3(4) and Vn V2 is A6/Es,. (3) .£1 n V is the set of long root groups of V, .£1 n V n V, = 0 and (.£1 n V n G V2) EA. As G is oftype M c, H has a permutation representation on a set n = n(t) of order 8 with (t). Set H* = H/(t) and regard H* as A1t(Q). We let x:::; H be a member of .£1; then x* is generated by a 3-cycle. Lemma 3.5. (1) x is inverted by no involution in G. (2) For each involution i E H - (t), i* is fixed point free on n. Proof. Part (2) is a property of the covering group of As. If (1) fails then x is inverted by an involution in H. But by (2), no involution in H inverts x. Lemma 3.6. Each pair of adjacent vertices in .£1 is contained in a unique conju- gate of A and each triangle in .£1 is fused into A under G.

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Proof. By Lemma 3.2(6), each edge in .:l is conjugate to (x, y) for some fixed y E .:lnA. By Lemma 3.2(5), y i. R, so as IT: RI = 3, T is the unique Sylow 3-subgroup of Gx containing xy, and hence A is the unique member of A G containing xy. As .:l is the commuting graph, each triangle of .:l generates a 3-group, so as A = (.:l n T) , each triangle of G is fused into A. Lemma 3.7. NG(X) is 2-transitive on N",(X) of order 25 with kernel X and for each pair x, y E N",(X) , O(NG(X)) = (x, y) and there exists an involution in G inverting X, x , and y. Proof. Let Y = NG(X) and Y = Y/X. Let y E .:l4(X) as in Lemma 3.2(6). By Lemma 3.3, Z«(x, y}) E XG so without loss X = Z«(x, y}). Indeed by Lemma 3.3, O(Y) = xP with x E SyI3(xP) , so P is transitive on N",(X) , and hence IN",(X)I = IP : Pxl = IP : XI = 25. Also Y = PYx with Yx a complement to P, so the action of Yx on N", (X) is equivalent to its action on P by conjugation. By Lemma 3.3, Yx ~ Z8/Z3 is faithful on P and hence regular on p", so Y is 2-transitive on N",(X). Finally by Lemma 3.3(5) there exists an involution inverting X and 5 members of N", (X), so the proof is complete. Lemma 3.8. For s EGan involution, XS ¢. .:l3(X).

Proof. If so there is an involution in L = (x, x s ) that we may take to be t and hence s E H . Now L * ~ A5 is the stabilizer of 3 points of n so as s acts on L it must fix one of these 3 points. This contradicts Lemma 3.5(2). Lemma 3.9. (1) Fix~(t) is of order 35. (2) For U E Fix~(t), (H n U)* is the stabilizer of a partition of Q into two halves. (3) Each dihedral subgroup of G fixes a point of '!/. (4) For y E.:l, (x, y) fixes a point of '!/ if and only if y E x.l U .:l2(X). (5) If y E.:l then (t, y) fixes a point of '!/ if and only if / E y.l u .:l2(y). (6) If y E.:l with / ¢. .:l4(y) and K :::; H with K* ~ A 6 , then there exists v E K n (y.l u .:l2(y)) . (7) If s is an involution in G and K :::; H with K* ~ A 6 , then there exists v E .:l n K with VS E v.l u .:l2(V). Further if K, :::; CG(s) with K,/(s) ~ A 6 , then there is u E.:l n K, with u E v.l u .:l2(V).

Proof. Let t E U. Then U has one class of involutions and Cu(t) ~ E4/ SL2(3) * SL2 (3) , so H is transitive on Fix(t) = Fix~(t) of order IH : Cu(t)1 = 35. Further the unique subgroup of H of index 35 is the stabilizer in (2), so (1) and (2) are established. Let D be a dihedral subgroup of order 2m. If m is even we may take t E Z (D). But then using Lemma 3.5(2) and (2) we check D fixes a point of '!/ . So m is odd and hence by Lemma 3.1(6), m = 3 or 5. If m = 3 then by Lemmas 3.5( 1) and 3.2, OeD) is determined up to conju- gation in G. Further using Lemma 3.5(2), H is transitive on its S3-subgroups

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and on O(D)G nH ,so G is transitive on its S3-subgroups. Similarly by Lemma 3.3, G has two classes of elements of order 5, X is not inverted by an involu- tion of G, and NG(X2 ) is transitive on the involutions in G inverting X 2 , so D is determined up to conjugacy in G. Finally U has dihedral subgroups of order 6 and 10, so (3) holds. As the members of A are long root elements of U4 (3) , if x E U then y E x.L U A2 (x) for all y E U, and U contains elements in Aj(x) for i::; 2. Thus (4) holds. Moreover (4) implies half of (5). Further by Lemma 3.4(3), if Y E A2(x) then (x, y) fixes a unique point of '!I, so if / E A2(y) then t fixes the unique point fixed by (y, yt). If / E y.L then (y, t) centralizes an involution s E G, and we check that (y, t) fixes a partition of O(s) into two halves and hence a member of '!I. So (5) holds. Let K ::; H with K* ~ A 6 • Then K is the stabilizer of 2 points of n, which allows us to check that each partition of n into two halves is fixed by some member of An K. Hence by (2), each member of Fixzr(t) is fixed by some member of An K. This observation together with (5) and Lemma 3.8 implies (6). Similarly under the hypotheses of (7), (3) says (s, t) fixes some U E '!I and by the observation there is v E An K, u E An K\ fixing U. Then u, v work in (7) by (4). In the remainder of this section let 8 be the graph on A with x adjacent to y if and only if y E A4(X). We wish to show

Theorem 3.10. If y E A4 (x) then A n (x, y) is a maximal clique in 8

The proof involves a number of lemmas. Let y E A4 (x); we may assume X = Z({x, y)). Set Y = NG(X) and Y = N(j(X). We must show the set 8(An Y) of vertices adjacent in 8 to each member of An Y is equal to An Y . By Lemma 3.7, AnY ~ 8(A n Y), so it remains to show that if Z E 8(A n Y) then z::; Y. Lemma 3.11. Let a and b be distinct elements of A. Then (1) If (a, b)::; A then Fixzr«(a, b)) = Fixzr(A) is of order 2. (2) If (a, b)::; U E '!I then A(a, b) ~ U. (3) IA(a, b)1 ::; 8. (4) I Fixzr(a) I = 5 and Fixzr(a) and Movzr(a) are orbits of Ga, with IA(a)n VI = 1 for V E Movzr(a).

Proof. Let U E '!I . As U is transitive on edges of A in U, Gab is transitive on Fixzr( (a, b)) if (a, b) ::; A. Similarly NG(A) is transitive o~ Fixzr(A) and Ga is transitive on Fixzr(a). Next Ga,b ::; NG(A) by Lemma 3.6, so Fixzr({a, b)) = Fixzr(A) and by Lemmas 3.4(3) and 3.2(3), ING(A) : Nu(A)1 = 2 for U E Fixzr(A). So (1) holds. Similarly I Fixzr(a) I = IGa : Ual = 5. Also by Lemma 3.6, A(a, b) ~ A so (2) and (3) hold in this case. If (a, b) ::; U but b ¢. A(a) , then by Lemma 3.4(3), {U} = Fixzr( (a, b)), so (2) holds in this case too, and implies (3). If Fixzr( (a, b)) = 0 then A(a, b) ~ CG(Z({a, b))), and we check IA(a, b)l::; 1 inside the later group. So (3) is established.

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We have seen Fix9l(a) is an orbit of Ga of length 5. Further by (1), each bE Ll(a) fixes exactly 3 points of Mov9l (a) , while by (2), Mov9l(a) nFix«(b , c}) = o for distinct b, c E Ll(a). Thus there are 3 'ILl(a)1 = 270 members of Mov9l(a) containing a member of Ll(a) and that member is unique. Finally IWI = 275, Ga is transitive on Ll(a) , and a is transitive on Fix9l(b)-Fix(ab) , so (4) is established.

Lemma 3.12. (1) Y has two orbits WI and W2 on W of lengths 125 and 150, respectively. (2) UnYnLl1=0 ifandonlyifUEWI,inwhichcase YnU isacomplement to P in Y. Proof. By Lemma 3.11(4), X is regular on Fix9l(x) , so for U E Fix9l(x) , UY is of order 125 with Y n U a complement to P in Y and consists of those members of W fixed by some member of Ll n Y. Similarly X 2 fixes some V E Wand N V (X2 ) = Y n V is a of order 20, so IVYI = IY: Y n VI = 150, and hence VY = W - UY. In the remainder of this section let A = O(Y n Ll) - (Y n Ll) and bE A.

Lemma 3.13. Yb = 1 .

Proof. Suppose g E Yb is of prime order p and let C = CG(g). If p = 5 then g rt X as b i. Y, and then Lemma 3.3 contradicts b ~ C. If p = 3 then (g) E Ll(b) nY, contradicting bE A. So p = 2. If g E G then g centralizes some x E Lln Y while by Lemma 3.5(1), b ~ C, so (x, b) ~ C, contradicting bE Ll4 (x) . _ Thus g E Y - Y is an involution. Then g inverts 5 members y j' 1 ~ i ~ 5 , of Y nLl and b. By Lemma 3.1 (3), C ~ Mil acts faithfully on a set 3 of order 11, and by Lemma 3.2(7), Cy ; ~ M9 is the stabilizer in C of some 2-subset

eU) of 3 and similarly Cb = C~(b)' If Cy; n CYj 1= 1 then there is some d of order 2 or 3 in Cc«(Yj, y)), contradicting Yj E Ll4 (Yj)' So eU) n eU) = 0, and similarly eU) ne(b) = 0. But then the six 2-subsets eU), e(b), 1 ~ i ~ 5, are pairwise disjoint, contradicting 131 = 11 .

Lemma 3.14. Let b ~ U E W, A( U) = A n U, and e( U) = U n Ll - A( U) . Then (1) U E W2 in the notation of Lemma 3.12. (2) IA(U)I ~ 200 and le(U)1 ~ 80.

Proof. Part (1) follows from Lemmas 3.12(2) and 3.9( 4). For the first inequality in (2), count the set :T of pairs (a, V) with a E bY, V E W, and a ~ V in two ways. By Lemma 3.13 there are IYI = 6,000 choices for a E bY and by (1) and Lemma 3.12( 1) there are 150 choices for V E W2 and a is contained in 5 members of W2 ,so U contains 200 Y-conjugates of b. Finally as IU n Lli = 280, the first inequality implies the second.

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We now complete the proof of Theorem 3.10 by proving le( V) I > 80, con- tradicting Lemma 3.14. Define .Si1' = U d(y) n V . yEYn~

For u E.Si1' , Id(U)nYI = 1 and d(u)nV ~ e(V) as z ~ d 4 (X) for x E d(u)nY and Z E d(U). Further by Lemmas 3.11(4) and 3.14(1), Id(X) n VI = 1, so 1.Si1'1 = 25. However for each u E.Si1', Id(U) n VI = 36 so by Lemma 3.11(3), l(d(U) n V) U (d(V) U V) U (d(W) n V)I ~ 3·36 - 3·8 = 84,

for distinct u, v , W E .Si1' . Thus the proof of Theorem 3.10 is at last complete.

4. SOME LOCAL SUBGROUPS OF GROUPS OF TYPE Ly In this section we assume G is a group of type Ly. Thus t EGis an involution, H = CG(t) , t is not weakly closed in H with respect to G and H is the covering group of A l1 . Thus we have a permutation representation of H on a set 0 = O(t) of order 11 with kernel (t). Set H* = H/(t) , and regard H* as Alt(O). For h E H, let Fixn(h) and Mov(h) denote the members of o fixed and moved by h, respectively. We take 0 = {I , ... , II} and let Xi' 1 ::; i ::; 3, be elements of order 3 in H such that X~ = (1, 2, 3), x; = (4,5,6), x; = (7, 8, 9) . Let d be the commuting graph on (Xj) G ; we also write d for the vertices of this graph. Lemma 4.1. (1) H is transitive on involutions in H - (t) . (2) If s E H - (t) is an involution then s* has 4 cycles of length 2 on O. (3) G has one conjugacy class of involutions. ( 4) G is simple.

Proof. Parts (1) and (2) are well-known properties of the covering group H of Al1 . Now by hypothesis t is not weakly closed in H with respect to G, so (1) implies (3). Then Lemma 2.1 implies (4). Lemma 4.2. Xj is not conjugate to XjX2 or XjX2X3 in G.

Proof. Let y = x j x2 and T, S be Sylow 2-subgroups of C H (x j ), C H (y) , re- spectively. Then (t) = Z(T) = Z(S) and S is semidihedral of order 16. As (t) is the center of T, TESyI2(CG(xj)) and similarly SESyI2(CG(Y))' Thus as ITI =I- lSI, x is not conjugate to y. Similarly let u = yx3. Then a Sylow 2-subgroup of CH(u) is a 4-group, so by a lemma of Suzuki (cf. Exercise 8.6 in [I]), a Sylow 2-subgroup of CG(u) is dihedral or semidihedral, so again u and Xj are not conjugate. Lemma 4.3. CG(xj)/(xj ) ~ Mc and NG«(xj»)/(xj) ~ Aut(Mc).

Proof. Let M = CG(x j ), M = NG«xj») , and M* = M/(xj ). First CM(t) = C H(X j ) = (Xj) xL, where L is the covering group of As' Further by Lemma

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4.2, x~ n H = x~, so M has one class of involutions. Thus M* ~ Mc by Lemma 3.1. As an element h of H inverting Xl induces an outer automor- phism on L, h induces an outer automorphism on M*; then as I Out( M c) I = 2, M-* ~ Aut(Mc) . Continue the notation of the proof of Lemma 4.3 for a few more moments, and let A* be the weak closure of x; in a Sylow 3-subgroup T* of M* . By Lemmas 4.3 and 3.2(3), A* ~ E 8l , N M" (A*) is the split extension of A*

by Z2 x MIO with ~A = (X2 )NM (A) U {(Xl)} of order 11 and some involution S E if - M inverts A* . Then s inverts A, so A ~ E35 • Each element of M* of order 3 is fused to x; or x;x; , so by Lemma 4.2, ~A = ~nT. As I~AI = 11 is prime to 3, a Sylow 3-subgroup of NG(T) fixes some member of ~A' and hence T E SyI3(G) and NG(A) controls fusion in A. In particular, NG(A) is transitive on ~A' so as NM(A)j(s)A is sharply 3-transitive on ~A - {(Xl)}' NG(A)j(s)A is sharply 4- transitive on ~A • It follows that NG(A) is the split extension of A by Z2 xM11 . We have shown

Lemma 4.4. Let T E SyI3(G) and A = (~n T). Then A ~ E 35, I~ n TI = 11, and NG(A) is the split extension of A by Z2 X M11 .

Lemma 4.5. (1) T = RA where R ~ 32+4 . (2) All elements of order 3 in T are contained in R or A. (3) {(Xl)' (X2 )} = ~ n R. (4) G has two classes of elements of order 3 with representatives Xl and X 1X 2 · (5) CG(xl ) is quasi-simple and the covering group of Mc.

Proof. Let M* = Mj(xl ). By Lemmas 4.3 and 3.1(4), T* = R* A* with R* ~ 31+4, all elements of order 3 in T* contained in R* or A* ,and (X;)M nR = (x;) . Therefore (2) is established. By (2), R is characteristic in T. Further (x;) = Z(R*) and x 2 is inverted in NM(R) , so x 2 E (R). Then as (Xl) is conjugate to (x2 ) in NG(T) , (Xl' X2) = (R). Therefore R ~ 32+4 and (1) holds. Also by Lemma 4.3, CG(xl ) is quasi-simple and hence (5) holds by Lemma 3.1(4). As each element of order 3 in M is fused under M into A and Mil has two orbits on A#, parts (3) and (4) hold by Lemma 4.2.

Lemma 4.6. (1) NG«(x l ,x2 )) is of index 2 in NG«(x l , x 2 )). (2) NG«(xl , x 2)) is the split extension of R by a complement to (Xl' X2) in N H( (Xl' x 2)), with t inverting Rj (Xl' X2) .

Proof· Let y = X1X2' E = (Xl' x 2), N = NG(E), and D = CG(y). By Lemmas 4.2 and 4.5(4), H has two orbits on yG n H, so D has two classes of involutions. However we saw during the proof of Lemma 4.2 that S E SyI2(CD(t)) is semi-dihedral and Sylow in D, so by Thompson transfer D, has a subgroup of index 2 with quaternion Sylow 2-subgroups, and then by a

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result of Brauer and Suzuki [3], D = O(D)CD(t). Now E = O(CD(t»:::; O(D). Conversely if s is an involution in CD(t) - (t) then CO(D)(S) :::; O(CG((s, y))), which is a 3-group, so O(D) is a 3-group. Then (d n O(D») :::; T, while R = O(CM(X2)) , so O(D) :::; R. Then by Lemma 4.5(3), {(XI)' (x2)} = (dnO(D») ~ D, so (1) holds and IN: N n MI = 2 and hence indeed R = O(D) .. Similarly as N n M = R(CNnM(t», N = RCN(t). Hence as N n H splits over E, the proof is complete. Lemma 4.7. Let X :::; H be of order 5 with X* = ((7, 8, 9, 10, 11»). Then NG(X) is the split extension of P = 0s(NG(X)) by a complement to X in NH(X) and P ~ 51+4 . Proof. Let L = CH(X) , Y = CG(X) , and P = O(Y). Then L ~ SL2(9) and E = (XI' x 2) E SyI3(L). By Brauer-Suzuki [3], Y = PCy(t). Now from Lemmas 4.3 and 3.3(2), O(CG(X(x))) = EPi with Pi ~ 51+2, so in particular Cp(xi ) = Pi as En O(Cy(t)) = 1 and t inverts PdX. Similarly by Lemma 4.6, O(CG((XIX2)X» = XE, so Cp(X IX2) = X. It follows that P is the central product of PI and P2 and hence P ~ 51+4 • The lemma follows. Lemma 4.8. Let X2 :::; H be of order 5 with X; = ((2, 3,4,5,6)(7,8,9, 10, G 11»). Then X 2 :::; 0s(NG(XI» for some XI E X and NG(X2) :::; NG(XI ). Further CG(X2) is the extension of Zs x 51+2 by S3 with a Sylow 3-subgroup of CG (X2 ) in d. Proof. Continue the notation of Lemma 4.7. There exists an involution s E H inverting X and (XI) and as t inverts Cp(xI)/X, Cp((XI' r) =f. 1 for r = s or st, say r = s. Let U be of order 5 in Cp((XI' s). Now by Lemma 3.3(4), X is weakly closed in NM(X) with respect to M and by Lemmas 4.2 and 4.7, d n Y = (xl)y, so X is also weakly closed in NM(X) with respect to G. In particular U i XG. Hence as H has two classes of subgroups of order 5 with representatives X and X 2 ' we conclude G UEX2 • Next CH(X2) = XX2(t) , so (I) E SyI2(CH(X2» and thus (t) E SyI2(CG(X2)). Therefore (s) E SyI2(CG(U» , so by Thompson transfer CG(U) = O(CG( U»(s) . Also as CH(t) = XX2(1) , CO(CG(U»(s) = Cp(s). By Lemma 3.3(3), Cp((XI' U)(xl ) = CM(U) , so (XI) E SyI3(CG(U)) and hence O(CG(U)) = O{2,3}'(CG(U»(xI ) and CQ(xl ) = 1 = CQ(s) for each (XI' s)-invariant Sylow p-subgroup of O{2,3}'(CG(U» and each prime p =f. 5. As (XI' s) ~ S3 it follows that O{2, 3}' (CG ( U» is a 5-group. Finally l1y(xI' 5) = {P}, so Cp(U) = Os(NG(U)) n NG(X). Also Cp(U) ~ Zs x 51+2 with X = (Cp(U» , so Cp(U) = 0s(NG(U». Thus the lemma is established.

5. THE LYONS GRAPH AND THE GROUP ORDER In this section we continue the hypotheses and notation of §4. In particular recall d is the commuting graph on the conjugates of (XI). We often write

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x for a member of ..1. Note this means x is a subgroup of order 3, not an element of order 3. For x E ..1, we define three subsets ..1;(x) of ..1 as follows: ..1; (x) consists of all YELl such that (x, y) ~ SL2(3), SL2(5), Z3/5'+2, for i = 1,2,3, respectively. Write Gx for NG(x). Lemma 5.1. For each x ELl: (1) ..1(x) and ..1;(x) are orbits of Gx on ..1. (2) The map y 1--+ xy/x is a bijection between ..1(x) and the set of 3-central subgroups of Gx/x of order 3. 3 2 2 4 (3) 1..1(x) I = 2 .5 .7.11 = 15,400 and Gx ,y ~ Z2/(Z4 * SL2(5))/3 + . (4) l..1i(x)I = 23. 35 • 52.11 = 534,600 and Gx,y ~ S7/Z2' (5) 1..1;(x)I = 22.35 .52.7.11 = 1,871,100 and Gx,y ~ S6/Z2' (6) 1..1;(x)1 = 27.36.7.11 = 7,185,024, Gx,y ~ S3/5'+2, and for each pair of distinct u, v E ..1 (x , y), v E ..1; (u) with ..1 (x , y) consisting of the 25 members of (u, v) n..1.

Proof. Without loss x = (x,). Let M = Gx ' M = CG(x), and M* = M/x. Now Z = (x2) E ..1(x) and by Lemma 4.5, (2) holds and ..1(x) is an orbit of M. Next by Lemma 3.2(2), NM.(z*) is the split extension of R* ~ 3'+4 by Z2/(Z4 * SL2(5)). Further by Lemma 4.5, R ~ 32+4 and NM.(z*) = G:,z' so Gx, z is as claimed in (3). So (3) is established. Next if y E ..1;(x) for i = 1 or 2, then (x, y) contains a unique involution, and as M has one class of involutions, we may choose that involution to be t. Then Gx, y = Hx, y' so the structure of Gx, y is easily calculated from its action on n. By 9-transitivity of H on n and Lemma 4.2, Hx is transitive on ..1;(x) n H, so ..1;(x) is an orbit of M and (4) and (5) are established. Finally let y E ..1;(x) and L = (x, y). Then Z(L) ~ Z5 and as M has two classes of elements of order 5, we may take Z(L) = X or X 2 , discussed in Lemmas 4.7 and 4.8, respectively. But by Lemma 4.8, X 2 i (Q) Q E Sy15(CG(X2))' so Z(L) = X. By Lemma 4.7, I1C(X) (x , 5) ~ P = 05(CG(X)), so 05(L) :::; P. Now from the proof of Lemma 4.7, [P, xl = Cp (x2) ~ 5'+2, so 05(L) = [P, xl. Also K = (x2)[P, x2l :::; CG(L) with K = (X2' v) for any (x2) =f. v E x: and v E ..1;«(x2)). It remains only to observe that NG(X(x)) is transitive on the 24 members of ..1 n L - {x} , so that indeed ..1;(x) is an orbit of M and that Gx, y = N G (X) x, y = K (s) , where s is an involution inverting x 2 • Hence (6) holds and the proof is complete. Lemma 5.2. For each x E ..1, 2 2 2 2 ..1 (x) =..1, (x) U ..12 (x) U ..13 (x) . Proof. This is a consequence of Lemmas 5.1(2) and 3.1(6). Lemma 5.3. Let Y = CG(t) or NG(X), Then for all x, y E ..1nY, d(x, y) :::; 2.

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Proof. If Y = CG(t) this is Lemma 2.2. So let Y = NG(X) , P = °5(Y) , and Y* = Y / P. We saw during the proof of Lemma 5.1 that if y * = x * then y E ~;(x), while if y* =1= x* but [x*, y*] = 1 then y E ~(x). Further by 3 Lemma 4.7, 0 ' (Y*) ~ SL2 (9), so if [x* , y*] =1= 1 then (x*, y*} ~ SL2 (3) or SL2 (5). We claim (x, y} ~ SL2 (3) or SL2 (5) in the respective case, so that y E ~2(X) and the lemma holds. The claim follows from an easy counting argument. Namely IP: Cp(x)1 = 25, so there are 54 pairs (u, v) with u, v E ~n Y, u* = x*, and v* = y*. But if L ~ Y with L a complement to P in the preimage of (x, y}* then X = Np(L) , so there are 54 conjugates of L in LP, and hence these are the 54 subgroups (u, v} .

Lemma 5.4. Let x E ~ and y E ~(x). Then (1) y is not inverted by an involution centralizing x. (2) Gx is transitive on involutions s E Gx - CG(x) and CG«x, s}) ~ M ll . (3) If s inverts u E ~ then d(x, u) ~ 2.

Proof. Part (1) follows from Lemmas 3.5 and 5.1(2). Part (2) is Lemma 3.1(3). So assume u E ~ is inverted by s, and let L = CG( (s, x}) and K = CG( (s, u}) . Without loss s = t. Now L ~ K ~ MIl by (2), and IMlll2 > 21Alll, so L n K =1= 1. So let gEL n K be of prime order p. Then x, U E CG(g). If p = 2 then d(x, u) ~ 2 by Lemma 5.3. If p = 3 then by Lemma 4.5(4), g is conjugate to XI or X I X 2 . In either case Lemma 4.6 says 02(CG (g)) ~ Gv for some v E ~ so v E x..L n u..L and clearly d(x, u) ~ 2. If p = 11 then Nw«g*}) ~ L* , so L* is the unique conjugate of L* containing g * , and hence L = K and x = u. This leaves the case p = 5. Then as gEL, (g} is conjugate to X 2 • Hence by Lemma 4.8, CG(g) is contained in some conjugate of NG(X) , so Lemma 5.3 completes the proof.

Lemma 5.5. If x, y E ~ with IGx ,) even then d(x, y) ~ 2.

Proof. Let t be an involution in Gx,y . By Lemmas 5.3 and 5.4 we may assume x E H but t inverts y. Thus we may take x = (XI} and by Lemma 5.4, L = CG ( (y , t}) ~ MIl' Now each element a of order 3 in L has three 3-cycles on n and by 4-transitivity of L on n, we may assume XI is one of those 3-cycles. Thus x, y E CG (a) , and hence as in the proof of the previous lemma, d(x,y)~2.

Lemma 5.6. Let y E ~, u E ~(y), X E ~7 (y) for i = 1 or 2, and t the involution in (x, y}. Then t 2 (1) u rf- ~2(u). (2) If ut E u..L U ~~(u) then there exists v E ~(x, y) with v E u..L U ~~(u).

Proof. Part (1) follows from Lemma 3.8. Similarly as x E ~7 (y) for i = 1 or

2, Hy contains a subgroup K with K* ~ A 6 , so Lemma 3.9(6) implies (2).

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Lemma 5.7. If p = xeyu is a path in ~ with y E ~i(x) and i is the involution in (x, y) then one of the following holds: ( 1) IG x , y , u I is even. (2) x.l n y.l n u.l # 0. 2 . 2 (3) u E ~2(X) and u l E ~3(U).

Proof. Without loss x = (Xl) and t is the involution in (x, y). Assume neither (1) nor (2) holds. Then as (1) fails, u # ut . Suppose ut E ~(u). Then B = (y, U, ut ) :::; T E SyI3(G) and thus B :::; A = (~n T). Then NG(A) is described in Lemma 4.5 and by Lemma 3.6, A is the unique conjugate of A containing B, so t E NG(A). As [y, t] = 1, CA(t) ~ E27 contains exactly 3 members y, v, w of ~. As y E ~i(x), I Mov(x) nMov(y)1 = 2, so Mov(x) nMov(z) = 0 for z = v or w, and hence z E ~(x, y, u) . But then (2) holds. Assume next that u t E ~i(u). Let r be the involution in (u, ut ) and notice r E Hand Fixn(r) = Mov(y). Thus I Mov(x) n Fix(r)1 = 2. Hence as x # y, xr E ~i(x) .

Let L = 02(CG((t, r))). Then L* = y* x L~ with L~ ~ Z3/21+4 the centralizer of r* in Alt(Mov(r)). Let K = Lx; then K ~ SL2(3) and there exists an involution s E CH(r) inverting x and y and with K(s) ~ GL2(3). Now consider the action of yK(s) on n(r). As ut E ~i(u), Movn(r)(u) = {a, b, c} with (a, b) a cycle of t on n(r). As K(s) ~ GL2(3) is faithful on n(r) , K is 2-transitive on the cycles of t on !l(r) , so we may pick s to have cycle (a, b) and fix c and ct. Thus s inverts y, u, and x, so that (1) holds. So by Lemma 5.6, ut E ~;(u). Let Xl = Z( (u, ut )) and observe that Xl :::; Hy • Let Y = NG(Xl)oo. By Lemma 4.7, Y/Os(Y) ~ SL2(9) , so y* = Y / O{S, 2} (Y) is faithfully represented as the alternating group on a set r of order 6 with Movr(y) of order 3 and Movr(u) = r - Movr(y). If Xl :::; Hx then as (y, x) ~ SL2(3), (y, x) induces A4 on r, so I Movr(x) nMovr(y)1 = 2, and hence I Movr(x) n Movr(u)1 = 1. Hence (x, u) induces As on r, so by Lemma 5.3, x E ~;(u). But then (3) holds. So Xl i Hx' Thus Fixn ( (x, y)) ~ Fix( (Xl' y)), so the element v E H n ~ with Mov( v) = Fix( (Xl' y)) is contained in C H( (Xl' x, y)). Let S E SyI2(G y,v n N(Xl )). By Lemma 3.7, Sv contains an involution s inverting u. B~t yXlS :::; Hy,v n N(Xl ) and some Sylow 2-subgroup of Hy,v n N(Xl ) normalizes x, so we may take s E Gx,y,u' That is (2) holds.

Lemma 5.8. Let p = xeyu be a path with y E ~;(y) and t E (x, y). Then

.1 2 .1 2 .1 H n x n (~l (u) U U ) n (~2(Y) U Y ) # 0.

Proof. If ut E u.1. U ~i(u) then the lemma holds by Lemma 5.6(2). Therefore by Lemma 5.6(1), we may assume ut E ~;(u). Let Xl = Z( (u, ut )); again Xl :::; H. Let Y = NG(Xl)oo. As in the

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proof of the previous lemma, Y* = Y/O{5,2}(Y) is faithfully represented as the alternating group on a set r of order 6 with Movr(y) of order 3 and Movr(u) = r - Movr(y). Suppose Xl ~ Hx . Then as (x, y) 2:! SL2(5), I Movr(x) n Movr(y)I = 1, so I Movr(x) n Movr(u)I = 2, and hence x E L1i(u); so that the lemma holds. Thus Xl i H x ' so Fix( (Xl' x, y)) contains at least 2 points a, b, and hence we can pick v E H n L1 with Mov(v) = {a, b, c} and c E Fix(x) n Mov(y). Thus v E C((Xl' x)) and (y, v) 2:! SL2(5). As (y, v) 2:! SL2(5), I Movr(v) n Movr(y)I = 1 and hence I Movr(v) n Movr(u)1 = 2. Therefore (v, u) induces A4 on r, so by Lemma 5.3, v E 2 L1 1(u). Theorem 5.9. L1 has diameter 2. Before proving Theorem 5.9 we record one of its corollaries.

Theorem 5.10. IGI = 28 • 37 .56 .7.11.31.37.67. Proof· IGI = IGxllL11 and 8 7 3 IGxl=2·3·5 ·7·11,

so it remains to calculate 1L11. But by Theorem 5.9, L1 = {x} u L1(x) U L12(X) , so by Lemmas 5.1 and 5.2, 1L11=9, 606,125=53 .31.37.67, completing the proof. The remainder of this section is devoted to a proof of Theorem 5.9. We first show Lemma 5.11. L1 is connected. Proof. If not then the stabilizer L of the connected component of L1 contain- ing a member of L1 n H is of type Ly and hence contains H, has one class of involutions, and by induction on the order of G, has the order n listed in Theorem 5.10. So L is strongly embedded in G and in particular by a stan- dard result (cf. Exercise 16.5 in [1]) there is a subgroup D of L of odd order transitive on the involutions of L. This is a contradiction. For example as D is of odd order, D is solvable and hence has a Hall {3, 67}-subgroup K. As L has n/IHI involutions, IKI = 3k ·67 with 3 ~ k ~ 7, so K has an element of order 3·67, contrary to Lemmas 4.5(4), 4.3, and 4.6. Let p = YOYlY2Y3 be a path in L1 of length 3; by Lemma 5.11, it suffices to show d(yo' Y3) ~ 2. Assume otherwise. Then Y2 E L1~(Yo) and Yl E L1~(Y3) for some i, j. Let x = YO' Y = Y2 ' and u = Y3' We first observe Lemma 5.12. We can choose Yl and Y2 so that i or j is not 3.

Proof. Suppose Y E L1;(x). Then by Lemma 5.1, L1(x, y) = L1 n (Y 1 ' v) and v E L1(x, y) n L1~(Yl)' Then by Theorem 3.10, there exists d E L1(x, y) such that d tI. L1;(u). Pick Yl = d to insure j -I- 3.

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By Lemma 5.12 and reversing the roles of x and u if necessary, we may assume i = 1 or 2. By Lemma 5.5, IGx ul is odd, while by hypothesis u ¢. d2(X). Thus ii-I by Lemma 5.7, so' i = 2. Then by Lemma 5.8 there exist v E d(X) n d~(U). But now replacing p by the path uwvx for some w E d( u, v) , we reduce to the case i = 1 , which we have already handled. This completes the proof of Theorem 5.9.

6. THE COMMUTING GRAPH ON SUBGROUPS OF Ly OF ORDER 3 In this section we continue the hypothesis and notation of §§4 and 5. In addition let A be the commuting graph on subgroups of G of order 3; we also write A for the vertices of this graph. Thus d is a subgraph of A. Lemma 6.1. (1) d and A - d are the orbits of G on A. (2) For each a E A - d there is a subset ~(a) of d of order 2 invariant .L.L ~ under Ga such that a <; x for each x E ¢(a) . (3) If a E A - d and x, y E A(a) n d then y ¢. d;(X). Proof. Part (1) is Lemma 4.5(4). Parts (2) and (3) follow from Lemma 4.6.

For each a E A - d, pick a member ¢(a) E ~(a). For XEd, let ¢(x) = x. Then by Lemma 6.1,

Lemma 6.2. ( 1) ¢: A -+ d is a morphism of graphs. (2) ¢ 0 I = idA' where I: d -+ A is the inclusion. (3) For each a E A, a.L <; ¢(a).L . Lemma 6.3. (1) ~(A) n P(d) <; ~(d). (2) If S is a set of cycles of A with ¢(S) <; ~(d) then P(d) n (S) <; ~(d). Proof. This follows from Lemmas 1.2, 1.3, and 6.2. Lemma 6.4. Let p = Yo··· Y4 be a square in d such that Y2 E d~(yo) for i = 1 or 2. Then p E ~(d). Proof. Let t be the involution in L = (Yo' Y2). Then d(yo' Y2) consists of those y E H n d such that Mov(y) <; Fix(L). In particular if i = 1 then d(yo' Y2) is connected, while if i = 2 then the graph on d(yo' Y2) obtained by joining x to y if x E y.L Udi(y) is connected. Thus Lemma 1.4 completes the proof. Lemma 6.5. Let p = ao ... a4 be a square in A. Then

(1) If aj ¢. d for some i then ¢(p) E ~(d). (2) If P <; CG(t) for some involution t then ¢(p) E ~(d).

Proof. If a1 ¢. d then by Lemma 6.1(3), ¢(ao) ¢. d;(¢(a2)) , so ¢(p) E ~(d) by Lemma 6.4. Hence (1) holds. Further in proving (2), by (1) we may assume p <; d. Then as p <; CG(t), a2 ¢. d;(ao) ' so again Lemma 6.4 completes the proof. Define a path p = ao··· an in A to be alternating if for all i such that a j ¢. d, a j +1 Ed.

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Lemma 6.6. If p is an alternating cycle in CA (t) then cjJ(p) E ~ (L\) . Proof. Write f'V,:= for the invariant relation generated by triangles on A, L\, respectively. Let p = ao··· an. We induct on the number m of indices i such that aj ¢. L\. If m = 0 the lemma holds by Lemma 2.2( 1). So let aj ¢. L\. Then as p is alternating, aj_1 and aj+1 are in L\. Let c be a cycle of aj on n and x E L\ with x* = c*. Then x E L\(aj_1 , a j +1) so p f'V q = ao··· aj_Ixai_ 1 ••• an. Hence by Lemma 1.2(3), cjJ(p) := cjJ(q). But by induction on m, cjJ( q) := 1 , so the proof is complete. Lemma 6.7. Let Mil ~ K:::; H and a, b E K nA. Then (1) There exists a path p = ao· .. a2n in H n A from a to b such that for all i, a2j :::; K and a2j+1 E L\. (2) If x E L\ with a, b, t E Gx then there exists an alternating path p in H n A from a to b such that q = pbxa E ~(A) and cjJ(q) E ~(L\). Proof. Part (1) is Lemma 2.2(2). So assume the hypotheses of (2). If x :::; H then p = axb works, so assume t inverts x. Then t induces an outer automorphism on CG(x)/x, so by Lemma 3.1(3), CG(x(t)) ~ Mil. Thus we take K = CG(x(t)) and apply (1) to get a path p = ao ... a2n . Let q = pbxa . Now qj = xa2ja2j+la2(i+I)x is a square for each i, and as a2j ¢. L\, cjJ(q) E ~(L\) by Lemma 6.5(1). Hence as q is in the closure of the cycles qj' (2) holds. Lemma 6.S. If p = Yo··· Y4 is a square in L\ such that Y2 E L\~(yo). Then PE~(L\). Proof. We first prove (a) If t E GYi for each i, then the lemma holds. For by Lemma 5.4(1), either p ~ H or t inverts each Yj' and by Lemma

6.5(2) we may assume the latter. Now by Lemma 4.6, 03(GYj>Yi +l) is of class 2

and hence not inverted by t, so there exists aj E A(y j , Y j+ I) n H. Pick a path Pj from aj to a j+ 1 as in Lemma 6.7(2) and let qj = Pjaj+IYjaj. Observe that p is in the closure of the cycles qj and triangles YjYj+1 ajyj , 0 :::; i :::; 3, and the cycle q = po· ··P3. So as cjJ(q) E ~(L\) by the choice of Lemma 6.7(2), it remains by Lemma 6.3(2) to show cjJ(q) E ~(L\). But this follows from Lemma 6.6. So (a) is established. Now to the general case. Let x = YO' u = Y, ' Y = Y2' and v = Y3. By Lemmas 3.3 and 3.7, D = Gx,u,y ~ D30 with all involutions in D inverting x, y, u, while by Lemma 5.1, (L\(x, y)) = L ~ Z3/51+2. Let Q = 05(L) and X = Z(Q), so that also X = Z(L) and D = Xu(t) for any involution tED. We next observe that (b) 16 members of L\(x, y) are inverted by involutions in D. For t acts on L and inverts u, so CQ(t) ~ Z5 is regular on the members of L\(x, y) inverted by t. Further t and s invert the same W =1= u if and only if ts E CD(w) = X. Similarly there are 16 members of L\( x , y) inverted by involutions in Gx, Y ,v ' so there exist W E L\(x, y) inverted by involutions tED and s E Gx,y,v.

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Hence by (a), xvywx and xwyux are in ~(Ll), so p is too.

Lemma 6.9. ~(Ll) = ~(Ll). Proof. This is immediate from Lemmas 6.4 and 6.8.

Lemma 6.10. Let x Ell, Y E Ll2(X) , and s E Gx,y an involution. Then either s fixes a member of Ll(x, y) or y E Ll;(X) and s centralizes a member of A(x,y).

Proof. If t E (x, y) then we check the assertion via the action of (x, y, s) on Q; the exceptional case occurs when s centralizes x and inverts y or vise- versa. So assume y E Ll;(X). Then s acts on the 25 points of Ll(X, y), so as 25 is odd, s fixes some member.

Lemma 6.11. If p = Yo" . Y5 is a pentagon in Ll such that Gy y has even 0' 2'Y3 order then p E ~(Ll).

Proof. The proof is much like that of statement (a) of Lemma 6.8. Let rv be the invariant relation on paths of A generated by triangles and squares. By Lemmas 6.3 and 6.9 it suffices to prove p rv 1, or that p is in the closure of cycles q of A with ¢(q) E ~(Ll) . Let x = YO' Y = Y2' and u = Y3; without loss t E Gx,y,u' By Lemma 6.10, there exist a E A(x, y) fixed by t and either a Ell or [t, a] = 1. Similarly there is b E A(x , u) with the same properties. As p' = xayubx rv p , it suffices to show ¢(p') E ~ (Ll). Let bi be the ith term of p' . As in Lemma 6.8, t centralizes some a2 E A(y, u). Similarly if a E Ll then t centralizes at E A(x, a) , while if a 1- Ll we set at = a. We obtain ai E An C( (bi ' bi + t ' t)) in an analogous manner. Pick a path Pi from ai to

a i+ t as in Lemma 6.7(2) and let qi = Piai+tbiai' Then p' is in the closure of triangles of A, the cycles qi' °:s; i :s; 4, and q = Po'" P4' By the choice of qi' qi rv 1 for each i, while by Lemma 6.6, ¢(q) E ~(Ll). Thus the proof is complete.

7. THE LYONS GRAPH IS SIMPLY CONNECTED In this section we continue the hypothesis and notation of §§4 and 5. We prove Theorem 7.1. Let G be a group of type Ly and Ll the commuting graph on the 3-central subgroups of G of order 3. Then Ll is simply connected.

We begin a short series of reductions. Let rv be the invariant relation on the paths of Ll generated by all triangles of Ll. We must show each cycle in Ll is trivial with respect to rv, so let p = Yo ... y n be a nontrivial cycle of minimal length n. By Theorem 5.9, Ll has diameter 2, so by Lemma 3.3 in [2], p is an n-gon and n :s; 2 diam(Ll) + 1 = 5. Hence by Lemma 6.9, n = 5. Let x = YO' Y = Y2 ' and u = Y3' By Lemma 6.11

Lemma 7.2. Gx,y,u has odd order.

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Lemma 7.3. We may choose p so that y E ~;(x) for i = 1 or 2.

Proof. This is the same argument we used in Lemma 5.11. Namely if y E ~;(x) then by Lemma 5.1, ~(x, y) = ~ n (Y" v) for v E ~(x, y) - {y,} and v E ~;(Y,). Then by Theorem 3.10, there exists d E ~(x, y) with d ¢. ~;(u). Hence as XdYUY4X '" p, we may assume d = y, and then shift the labeling of p by 1.

From now on we use Lemma 7.3 to assume y E ~;(x) for i = 1 or 2; without loss t is the involution in (x, y). By Lemma 1.5 .1 .1 .1 Lemma 7.4. x n y n u = 0 . 2 Lemma 7.5. y ¢. ~,(x).

Proof. If so by Lemmas 5.7, 72, and 7.4, u E ~;(x) and ut E ~;(u). Similarly

Let s be the involution in (x, u); then by Lemma 5.1, (~(x, u))j(s) is an A6-subgroup of CG ( (x, s)), so by Lemma 3.9(7) there is v E ~(x, u) such that

Now as xy,yuvx '" p, we may assume v = Y4. But then (*) and (**) supply a contradiction.

By Lemma 7.5, i = 2, so Y E ~;(x). Then by Lemma 5.8

Lemma 7.6. There exists v E H n x.l n (~~(u) U u.l) n (~;(y) u y.l).

Let W Ev.lnu.l and u, Ey.lnv.l. Then as v E~~(U)Uu.l, q, =XVWUY4X and q2 = vwuyu,v are trivial by Lemma 7.5. Further q3 = xy,yu,vx is trivial by Lemma 7.2 as t E Gx,y,v. So as p is in the closure of qj' 1::; j::; 3, p is trivial. Thus the proof of Theorem 7.1 is complete.

8. THE UNIQUENESS OF GROUPS OF TYPE Ly In this section we complete the proof of the main theorem. Thus we continue to assume the hypothesis and notation of §§4 and 5. In addition let x = (x,) and y = (x2). By Lemma 4.4 there is T E Sy13 (G) with T ::; Gx ,y. Let ~K=~nT, A=(~K)' K=NG(A), I=Kx,and '!/=(G,K'~'~K)· We prove the main theorem by appealing to the corollary to Theorem 1 of [2]. In particular the reader is referred to [2] for the definition of various concepts like uniqueness system, etc. Lemma 8.1. '!/ is a uniqueness system.

Proof. By Lemma 5.1, ~(x) is an orbit under Gx ' so G is an edge transitive group of automorphisms of ~.

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Observe Go = (K, GJ is also of type Ly, as H = (HnK, Hx) and t is not weakly closed in Hx with respect to Gx . Thus by Theorem 5.10, IGI = IGol, so G = Go. Similarly Gx = (Gx,y' KJ by Lemma 3.1 and K = (K( {x, y}), Kx) as K is 2-transitive on 11K • Thus we have verified that "V is a uniqueness system. Assume G is a second group of type Ly with subgroup K and graph ~, etc. By Lemma 8.1 applied to G we have a uniqueness system (G, K , ~, ~K) . Further

Lemma 8.2. There exists an isomorphism C: K ---+ K with xC = x and yC = y.

Proof. By Lemma 4.4, K is the split extension of A by Z2 X Mil . Hence by Lemma 2.5, there is an isomorphism C: K ---+ K . Further 11K and ~K are the orbits of length lIon points of A and A, respectively, so by 2-transitivity of K on 11K we may pick xC = x and yC = y. Lemma 8.3. I = Aut(l) . Proof. We have just observed K is the semidirect product defined in Lemma 2.5, so the lemma is just Lemma 2.5(3).

Lemma 8.4. There exists an isomorphism O!: Gx ---+ Gx with IO! = 1 and yO! = y.

Proof. By Lemmas 4.3 and 4.5(5), Gx is the split extension of the covering group of M c by an involutory outer automorphism, so there exists an isomor- phism O!: Gx ---+ Gx . Further y is determined up to conjugation in Gx by the isomorphism type of Gx,y , so we can choose yO! = y. We can now complete the proof of the main theorem. First we observe that the hypotheses of Theorem 5 of [2] are satisfied with Z(x) = x with respect to the maps C and O! constructed in Lemmas 8.2 and 8.4. Namely hypotheses (1) and (2) of Theorem 5 of [2] are visibly satisfied, while the remaining two hypotheses follow from Lemma 8.3. Thus we conclude from Theorem 5 of [2] that "V and "V are similar. Next as T E SyI3(K) n SyI3(Gx) and Z(Kx ,y ) = 1, Lemma 1.1 of [2] estab- lishes the hypothesis of Theorem 2 of [2], so that Theorem says "V is equivalent to "V. By Theorem 7.1, 11 and ~ are simply connected. Further by Lemma 3.6, each triangle of 11 is G-conjugate to a triangle of 11K . Therefore the corollary to Theorem 1 of [2] completes the proof.

9. THE CLIQUE COMPLEX In this section A and 11 are graphs. The clique complex of 11 is the simplicial complex K(I1) whose vertices are the vertices of 11 and whose simplices are the cliques of 11. Observe that K is a functor from graphs to simplicial complexes where the simplicial map induced by a morphism d: A ---+ 11 of graphs is defined by d({xI' ... , x n}) = {d(xl ), ••• , d(xn)} for each simplex {XI' ... , xn} of K(A) .

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Define two morphisms d, e: A ---+ Ll to be contiguous if the induced maps of complexes are contiguous; i.e., for all simplices s of K(A) , d(s) u e(s) is a simplex of K(Ll). Write d:::: e to indicate d is contiguous to e. Recall that we also have a functor K f-+ IKI from simplicial complexes to topological spaces. Further if d is contiguous to e then Idl is homotopy equivalent to lei. Thus

Lemma 9.1. Let d: A ---+ Ll and e: Ll ---+ A be morphisms of graphs such that eod :::: idA and doe :::: id~. Then IK(A)I has the same homotopy type as IK(Ll)I, and hence IK(A)I and IK(Ll) I have the same homology andfundamental group. This leads us to define graphs Ll and A to have the same homotopy type if IK(A)I and IK(Ll) I have the same homotopy type. Further we define the homology of Ll to be the homology of IK(Ll)I. Finally we recall that by Remark 5 in [2], Ll is simply connected if and only if IK(Ll)1 is simply connected. Thus we can restate Lemma 9.1:

Lemma 9.2. Let d: A ---+ Ll and e: Ll ---+ A be morphisms of graphs with e 0 d :::: idA and doe :::: id~. Then A and Ll have the same homotopy type and homology and A is simply connected if and only if Ll is simply connected.

Lemma 9.3. If d: A ---+ Ll and e: Ll ---+ A are morphisms with a~ ~ e(d(a))~ for all a E A then e 0 d :::: idA. Lemma 9.4. Let Ll be a subgraph of the graph A and assume for each a E A - Ll there is some d(a) E Ll with a~ ~ d(a)~. Then Ll and A have the same homotopy type and homology and A is simply connected if and only if Ll is simply connected.

Proof. Extend d: A ---+ Ll by defining d(x) = x for x Ell, and let e: Ll ---+ A be inclusion. Then if x, YEA with x ..1 y, we have y E x~ ~ d(x)~ , so d(x) E y~ ~ d(y)~ , and hence d is a morphism. Then by Lemma 9.3, eo d :::: idA' while of course doe = id~, so the lemma follows from Lemma 9.2. Recall that each poset P may be viewed as a simplicial complex whose ver- tices are the members of P and whose simplices are the chains in P. Define the clique geometry sd(Ll) to be the geometry over 1= {O, ... ,d} whose ob- jects of type i are the i-dimensional simplices of K(Ll) and with incidence equal to inclusion, where d = dim(K(Ll)) . Observe (9.5) K(sd(Ll)) = sd(K(Ll)) , where sd(K) is the barycentric subdivision of K; that is, sd(K) is the sim- plicial complex of the poset of simplices of K partially ordered by inclusion. Thus as sd(K) has the same homotopy type as K, we have (9.6) Ll and sd(Ll) have the same homotopy type. For example let G be a finite group and p a prime. The Quillen complex ~ (G) for G at the prime p is the simplicial complex of the poset of all nontrivial elementary abelian p-subgroups of G partially ordered by inclusion (cf. [8]). Define ~ * (G) to be the simplicial complex whose vertex set consists of the maximal members of ~(G) and s ~ ~*(G) is a simplex if nAES A =I- 1.

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Lemma 9.7. Let G be ajinite group, p a prime, and A the commuting graph on subgroups of G oforder p. Then K (A), ~ (G), and ~ * (G) have the same homotopy type. Proof. The fact that K(A) ~ ~(G) was observed independently by Alperin. This proof is a variant of Alperin's proof; our original proof had a hole. Let ~; ~ be the covers of K(A); ~(G) consisting of all subcomplexes K(A); ~(A), A E ~*(G), respectively. Both covers are connically con- tractible, (cf. [8]) so by a standard lemma (cf. [8]) the corresponding nerves N(~); N(~) of the covers have the same homotopy type as K(A); ~(G), respectively. Finally visibly the maps A ~ K(A) and A ~ ~(A) are isomor- phisms of ~*(G) with N(~) and N(~), respectively. Lemma 9.S. Assume d is a graph and m an such that for all simplices s for K(d) of dimension at least m, there exists a unique simplex des) of dimension n containing s, where n = dim(K(d)). Let r be the geometry on I = {I , ... , m} whose objects of type i < m are the i-dimensional simplices of K(d), whose objects of type m are the n-dimensional simplices, and with incidence equal to inclusion. Then d and r have the same homotopy type. In particular Hj(d) = 0 for i> m. Proof. Apply Lemma 9.4 to the pair Sd(d) , r. We can now prove Theorem 2. Let G be of type Ly, d the commuting graph on 3-central subgroups of order 3, A the commuting graph on subgroups of G of order 3, and r the 3-10cal geometry of G. By Lemmas 6.2 and 9.4, d and A have the same homotopy type. By Lemma 9.7, A and ~ (G) have the same homotopy type. The geometry r is isomorphic to the subgeometry of Sd(d) of all simplices of dimension 0, 1, 10, and by Lemma 3.6, each simplex of dimension at least 2 is contained in a unique conjugate of A. Thus by Lemma 9.8, d and r have the same homotopy type and Hj(d) = 0 for i> 2. By Theorem 7.1, d is simply connected, so the fundamental group of K(d) is trivial. Thus as HI (d) is the abelianization of the fundamental group HI (d) = O. Of course as d is connected, the reduced homology group Ho(d) = O. Now Xed) = L(-I)jdim(C/d)) = L(-I)jdim(Hj(d)) = 1 + dim(H2(d)) , j j so dim(H2(d)) = Xed) - 1 . Let n = Idl, k = Id(X)I, and m = IAGI. Then we have dim(Co(d)) = n and dim(CI(d)) = nk/2. Further for i 2: 2, each simplex of dimension i is contained in a unique conjugate of And, so dim(K(d)) = 10 and

L(-I)l10" dim(Cj(d))=-m ( L(-I)JII "(11)) j =45m. 1=2 J=3 Notice m = 23 • 52 ·7 . nand k = 23 • 52 . 7 . 11 . Thus . 2 2 3 2 3 dlm(H2(L1))=n(I-2·5 ·7·11+2·3·5·7)-1 = 55, 301'n - 1 = 531,228,318,624, completing the proof.

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REFERENCES

1. M. Aschbacher, Finite , Cambridge Univ. Press, Cambridge, 1986. 2. M. Aschbacher and Y. Segev, Extending morphisms ofgroups and graphs, to appear in Ann. Math. 3. R. Brauer and M. Suzuki, On finite groups of even order whose 2-Sylow subgroup is a , Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 1757-1759. 4. L. Finkelstein, The maximal subgroups ofConway's group C3 and the McLaughlin's group, J. Algebra 25 (1973), 58-89. 5. G. James, The modular characters of the Mathieu groups, J. Algebra 27 (1973), 57-Ill. 6. Z. Janko and S. Wong, A characteristic of the McLaughlin's simple group, J. Algebra 20 (1972), 203-225. 7. R. Lyons, Evidence for a new finite simpLe group, J. Algebra 20 (1972), 540-569. 8. D. Quillen, Homotopy properties of the poset of nontriviaL p-subgroups, Adv. in Math. 28 (1978), 10 1-128. 9. C. Sims, The existence and uniqueness of Lyons' group, Gainesville Conference on Finite Groups, North-Holland, Amsterdam, 1973, pp. 138-141. 10. J. Conway, et aI., AtLas offinite groups, Clarendon Press, Oxford, 1985.

DEPARTMENT OF MATHEMATICS, CALIFORNIA INSTITUTE OF TECHNOLOGY, PASADENA, CALI- FORNIA 91125

DEPARTMENT OF MATHEMATICS, BEN GURION UNIVERSITY, BEER SHEVA 84105, ISRAEL

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