Simple Connectedness of the 3-Local Geometry of the Monster

194, 383]407 Ž1997. JA977026

JOURNAL OF ALGEBRA ARTICLE NO.

Simple Connectedness of the 3-Local Geometry of the Monster A. A. Ivanov* Department of Mathematics, Imperial College, London, SW7 2BZ, United Kingdom

and U. Meierfrankenfeld † Department of Mathematics, Michigan State Uni¨ ersity, East Lansing, Michigan 48824 Communicated by Gernot Stroth Received August 5, 1996

We consider the 3-local geometry M of the Monster group M introduced as a Ž . locally dual polar space of the group Vy 8 3 and independently in the context of minimal p-local parabolic geometries for sporadic simple groups. More recently the geometry appeared implicitly within the Z3 -orbifold construction of the Moonshine module V h. In this paper we prove the simple connectedness of M . This result makes unnecessary the refereeing to the classification of finite simple groups in the Z3 -orbifold construction of V h and realizes an important step in the classification of the flag-transitive c-extensions of the classical dual polar spaces. We make use of the simple connectedness results for the 2-local geometry of M and for a subgeometry in M which is the 3-local geometry of the Fischer group M Ž24.. Q 1997 Academic Press

1. INTRODUCTION The Monster group M acts flag-transitively on a diagram geometry M which is described by the diagram c* o o o o . 3

3

9

1

* E-mail address: [email protected]. † E-mail address: [email protected]. 383 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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The elements of M corresponding to the nodes from the left to the right on the diagram are called points, lines, planes, and quadrics, respectively. The residue of a quadric is the classical polar space associated with the Ž . group Vy 8 3 . The quadrics and planes incident to a line form the geometry of vertices and edges of a complete group on 11 vertices. The existence of M was independently established in wBF, RSx. We follow wBFx to review briefly the construction of M and to formulate its basic properties. The starting point is the description of conjugacy classes of the subgroups of order 3 in the Monster wAtx. LEMMA 1.1. In the Monster group M e¨ ery element of order 3 is conjugate to its in¨ erse and there are exactly three conjugacy classes of subgroups of order 3 with representati¨ es s , m , and t , so that Ža. NM Ž s . ; 31q 12 ? 2 ? Suz ? 2, where Suz is the Suzuki sporadic simple group; Žb. NM Ž m . ; 3 ? M Ž24., where M Ž24. is the largest sporadic Fischer 3-transposition group; Žc. NM Žt . ; SymŽ3. = F3 , where F3 is the sporadic simple group disco¨ ered by Thompson. We define a subgroup of order 3 in M to be of Suzuki, Fischer, or Thompson type if it is conjugate to s , m , or t from Lemma 1.1, respectively. A crucial role in the construction of M is played by a subgroup Ž . Ž . M8 ; 3 8 ? Vy 8 3 ? 2 in M . If Q8 s O 3 M8 then M8 rQ8 is an extension of Ž . the simple orthogonal group Vy 3 by an automorphism of order 2, Q8 is 8 the natural orthogonal module for M8rQ8 , and NM Ž Q8 . s M8 . LEMMA 1.2. Let w be the orthogonal form of minus type on Q8 preser¨ ed by M8rQ8 . Then M8rQ8 acting on the subgroups of order 3 in Q8 has two orbits I and N such that Ža. < I < s 1066, the subgroups in I are isotropic with respect to w and of Suzuki type in M; for s g I we ha¨ e NM 8 r Q 8Ž s . ; 3 6 ? 2 ? U4Ž3. ? 2 2 ; Žb. < N < s 2214, the subgroups in N are non-isotropic with respect to w and of Fischer type in M; for m g N we ha¨ e NM 8 r Q 8Ž m . ; Q7 Ž3. ? 2. Ž . Proof. Under the action of O 2 Ž M8rQ8 . ; Vy 8 3 the set of order 3 subgroups in Q8 splits into three orbits I, N1 , N2 with lengths 1066, 1107, 1107 and stabilizers isomorphic to 3 6 ? 2 ? U4Ž3. ? 2, V 7 Ž3. ? 2, V 7 Ž3. ? 2, respectively Žcf. wAtx.. As 317 divides the order of each of the stabilizers there are no Thompson type subgroups in Q8 and as the elements of I are 3-central they are of Suzuki type. By Lagrange 3 7 V 7 Ž3. is not involved in Suz and since 31q 12 has no elementary abelian subgroup or order 3 8 , N1

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and N2 consist of Fischer type subgroups. Finally, in M Ž24. all subgroups of order 3 7 whose normalizers involve V 7 Ž3. are conjugated. Hence N1 and N2 fuse into a single M8rQ8-orbit. By Lemma 1.2 the polar space acted on flag-transitively by M8rQ8 can be identified with the Suzuki-pure subgroups in Q8 with two subgroups being incident if one of them contains the other one. Let Q1 , Q2 , Q3 be Suzuki-pure subgroups in Q8 with Q1 - Q2 - Q3 , so that < Q i < s 3 i for 1 F i F 3. Then the points, lines, planes, and quadrics in M are defined to be the subgroups in M conjugate to Q1 , Q2 , Q3 , and Q8 , respectively, with F s  Q1 , Q2 , Q3 , Q8 4 being a maximal flag. Let Mi s NM Ž Q i . be the maximal parabolic subgroup corresponding to the flag-transitive action of M on M . Then M8 is as above while M1 is the normalizer of a Suzuki type subgroup Q1 Žwhich we will also denote by s . and M1 ; 31q 12 ? 2 ? Suz ? 2 by Lemma 1.1Ža.. The stabilizer of F in M contains a Sylow 3-subgroup of M. Hence for two elements of M to be incident it is necessary for their common stabilizer in M to contain a Sylow 3-subgroup. Let Pi s O 3 Ž Mi ., PiU be the kernel of the action of Mi on the residue of Q i in M and Mi s MirPiU for i s 1, 2, 3, and 8. It is clear that Q i F Pi and that Q8 s P8 s P8U . For i s 1, 2, 3, and 8 we denote by Mi the set of points, lines, planes, and quadrics in M , respectively. For an element a in M we denote by Mi Ž a . the set of elements in Mi incident to a . Let S be the graph on the Suzuki type subgroups in Q8 in which two subgroups are adjacent if they are orthogonal with respect to w . Then S is strongly regular with parameters ¨ s 1066,

k s 336,

l s 729,

l s 92,

m s 112

Žthat is, S has ¨ s 1066 vertices, every vertex has k s 336 neighbors and l s 729 vertices in distance two, two adjacent vertices have l s 92 common neighbors, and two vertices of distance two have m s 112 common neighbors.. The quotient M8rQ8 induces a rank 3 action on S, so that if s g S then NM 8 r Q 8Ž s . acts transitively on the set S 1Ž s . of points adjacent to s in S and on the set S 2 Ž s . of points at distance 2 from s . The next statement follows from standard properties of classical groups. LEMMA 1.3. Then

Let L s NM 8 r Q 8Ž s . and z be an in¨ olution from O 3, 2 Ž L..

Ža. L ; 3 6 ? 2 ? U4Ž3. ? 2 2 and Q8 , as a module for L, has a unique composition series: 1 - s -² S 1 Ž s .: s s H - Q8 ;

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Žb. both s H rs and O 3 Ž L. are isomorphic to the natural orthogonal Ž . module for O 2 Ž L.rO 3 Ž L. ; 2 ? U4 Ž3. ; 2 ? PVy 6 3 ; Žc. O 3 Ž L. acts regularly on S 2 Ž s .; Žd. z acts fixed point-freely on s H rs and on O 3 Ž L.; it centralizes a unique subgroup « g S 2 Ž s . and CQ 8Ž z . s ² s , « : is 2-dimensional containing two subgroups of Suzuki and two subgroups of Fischer type. Since NM 8Ž s . contains a Sylow 3-subgroup of M, it contains P1 s O 3 Ž NM Ž s ... By Lemmas 1.1Ža. and 1.3, we have P1 l Q8 s s H , P1 Q8 s O 3 Ž NM 8Ž s .. and z acts fixed-point freely on P1rQ1. This shows that all the points collinear to s are contained in P1 and that P1U s P1² z :. Let « be as in Lemma 1.3Žd.. Then Q8 s ² « , CP1Ž « .: is uniquely determined by « and s . So if QU8 is another quadric containing s then Q8 l QU8 is a point, a line, or a plane. Furthermore, the image d of Q8 in M1 s M1rP1U ; Suz ? 2 is a subgroup of order 3. Moreover, NM 8Ž s .rP1U s NM 1Ž d . ; 3 ? U4 Ž3. ? 2 2 and by wAtx is a maximal subgroup in M1. Thus the quadrics from M8 Ž s . correspond to 3-central subgroups of order 3 in M1. The next lemma Žcf. wBCN, Sect. 13.7x. describes the action of M1 on its 3-central subgroups of order 3. LEMMA 1.4. The group S ; Suz ? 2 acting on the set D of its subgroups of order 3 with normalizer U ; 3 ? U4Ž3. ? 2 2 has rank 5 with subdegrees 1, 280, 486, 8505, and 13,608. If D denotes also the graph of ¨ alency 280 in¨ ariant under this action, then: Ža. two distinct ¨ ertices of D commute Ž as subgroups in S . if and only if they are adjacent; Žb. D is distance-transiti¨ e with distribution diagram gi¨ en on Fig. 1 and S is the full automorphism group of D. Žc. if K is a maximal clique in D then < K < s 11, the setwise stabilizer T of K is a maximal subgroup in S, and T ; 3 5 ? Ž2 = Mat11 ., so that O 3 ŽT . is generated by the subgroups from K and TrO 3, 2 ŽT . ; Mat11 acts 5-transiti¨ ely on the ¨ ertices of K while O 3 ŽT . fixes none of the ¨ ertices outside K; Žd. let d be the ¨ ertex of D stabilized by U, then the geometry of cliques and edges containing d with the incidence relation ¨ ia inclusion is isomorphic

FIG. 1. Distribution diagram of D.

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to the geometry of 1- and 2-dimensional totally isotropic subspaces in 6dimensional orthogonal GF Ž3.-space of minus type and it is acted on flagŽ . transiti¨ ely by O 2 ŽU .rO 3 ŽU . ; U4 Ž3. ; PVy 6 3 ; Že. if D is a ¨ ertex at distance 2 from d in D then the subgraph induced on the ¨ ertices adjacent to both d and D is the complete bipartite graph K 4, 4 ; Žf. if D is a ¨ ertex of distance 2, 3, or 4 from d then ² d , D : is isomorphic to SL 2 Ž3., AltŽ5., and AltŽ4., respecti¨ ely. Next we make use of the following information about the action of M1 on the set of subgroups of order 9 in P1 containing s Žcf. wWix.. LEMMA 1.5. M1 has two orbits L and K on the set of subgroups of order 9 in P1 containing s , moreo¨ er Ža. if l g L then NM Ž l . ; 3 5 ? Ž2 = Mat 11 . is the stabilizer of a maxi1 mal clique in the graph D as in Lemma 1.4Ž c . and all subgroups of order 3 in l are of Suzuki type; Žb. if k g K then NM Ž k . ; U5 Ž2. ? 2 and all subgroups of order 3 in k 1 except s are of Fischer type. Since P1 is extraspecial, it follows from the above lemma that the subgroups of order 3 in P1 other than s form exactly two conjugacy ˜ and K˜ of M1 with normalizers classes L

Ž 3=31q 10 . ?2?3 5 ? Ž 2=Mat 11 .

and

Ž 3=31q 10 . : Ž 2=U5 Ž 2 . ? 2 . ,

respectively. It is clear that the subgroups from L in Lemma 1.5 are exactly the lines from M2 Ž s .. Comparing Lemmas 1.5Ža. and 1.4Žc. we can identify M2 Ž s . with the set of cliques in the graph D on M8 Ž s .. Since a flag of M is stabilized by a Sylow 3-subgroup of M, it follows from Lemma 1.4Žc. that a line l g M2 Ž s . and a quadric d g M8 Ž s . are incident if and only if l, as a clique of D, contains d . By Lemma 1.4Žd. two cliques l 1 and l 2 of D of maximal possible intersection have exactly two vertices, say d 1 and d 2 in common. Then the lines l 1 and l 2 are in two different quadrics and hence they generate an element of M which has to be the plane p which is the intersection of d 1 and d 2 . This enables us to identify p with the edge  d 1 , d 2 4 of D. Thus the elements from Mi Ž s . for i s 2, 3, and 8 can be considered as cliques, edges, and vertices of the graph D with the natural incidence relation. In particular the planes and quadrics incident to a given line are edges and vertices of the corresponding clique of size 11 in D. Hence we

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have that the diagram of M is as given above and also Žcompare wWix. that M2 ; 3 2q 5q10 : Ž GL2 Ž 3 . = Mat 11 . , M3 ; 3 3q 6q8 : Ž L3 Ž 3 . = D 8 : 2 . ,

M2 ; Sym Ž 4 . = Mat 11 ; M3 ; L3 Ž 3 . = 2.

The main result of the paper is the following. THEOREM 1.6. The 3-local geometry M of the Monster is simply connected, equi¨ alently, M is the uni¨ ersal completion of the amalgam of maximal parabolic subgroups M1 , M2 , M3 , and M8 corresponding to the action of M on M . Here and elsewhere a tuple  Hi Ng I 4 of subgroups in a group will also be viewed as the amalgam obtained by considering the intersection of the Hi and the inclusion maps. To prove the theorem we define G to be the universal completion of the amalgam Ž Mi N i s 1, 2, 3, 8.. Identify Mi with its image in G. Then there is a unique homomorphism x of G onto M with x M i s id M i for all i. We will show eventually that x is an isomorphism. The second author thanks the Imperial College, the Universitat ¨ Bielefeld, and the Martin Luther-Universitat ¨ Halle-Wittenberg for their hospitality.

2. M Ž24.-SUBGEOMETRY In this section we discuss a subgeometry M Ž m . in M stabilized by a subgroup F [ NM Ž m . ; 3 ? M Ž24., where m is a subgroup of order 3 of Fischer type in M. The elements of M Ž m . are some Žnot all. elements of M centralized by m and the incidence relation is induced by that in M . As above, let F s  Q1 , Q2 , Q3 , Q8 4 be a maximal flag in M with Q1 s s and let m be a Fischer type subgroup of Q8 contained in Q3H . Define M Ž m . to be the subgeometry in M induced by the images under F s NM Ž m . of the elements in F. We discuss the diagram of M Ž m . and the structure of the parabolic subgroups Fi [ NF Ž Q i . s NM iŽ m . corresponding to the action of F on M Ž m .. Since m is non-isotropic with respect to w we have Q8 s m [ m H , where m H is the natural orthogonal module for F8rQ8 ; V 7 Ž3. ? 2. Moreover, Q1 , Q2 , and Q3 are contained in m H and form a maximal flag in the

3-LOCAL GEOMETRY OF THE MONSTER

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polar space defined on m H . Thus F8rm ; 37 ? V 7 Ž3. ? 2 and the residue of Q8 in M Ž m . is the non-degenerate orthogonal polar space in dimension 7 over GF Ž3.. By Lemma 1.5 we have F1rm ; 31q 10 ? Ž2 = U5 Ž2. ? 2. and one can see that the image of Q8 in F1 s F1rŽ F1 l P1U . ; U5 Ž2. ? 2 is a 3-central subgroup of order 3 with the normalizer isomorphic to Ž3 = U4Ž2.. ? 2. Let Q be the graph on all these subgroups of order 3 in F1 in which two subgroups are adjacent if they commute. Then Q is strongly regular with parameters ¨ s 176,

k s 40,

l s 12,

l s 135,

ms8

and clearly it is a subgraph in the graph D as in Lemma 1.4. In these terms the quadrics, planes, and lines in M Ž m . incident to s are the vertices, edges, and cliques Žof size 5. in Q with the natural incidence relation. This shows that the diagram of M Ž m . is c* o

o

o

o

3

3

3

1

.

It is easy to deduce the structure of two other parabolic subgroups Žcompare wRSx.: F2rm ; 3 2q 4q8 ? Ž GL2 Ž 3 . = Sym Ž 5 . . ,

F3rm ; 3 3q 7q3 ? 2 ? Ž L3 Ž 3 . = 2 . .

In wISx the geometry M Ž m . was proved to be simply connected. LEMMA 2.1. The geometry M Ž m . is simply connected and hence 3 ? M Ž24. is the unique faithful completion of the amalgam consisting of the subgroup F1 , F2 , F3 , and F8 . This immediately implies the following. LEMMA 2.2. Let X be a faithful completion of the amalgam consisting of the Monster subgroups Mi , i s 1, 2, 3, 8. Let m be a non-isotropic subgroup of order 3 in Q8 contained in Q3H . Then X contains a subgroup Mm ; 3 ? M Ž24., which normalizes m , such that Mm l Mi s NM iŽ m . s NMmŽ Q i . for i s 1, 2, 3, and 8. If X s M then Mm s NM Ž m .. A subgroup m as in the above lemma will be said to be of Fischer type. We remark that the subgroup Mm of X does not only depend on m but a priori also on the flag Ž M1 , M2 , M3 , M8 .. But as the reader might check Mm is already determined by m together with any one of the Mi ’s, i s 1, 2, 3, or 8.

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3. THE 2-LOCAL GEOMETRY OF THE MONSTER There are exactly two classes of involutions in M, called the Conway type and Baby Monster type involutions with representatives z and t, such that 1q 24 CM Ž z . ; 2q ? Co1

CM Ž t . ; 2 ? F2 ,

and

where Co1 is the first Conway sporadic simple group and F2 is the Fischer Baby Monster group wAtx. Let C s CM Ž z .. Then for i s 4 and 8 up to conjugation in C there is a unique Conway-pure subgroup Ei of order i in O 2 Ž C . containing z, whose normalizer in M contains a Sylow 2-subgroup of M. Moreover these two subgroups can be chosen so that E4 - E8 and we will assume that the inclusion holds. Let N s NM Ž E4 . and L s NM Ž E8 .. Then 1q 24 C ; 2q ? Co1 ,

N ; 2 2q 11q22 ? Ž Sym Ž 3 . = Mat 24 . ,

L ; 2 3q 6q12q18 ? Ž L3 Ž 2 . = 3 ? Sym Ž 6 . . . Furthermore C, N, and L are the stabilizers of a point, a line, and a plane from a maximal flag in the 2-local minimal parabolic geometry of the Monster group wRSx having the diagram ; o

o

o

o

o

2

2

2

2

2

.

This geometry was proved to be 2-simply connected in wIv1x and by standard principles this result is equivalent to the following. LEMMA 3.1. The Monster group M is the uni¨ ersal completion of the amalgam of its subgroups C, N, and L defined as abo¨ e. Our strategy to prove Theorem 1.6 is to show that the universal completion G of the amalgam of the 3-local parabolics Mi is also a completion of the amalgam consisting of the subgroups C, N, and L as in Lemma 3.1. LEMMA 3.2. Let m be a subgroup of Fischer type in M. Then Mm has exactly four classes of in¨ olutions and for an in¨ olution t g Mm exactly one of the following holds: Ža. t in¨ erts m , t is of Baby Monster type, and CM Ž t . ( M Ž23. = C2 . m Žb. t centralizes m , t is of Baby Monster type, and CM Ž t . ; m 3 ? 2 2 M Ž22. ? 2. Žc. t in¨ erts m , t is of Conway type, and CM Ž t . ; 2 3 ? U6 Ž2. ? SymŽ3.. m

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391

1q 12 Žd. t centralizes m , t is of Conway type, and CM Ž t . ; 3.2q ?3? m 2 U4 Ž3. ? 2 .

Proof. By wAtx, Mm has four classes of involutions with centralizers as given. By Lagrange, Co1 involves neither M Ž23. nor 3 ? M Ž22. ? 2 and so the first two classes are of Baby Monster type. Since Conway type involutions both invert and centralize groups of Fischer type the remaining two classes must be of Conway type. LEMMA 3.3. Let z be an in¨ olution from P1U s O 3, 2 Ž M1 .. Then e¨ ery in¨ olution in M1 is conjugated to an in¨ olution s g CM 1Ž z . and one of the following holds: Ža. s s z, CM Ž s . ; 6 ? Suz ? 2, and s is of Conway type in M; 1 Žb. s in¨ erts s , CM Ž s . P1U rP1U ; 2 ? Mat 12 ; s and sz are conjugated in 1 M1 and the centralizer of s in P1rs has order 3 6 ; 1q6 Žc. s centralizes s and CM Ž s . P1U rP1U ; 2y ? O6y Ž2. Ž two conjugacy 1 classes.. Proof. The conjugacy classes of involutions in M1rP1 ; 2 ? Suz ? 2 can be read from wAtx. Since z centralizes s and acts fixed point-freely on P1rs , the structure of CM 1Ž s . in Ža. follows. Since the Baby Monster has no elements of order 3 with normalizer of the shape 3 ? Suz ? 2, z is of Conway type. In Žb. we have CP1 r s Ž s . s w P1rs , sz x and since s and sz are conjugated, both subspaces have dimension 6.

4. THE 3-LOCAL GEOMETRY FOR Co1 In wIv2x, a relationship between the 3- and 2-local geometries of the Monster via a 2 24 -cover of the 3-local geometry of the Conway group wBFx was noticed. Let X be an arbitrary faithful completion of the amalgam Ž M1 , M2 , M3 , M8 . of the 3-local parabolics in M which has M has a quotient and let X be the geometry whose elements are the cosets in X of Mi for i s 1, 2, 3, 8 and where two cosets are incident if their intersection is non-empty. If X s M or X s G where as above G is the universal completion of the amalgam, then X is M or the universal cover G of M , respectively. For an element x of X let M x denote the stabilizer of x in X which is a conjugate of Mi for i s 1, 2, 3, or 8 depending on the type of x. If x s Mi g put Q x s Q ig , Px s Pi g , and PxU s PiU g . When working in the residue of an element we can and will identify x with Q x . If m is a subgroup of order 3 of Fischer type in Q8g , then Mm denotes the subgroup as in Lemma 2.2, i.e., if m g Q3g H then Mm s ² NM ig Ž m . N i s 1, 2, 3, 8:.

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Let us pick an involution z from P1U s O 3, 2 Ž M1 .. Then by Lemma 3.3Ža., CM 1Ž z . ; 6 ? Suz ? 2. Let J s J X be the set of points of X such that x g J if and only z g O 2, 3 Ž M x .. Let J denote also the graph on J in which two points are adjacent if they are incident to a common quadric. It is clear that C X Ž z . preserves J as a whole as well as the adjacency relation on J. LEMMA 4.1. Locally J is the commuting graph D of 3-central subgroups of order 3 in M1 ; Suz ? 2 as in Lemma 1.4. Let V be a maximal clique in J containing s and H be the setwise stabilizer of V in C X Ž z .. Then < V < s 12 and there is a unique point a collinear to s such that H s CMaŽ z .. Moreo¨ er, H ; 2.36 ? Ž2 ? Mat 12 ., O 3 Ž H . s Pa l H, H induces the natural action of Mat 12 on the ¨ ertices of V, and O 3 Ž H . is an irreducible GF Ž3.-module for HrO 3 Ž H .² z : ; 2 ? Mat 12 . Proof. Abusing the notation we denote by s the point stabilized by M1 so that s g J. By Lemma 1.3Žd. every quadric incident to s contains besides s exactly one point « centralized by z and « is not collinear to s . This means that the set JŽ s . of points adjacent to s in J is in a natural bijection with the set of quadrics incident to s , i.e., with the vertices of the graph D as in Lemma 1.4. Moreover, if d g D then there is a unique point centralized by z which maps onto d under the homomorphism of M1 onto M1. We will identify d with this unique point. By definition if x and y are adjacent points in J then w Q x , Q y x s 1. Hence if d 1 , d 2 g JŽ s . are adjacent in J, then the corresponding vertices of D are adjacent. In particular a maximal clique in J contains at most 12 vertices. We are going to show that this bound is attained. Let l be a line incident to s and let s , a , b , and g be all the points incident to l. Since z acts fixed-point freely on P1rs ; 312 , we can choose our notation so that z inverts a and permutes b and g . So on every line incident to s there is exactly one point which is inverted by z. Since CM 1Ž z . P1 s M1 , CM 1Ž z . permutes transitively the lines incident to s and hence also the points collinear to s and inverted by z. This implies that CMaŽ z . permutes transitively the points collinear to a and centralized by z. Let Q8 denote a quadric incident to l and let « be the point in Q8 other than s centralized by z. Then « is collinear to exactly one point on l. We know that s and « are not collinear and since b and g are permuted by z, « is collinear to a . Thus in every quadric incident to l besides s there is exactly one point collinear to a and centralized by z. By the diagram of X there are exactly 11 such quadrics which correspond to a clique K of D. Let V s  s 4 j K and H be the setwise stabilizer of V in C X Ž z .. Since locally J is D, K is a maximal clique in D, CM 1Ž z . acts transitively on the set of cliques in D and since C X Ž z . is vertex-transitive on J, we see that H

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393

acts transitively on V. Since a is the only point which is collinear to every point in V, it is clear that H F CMaŽ z .. Since z acts fixed-point freely on P1rs , CP1U Ž z . s s = ² z :. By Lemma 1.4Žc. and the Frattini argument Ž H l M1 . P1U rP1U ; Ž Ml l M1 .rP1U ; 3 5 ? Ž2 = Mat 11 .. Since H l M1 induces the natural action of Mat 11 on the points in K, H induces on the points in V the natural action of Mat 12 . Thus O 3 Ž H . is elementary abelian of order 3 6 generated by the 12 points in V and HrO 3 Ž H .² z : ; 2 ? Mat 12 induces a non-trivial action on O 3 Ž H .. By wMoAtx, Mat 12 does not have a faithful GF Ž3.-representation of dimension less than or equal to 6 and the smallest faithful GF Ž3.-representation of 2 ? Mat 12 has dimension exactly 6. Thus we have shown that H ; 2.36 ? Ž2 ? Mat 12 . and by Lemma 3.3Žb., H s CMaŽ z .. In what follows we will need the detailed information on the structure of 6-dimensional GF Ž3.-modules of 2 ? Mat 12 contained in the following lemma. LEMMA 4.2. Let H ; 2 ? Mat 12 and A be a faithful irreducible 6dimensional GF Ž3. H-module. Then the following assertions hold: Ža. Žb.

H has a unique orbit A of length 12 on the 1-spaces of A. Any fi¨ e elements from A are linearly independent.

Žc. H has a unique orbit L of length less or equal to 12 on the hyperplanes of A. Moreo¨ er, < L < s 12 and if L g L then L contains no element from A. Žd. Let B be the set of 1-spaces of A of the form ² a1 q a2 :, where ² a1 : and ² a2 : g A are different elements of A. Then < B < s 132 and H acts transiti¨ ely on B. Že. If F g B then there exist unique elements D 1 and D 2 in A with F F D 1 q D 2 . If L g L and F˜ is the 1-space in D 1 q D 2 different from D 1 , D 2 , and F, then F F L if and only if F˜ z L. Žf. Define L g L and B g B to be incident if B F L. Then Ž L , B . is a Steiner system of type Ž5, 6, 12.. Žg. Let T ; L with < T < s 4 and put F s F T. Then F is a 2-subspace of A, all 1-spaces of F are in B, and NH Ž F .rCH Ž F . ( GL2 Ž3.. Proof. Let X and Y be two non-conjugate subgroups in H isomorphic to Mat 11 . Then every proper subgroup of index at most 12 in HrZŽ H . ( Mat 12 is conjugate to the image of either X or Y. Moreover, H s ² X, Y : and X l Y ( L2 Ž11.. Let Z be one of the subgroups X, Y and X l Y. By wMoAtx every faithful irreducible GF Ž3. Z-module is 5-dimensional. This means that Z normalizes in A at most one 1-subspace and at most one 5-subspace. Suppose that A contains a 1-subspace normalized by X and a

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1-subspace normalized by Y. Then both these 1-spaces are normalized by X l Y and hence this is the same 1-space, normalized by the whole H s ² X, Y :, a contradiction to the irreducibility of A. Applying the same argument to the module dual to A, we obtain that the subspaces in A normalized by X and Y have different dimensions and we can choose our notation so that X normalizes a 1-space D and Y normalizes a 5-space E. In this case A s D [ E as a module for X l Y. Moreover, A [ D H is the only orbit of length 12 of H on 1-spaces in A and L [ E H is the only orbit of length 12 of H on hyperplanes in A and Žc. holds. The actions induced by H on A and L are two non-equivalent 5transitive actions of Mat 12 . Since A is irreducible, A spans A and so there is a set of six linearly independent elements in A. Since H induces on A a 5-transitive action, every set of five elements in A is linearly independent and thus Žb. holds. Let D 1 / D 2 g A and let D 1 , D 2 , F, F˜ be the set of all 1-spaces in D 1 q D 2 . Then F, F˜ g B. We are going to show that B satisfies the properties stated in Žd. ] Žf.. If there are Di , Dj g A with  i, j4 /  1, 24 such that F is contained in Di q Dj then the set  D k N k s 1, 2, i, j4 of size at most four in A would be linearly dependent, a contradiction to Žb.. Hence the pair  D 1 , D 2 4 is uniquely determined by F. Let L g L . Since L is a hyperplane in A, its intersection with D 1 q D 2 is at least 1dimensional. By Žc. neither D 1 nor D 2 are in L, hence Že. follows. Moreover, F or F˜ is contained in at least 6 elements of L . Since the action of H on L is 5-transitive, we conclude that the intersection of any five elements of L is in B. Let D be the set of elements of L containing F. Suppose that < D < G 7. Then by 5-transitivity of H on L there exists h g H with < D l D h < G 5 and D / D h. But then the intersection of the elements on D, D l D h , and D h , respectively, are all equal to F, a contradiction to D / D h. Hence < D < F 6 and both F and F˜ are contained in exactly six elements of L . Thus Žf. holds. As H acts transitively on the blocks of any associate Steiner systems, Žd. follows. By Žf., T is incident to exactly four elements say B1 , B2 , B3 , B4 of B. By the dual of Žb., F is a 2-space and so B1 , B2 , B3 , B4 are exactly the 1-spaces of F. Since NH Ž T . induces SymŽ4. on  B1 , B2 , B3 , B4 4 we conclude NH Ž F .rCH Ž F . ( GL2 Ž3.. By Lemmas 4.1 and 1.4Žd. two maximal cliques in J are either disjoint or have intersection of size 1, 2, or 3. Moreover, if C s C X is a geometry whose elements are maximal cliques, triangles, edges, and vertices of J X with respect to the incidence relation given by inclusion, then C corresponds to the diagram c* c* o

o

o

o

3

9

1

1

.

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The geometry C is connected precisely when J is connected. Let s s V 1 ; V 2 ; V 3 ; V 12 s V be the maximal flag in C . Then V i is a complete subgraph of size i in J. Let Ci denote the stabilizer in C X Ž z . of V i . Then C1r² z : ; 3 ? Suz ? 2,

C2r² z : ; 3 2 ? U4 Ž 3 . ? D 8 ,

C3r² z : ; 3 3q 4 ? w 2 3 x ? S4 ? S3 ,

C12r² z : ; 3 6 ? 2 ? Mat 12 .

Consider the situation when X s M. By Lemma 3.3Ža., z is of Conway 1q 24 type and CM Ž z . s C ; 2q ? CO1. Put R s O 2 Ž C .. LEMMA 4.3. The graph J M is connected. Proof. Let A be the setwise stabilizer in CM Ž z . of the connected component of J M which contains s . Then A contains C1 ; 6 ? Suz ? 2. Let « be a vertex adjacent to s in J M . Then w s , « x s 1 and since s acts fixed-point freely on Rr² z :, we have s R / « R. Since C1 R is maximal in C, this means that AR s C. Finally, Cr² z : does not split over Rr² z : and hence A s C and J M is connected. The homomorphism x : G ª M induces morphisms G ª M and C G ª C of geometries which will be denoted by the same letter x . Our goal is to show that the restriction of x to the connected component of C G containing s is an isomorphism onto C M. This will immediately imply that the setwise stabilizer in CG Ž z . of the connected component of C G maps 1q 24 isomorphically onto C ; 2q ? Co1. An important role in the realization of this step will be played by a simply connected subgeometry in G . Let m be a subgroup of Fischer type as in Section 2. Then k [ sm is a subgroup of order 9 in P1 which is not a line Žso that k is as in Lemma 1.5Žb... Since z acts fixed-point freely on P1rQ1 , as in the proof of Lemma 4.1 we have a unique subgroup of order 3 in k which is normalized and inverted by z. Hence we can and do choose m so that z inverts m. By Lemma 2.2 there is a subgroup Mm ; 3 ? M Ž24. in X which normalizes m such that Mm l Mi s NM iŽ m . for i s 1, 2, 3, and 8. Let W s CMmŽ z . and let C be the orbit of W on J which contains s . M

LEMMA 4.4. Ža. < C < s 2688 and Wr² z : ; 2 2 ? U6 Ž2. ? SymŽ3. acts faithfully on C; Žb. locally C is the commuting graph Q on the 3-central subgroups of order 3 in U5 Ž2. ? 2. Proof. By Lemma 1.5Žb. and since Mm l M1 s NM 1Ž m ., CMmŽ z . l M1 ; 2 ? Ž3 = U5 Ž2.. ? 2. By Lemma 3.2 and since z is of Conway type and inverts m , W ; 2 3 ? U6 Ž2. ? SymŽ3.. Thus Ža. holds. For Žb. we may by Ža. assume that X s M. The subgroups of Fischer type in P1 normalized by z, are permuted transitively by CM 1Ž z . and hence

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C contains a vertex x of J if and only if m is contained in Px , or equivalently if x is contained in M Ž m . and hence Žb. follows. Since C is locally Q, its maximal cliques have size 6 and two such cliques are either disjoint or have intersection of size 1, 2, or 3. Define U to be a geometry whose elements are maximal cliques, triangles, edges, and vertices of C with the natural incidence relation. Since C s J l M Ž m ., it is easy to see that the diagram of U is c* c* o

o

o

o

3

3

1

1

.

As follows from Lemma 4.4, the isomorphism type of U is independent on whether X s M or X s G, since U is contained in M Ž m . which is simply connected. It is worth mentioning that U itself is simply connected as proved in wMex and that C is distance-transitive with the distribution diagram given on Fig. 2. 5. A CHARACTERIZATION OF C M It is not known whether the geometry C M is simply connected. In this section we establish a sufficient condition for a covering of C M to be an isomorphism in terms of the subgeometry U and its images under CM Ž z .. Let R s O 2 Ž CM Ž z .. which is extraspecial of order 2 25 . We start by defining the folding C of C M with respect to the action of R. The kernel of the action of C s CM Ž z . on C M is ² z : and since O 2 Ž Cir² z :. s 1 for i s 1, 2, 3, and 12, the action of Rr² z : is fixed-point free. Let C be the folding of C M with respect to the action of R. This means that C is a geometry whose elements are the orbits of R on C M with two such orbits O1 and O 2 incident if and only if an element from O1 is incident in C M to an element from O 2 . Since Rr² z : acts fixed-point freely on C M , it is easy to see that if O1 and O 2 are incident in C then each element from O1 is incident in C M to exactly one element from O 2 . Let J be the collinearity graph of C which is also the folding with respect to the action of R of the collinearity graph J M of C M.

FIG. 2. Distribution diagram of C.

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We put C s CrR and use the bar notation for the images of C of subgroups of C. Then s is a subgroup of order 3 in C and NC Ž s . ; 3 ? Suz ? 2 which is a maximal subgroup in C. This enables us to identify the vertices of J with the Suzuki-type subgroups of order 3 in C ; Co1. We will use the following properties of the action of C on J. LEMMA 5.1. Let C ( Co1 , J be the set of Suzuki-type subgroups of order 3 in C, s g J, and C Ž s . s NC Ž s . ; 3 ? Suz ? 2. Then C acts primiti¨ ely on J while C Ž s . has 5 orbits on J:  s 4 , J1Ž s ., J 2 Ž s ., J 3 Ž s ., and J 4Ž s . with lengths 1, 22,880, 405,405, 1,111,968, and 5346, respecti¨ ely. Let J denote also the graph on J in¨ ariant under the action of C, in which s is adjacent to the ¨ ertices from J1Ž s .. Let m i g J i Ž s . and Bi s C Ž s . l C Ž m i . for i s 1, 2, 3, 4. Then Ža. d g J _  s 4 is adjacent to s in J if and only if w s , d x s 1, so that J is the folding of J M with respect to the action of R; the distribution diagram of J is gi¨ en on Fig. 3; Žb. B1 ; 3 2 ? U4Ž3. ? 2 2 , locally J is the commuting graph D of central subgroups of order 3 in C Ž s .rs ; Suz ? 2; Žc. B2 ; 2 1q 6 ? U4Ž2. ? 2 acts transiti¨ ely on J i Ž s . l J1Ž m 2 . for i s 1, 2, and 3, the subgraph induced on J1Ž m 2 . l J1Ž s . is the disjoint union of 40 copies of the complete 3-partite graph K 4, 4, 4 , these copies are permuted primiti¨ ely by B2rO 2 Ž B2 . ; U4Ž2. ? 2, ² s , m 2 : ( SL 2 Ž3.;

FIG. 3. Distribution diagram of J.

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Žd. B3 ; J 2 : 2 = 2 acts primiti¨ ely on J1Ž m 3 . l J i Ž s . for i s 1, 4 and transiti¨ e for i s 2, ² s , m 3 : ( AltŽ5.; Že. B4 ; G 2 Ž4. ? 2 acts primiti¨ ely on J1Ž m4 . l J i Ž s . for i s 1 and 3, ² s , m4 : ( AltŽ4.; Žf.

the subgraph induced on J1Ž m i . l J1Ž s . is empty for i s 3 and 4;

Žg. each ¨ ertex from J1Ž m 3 . l J 3 Ž s . is adjacent to a ¨ ertex from J1Ž m 3 . l J1Ž s . or to a ¨ ertex from J1Ž m 3 . l J 2 Ž s .. Proof. The subdegrees, 2-point stabilizers Bi of the action of C on J and ² s , m i : are well known Žcf. Lemma 49.8 in wAsx or Lemma 2.22.1Žii. in wILLSSx.. The distribution diagram on Fig. 3 is taken from wPSx. This diagram and the structure of B1 show that the subgraph induced on J1Ž s . is isomorphic to the graph D as in Lemma 1.4 and that C Ž s . induces its full automorphism group. This means that B1 acts transitively on J1Ž m 1 . l J i Ž s . for i s 1, 2, 3, 4 and hence for every vertex g at distance 2 from s in J, C Ž s . l C Žg . acts transitively on J1Ž s . l J1Žg .. Let x i be the permutational character of C Ž s . on the cosets of Bi s for i s 1, 2, and 4. By Lemma 2.13.1 in wILLSSx the inner product of x 1 and x i is 5, 3, and 2 for i s 1, 2, and 4, respectively. This implies the transitivity statements in Žc., Žd., and Že.. By wAtx every action of B3 of degree 100 or 280 as well as every action of B4 of degree 2080 or 20,800 is primitive. Let d i g J1Ž s . l J1Ž m i . for i s 2, 3, 4. Then since locally J is D, the distance from s to m i in the subgraph induced on J1Ž d i . is i. Hence the subgraph induced by J1Ž m i . l J1Ž s . is empty for i s 3 and 4, while for i s 2 it is locally K 4, 4 Žcompare Lemma 1.4Že... It is well known and easy to check that K 4, 4, 4 is the only connected graph which is locally K 4, 4 and the structure of the subgraph induced on J1Ž m 2 . l J1Ž s . follows. Finally, every transitive action of B2 of degree 40 is primitive and has O 2 Ž B2 . in its kernel. Thus all statements except Žg. are proved. We will prove Žg. with the roles of s and m 3 interchanged. For this we first determine the orbits of B3 on J1Ž s .. Let A s ² s , m 3 :. Then A ( AltŽ5.. Note that there exist exactly two elements r g J l A such that ² r , s : ( AltŽ4. and ² r , m 3 : ( AltŽ5.. Without loss m4 is one of these two. Put J s NB 3Ž m4 . s B4 l B3 . Then J is of index two in B3 and J ; J 2 ? 2. Put K [ B4 . Then K ; G 2 Ž4. ? 2. As the main step in determining the orbits of B3 on J1Ž s . we compute the orbits of J by decomposing the orbits of K. By Že., K acting on J1Ž s . has two orbits, G1 s J1Ž s . l J 3 Ž m4 . and G2 s J1Ž s . l J1Ž m4 . with lengths 20,800 and 2080, respectively, moreover if K 1 and K 2 are the respective stabilizers, then K 1 ; U3 Ž3.: 2 = 2 and K 2 ; 3 ? L3 Ž4. ? 2 2 . Consider the graph S with 416 vertices of valency 100 on which K acts as a

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rank 3 automorphism group Žsee wBvLx.. Then the parameters of S are ¨ s 416,

k s 100,

l s 315,

l s 36,

m s 20.

It follows from the list of maximal subgroups in K, that G1 can be identified with the set of edges of S while J is the stabilizer in K of a vertex x of S. By well known properties of the action of K on S wBvLx the orbit of J on the edge-set of S containing an edge  y 1 , y 2 4 of S is uniquely determined by the pair  d1 , d 2 4 where d i is the distance from x to yi in S. This and the parameters of S given above show that under the action of J the set of edges of S Židentified with the set G1 . splits into four orbits V 1 , V 2 , V 3 , and V 4 corresponding to the pairs of distances  0, 14 ,  1, 14 ,  1, 24 , and  2, 24 and having lengths 100, 1800, 6300, and 12,600, respectively. Let V 5 s J1Ž s . l J1Ž m 3 . and g g V 5 . Note that J acts transitively on V 5 and < V 5 < s 280. By Ža., g commutes with m 3 . Thus g F J F K 9 and so g g G2 and V 5 is an orbit for J on G2 . Let K 2 be the stabilizer of g in K. Then g s O 3 Ž K 2 .. By Žf. all 280 vertices adjacent to g in the subgraph induced on J1Ž s . are in G1 and by Ža. these 280 vertices are fixed by g . Let SŽg . be the set of vertices in S fixed by g . Comparing the permutation characters of K on s with the permutational character of K on G2 , we see that K 2 has exactly two orbits on the vertex set of S. On one hand this means that under the action of J the set G2 splits into two orbits namely V 5 and an orbit V 6 of length 1800. On the other hand K 2rg ; L3 Ž4. ? 2 2 acts transitively on SŽg . and so < SŽg .< s 280 ? Ž< J
k s 10,

l s 45,

l s 0,

m s 2.

Hence we conclude that K 2 acts transitively on the set of edges in S fixed by g . Again since g is adjacent in J to exactly 280 vertices from G1 there are 280 edges in the subgraph of S induced on SŽg . and hence this subgraph is the Gewirtz graph rather than its complement. Note that V i , 1 F i F 6 are the orbits for J on J1Ž s .. If B3 normalizes K 9 then K 9 centralizes ² s , m 4B 3 : s A and so K 9 F B3 , a contradiction. Since K 9 is generated by the elements of G2 s J l K we conclude that B3 does not normalize G2 . Thus some of the orbits of J must be fuzed by B3 . Since J is normal in B3 , only orbits with the same lengths can fuse. Thus V 2 j V 6 is a single orbit of B3 . The distribution diagram of J enables us to identify V 5 , V 3 , V 2 j V 4 j V 6 , and V 1 with J1Ž s . l J i Ž m 3 . for i s 1, 2, 3, and 4, respectively. A vertex from G1 is adjacent to g in J if and only if the corresponding edge of S is fixed by g . The parameters of the Gewirtz graph imply that g is adjacent to 10, 90, 180

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vertices from V i for i s 1, 3, and 4, respectively. Since every vertex from G1 is adjacent to 28 s 280 ? < G2
k s 176,

l s 495,

l s 40,

m s 48.

The image W of W s CMmŽ z . in C is isomorphic to U6 Ž2. ? SymŽ3.. Since locally C Žas well as C . is the commuting graph Q of 3-central subgroups of order 3 in U5 Ž2. ? 2 which is strongly regular, it is easy to see that in terms of Lemma 5.1, C :  s 4 j J1Ž s . j J 2 Ž s .. ˜ ª C M be a covering of C M such that there is a flat-transitive Let D : C ˜ which commutes with D and whose induced automorphism group of C action on C M coincides with that of Cr² z :. In particular D can be the restriction to a connected component of C G of the morphism of C G onto C M induced by the homomorphism x : G ª M. In this case C˜ is the setwise stabilizer in CG Ž z .r² z : of that connected component. Let R˜ be ˜ be the kernel of the natural homomorphism of C˜ onto C˜ ; CrR. Let J ˜ so that there is a natural morphism of J ˜ the collinearity graph of C onto J. Let C and W be as in Lemma 4.4. Let C be the image of C in J and W be the image of W in C. Let C be a connected component of the ˜ be the stabilizer of C ˜ in the preimage preimage of C under D and let W ˜ of Wr² z : in C.

˜ is isomorphic to C, W˜ ; Wr² z : ; LEMMA 5.3. In the abo¨ e notation C ˜ l R˜ is elementary abelian of order 2 2 . 2 2 ? U6 Ž2. ? SymŽ3. and hence W Proof. The result follows from Lemma 4.4 and the fact that C is the collinearity graph of the geometry U which is simply connected by wMex. Let T Ž s . be the set of images of C under C which contain s . Equivalently we can define T Ž s . to be the set of images of C under

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401

˜. Let T˜Ž s˜ . be the set of connected NC Ž s .. Let s˜ be a preimage of s in C ˜ ˜ ˜ maps onto some F g T Ž s .. If subgraphs F such that s˜ g F and F ˜ ˜ ˜ ˜ ˜ ˜ in C, ˜ then by F g T Ž s˜ . and U [ C Ž F . is the setwise stabilizer of F 2 ˜ ˜ ˜ Lemma 5.3, O 2 ŽU . s U l R is of order 2 . Let ˜.. N F ˜ g T˜Ž s˜ . . R˜s s O 2 Ž C˜Ž F

¦

LEMMA 5.4.

;

˜ R˜s s R.

ˆ be the folding of J ˜ with respect to the orbits of R˜s . This Proof. Let J ˆ are the orbits of R˜s on the vertex set of J ˜ means that the vertices of J ˆ with the induced adjacency relation. Notice that in the way it is defined J is not necessary vertex-transitive although every automorphism from C˜ ˆ Nevertheless stabilizing s˜ can be realized as an automorphism of J. ˆ is equal to J and in particular it is eventually we will see that J vertex-transitive. Since the vertices of J can be considered as orbits of R˜ ˜ and R˜s is contained in R, ˜ there is a covering v : J ˆ ª J and on J ˜ ˜ ˆ Rs s R if and only if v is an isomorphism. Let sˆ be the image of s˜ in J. ˜ ˜ ˜ Since Rs is normalized by the stabilizer C Ž s˜ . of s˜ in C, there is a ˆ which stabilizes sˆ and subgroup CˆŽ sˆ . in the automorphism group of J maps isomorphically onto C Ž s . ; 3 ? Suz ? 2. We will identify CˆŽ sˆ . and ˆ let J ˆ 1Ž dˆ. be the set of vertices adjacent to dˆ in J. ˆ Since C Ž s .. For dˆ g J ˆ 1Ž dˆ. is isomorphic to D and if v is a covering, the subgraph induced on J dˆ s sˆ then CˆŽ sˆ . induces the full automorphism group of this subgraph. Hence CˆŽ sˆ . has exactly three orbits, on the vertices at distance 2 from sˆ . ˆ i Ž sˆ ., so that v ŽJ ˆ i Ž sˆ .. s J i Ž s . for 2 F i F 4. We denote these orbits by J ˆ ˆ Ž . Let m g J s and B be the stabilizer of m ˆi ˆ i in CˆŽ sˆ .. We assume that i ˆ i ˆ there is a vertex m ˆ 1 g J1Ž sˆ ., adjacent to m ˆ i for 2 F i F 4 and that m ˆ 3 is adjacent to m ˆ 2 and m ˆ 4 . Assuming also that v Ž m ˆ i . s m i , we can consider Bˆi as a subgroup in Bi , 1 F i F 4. Notice that Bˆi acts transitively on the set ˆ 1Ž sˆ . l J ˆ 1Ž m ˆ 1Ž sˆ . l J ˆ i .. Since v is a covering, the subgraph induced by J ˆ Ž . J1 m ˆ 2 is union of m disjoint copies of K 4, 4, 4 where 1 F m F 40. For ˜ g T˜Ž s˜ . the image F ˆ of F ˜ in J ˆ is isomorphic to C as in Lemma 5.2 F ˆ 1Ž sˆ . j J ˆ 2 Ž sˆ .. The parameters of C imply and is contained in  sˆ 4 j J that m G 3. Since B2 acts primitively on the 40 copies of K 4, 4, 4 as in Lemma 5.1Žc. we have m s 40 and Bˆ2 s B2 . By Lemma 5.1Žc., Bˆ2 has ˆ 1Ž m three orbits on the vertices from J ˆ 2 . with lengths 480, 5120, and ˆ i Ž sˆ . for i s 1, 2, and 3, 17,280, moreover, these orbits are contained in J respectively. In particular Bˆ2 l Bˆ3 has order divisible by 2 7. By Lemma ˆ 1Ž sˆ . l J ˆ 1Ž m 5.1Žd. the stabilizer in B3 of a vertex from J ˆ 2 . has order not divisible by 2 7 and so Bˆ3 l Bˆ1 is a maximal subgroup of B3 not containing Bˆ2 l Bˆ3 . Thus Bˆ3 s B3 . Arguing similarly Bˆ3 l Bˆ4 and Bˆ1 l Bˆ4 are two

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different maximal subgroups of B4 and so Bˆ4 s B4 . Let rˆ be a vertex adjacent to m ˆ i for i s 2 or 4. By Lemma 5.1Žc., rˆ is conjugate under CˆŽ sˆ . to m ˆ j for some 1 F j F 4, except maybe in the case where rˆ is adjacent to m ˆ 2 and rˆ maps onto an element of J 2 Ž s .. In the latter case we see from the distribution diagram of D that such a rˆ can already be found in the residue of m 1. Hence in any case a vertex adjacent to m ˆ i for i s 2 or 4 is in ˆ j Ž sˆ . for 1 F j F 4. Suppose that there is a vertex nˆ which is adjacent to J m ˆ 3 and whose distance from sˆ is 3. By Lemma 5.1Žg. there must be a ˆ 1Ž m ˆ j Ž sˆ . for j s 1, or 2 which is adjacent to nˆ. As we vertex in J ˆ3. l J have seen above, this is impossible. Hence there are no vertices at distance 3 from sˆ and v is an isomorphism. COROLLARY 5.5. C is the uni¨ ersal completion of the amalgam Ž C1 , C2 , C12 , W ..

6. CONSTRUCTION OF THE 2-LOCALS As above let G denote the universal completion of the amalgam Ž Mi N i s 1, 2, 3, 8. and x be the homomorphism of G onto M which is identical on this amalgam. We will consider the Mi ’s as subgroups both in M and G. The group G acts flag-transitively on the universal cover G of M . The points, lines, planes, and quadrics in G and M are the cosets of M1 , M2 , M3 , and M8 in G and M, respectively. We follow notation introduced in the beginning of Section 4, so that X stays for an arbitrary completion of the amalgam which has M as an quotient. Let s s M1 viewed as a point stabilized by M1 , d s M8 viewed as a 1q 24 quadric stabilized by M8 , z an involution from P1U , C s CM Ž z . ; 2q ? Co1 , and R s O 2 Ž C .. Our nearest goal is to construct in CG Ž z . a subgroup C˜ which maps isomorphically onto C. As above let J be the graph on the set of points t with z g PtU in which two points are adjacent if they are incident to a common quadric. We will obtain C˜ as the stabilizer in CG Ž z . of the connected component of J containing s . Let V be a maximal clique in J containing s , H be the setwise stabilizer of V in C X Ž z ., and put A s O 3 Ž H .. Then by Lemma 4.1, H ; ² z : = 3 6 ? 2 ? Mat 12 , moreover there is a unique point a collinear to s , and inverted by z, such that H s CMaŽ z . and O 3 Ž H . s Pa l H. We use notation introduced in Lemma 4.2, so that A and B are orbits of H s Hr² A, z : on the set of subgroups of order 3 in A with lengths 12 and 132, respectively, while L is the unique orbit of length 12 of H on the set of hyperplanes of A. Then it is straightforward to identify A with the vertices in V. Let  s , d 4 be the edge of V incident to d. Then ² s , d : s CQ dŽ z .. Besides s and d there are two subgroups, say r and r 9 of order 3 in

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CQ dŽ z .. These subgroups are of Fischer type, and lie in the orbit B. Since r F Pa we can define Mr as in Lemma 2.2. Since CM dŽ z . ; 2.3 2 ? U4Ž3. ? D 8 , we have CM dŽ z . l Mr ; 2.3 2 ? U4Ž3. ? 2 2 . Moreover by Lemma 3.2, z is a 2-central involution in Mr and 1q 12 CMrŽ z . ; Ž 3 = 2q . ? 3 ? U4 Ž 3. ? 2 2 .

Put C0 s CMrŽ z . and R 0 s O 2 Ž C0 .. Recall the choice of m and the definition of W before Lemma 4.4. In particular s , d F W and both s and d act non-trivially on O 2 ŽW .. Thus one of r and r 9 centralizes O 2 ŽW .. We choose notation so that r centralizes O 2 ŽW .. Recall the definition of Ci , i s 1, 2, 3, 12 before Lemma 4.3, where we choose V 2 s  s , d 4 . So C1 s CM Ž z ., C2 s CM Ž z ., and C12 s H s CM Ž z .. s s , d 4 a LEMMA 6.1. Ža. R s Ł L g L CR Ž L.; Žb. R 0 s Ł r F L g L CR Ž L.. Proof. The image in CrR ( Co1 of H is the full normalizer of the image of A which shows that R 0 F R and R 0 s CR Ž r .. Note that w Rr² z :, A x is a non-trivial GF Ž2.-module for H of dimension at most 24. The restriction of this module to A is a direct sum of irreducible 2-dimensional modules and the kernel of such a summand is a hyperplane. The hyperplanes appearing as kernels form a union of orbits under H. By Lemma 4.2 there are no orbits of length less than 12 and L is the only orbit of length 12. This implies Ža.. Since r acts fixed-point freely on RrR 0 , we have Žb.. PROPOSITION 6.2. C is the uni¨ ersal completion of the amalgam Ž C0 , C1 , C2 , C12 , W . of subgroups of C. Proof. Let C˜ be the universal completion of the amalgam and as usual ˜ By Lemma 4.4Žb., C2 l W ; 3 2 ? view the Ci and W has subgroups of C. U4 Ž2. ? 2 and so C2 l W normalizes no non-trivial 2-subgroup of O 2 Ž C0rR 0 .. Thus O 2 ŽW . F R 0 . Since H l W ; 3 4q 1 ? 2 ? SymŽ6. we conclude from Lemma 4.2 applied to the dual of A that Ž H l W . A s NH Ž A l W . and that there exists unique elements L1 and L2 in L with L1 l L2 F A l W. Let U s ² O 2 ŽW . A :. Then Ur² z : is a subspace in R 0r² z : of dimension at least 4 centralized by CAŽ O 2 ŽW ... Thus by Lemma 6.1Žb., CAŽ O 2 ŽW .. is the intersection of two members of L . Hence CAŽ O 2 ŽW .. s L1 l L2 , U s CR 0Ž L1 .CR 0Ž L2 ., r F L1 l L2 , and < U < s 2 5. Put V s CR 0Ž L1 .. We conclude from Lemma 6.1Žb. that NH l C 0Ž V . ; 3 6 ? 2 ? SymŽ5.. On the other hand Ž H l W .` is normal in Ž H l W . A and so Ž H l W .` centralizes all conjugates of O 2 ŽW . under A. Thus

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Ž H l W .` F NH Ž V .. It follows that NH Ž V . s ² NH l C Ž V ., Ž H l W .` : ; 0 3 6 ? 2 ? Mat 11 . In particular, H acts doubly transitive on the 12 elements of 1q4 V H and since VV h ( 2q for h g H l C0 _ NH Ž V . we conclude that H 1q24 ˜ R [ ² V : ( 2q . ˜ By definition H normalizes R. ˜ We claim that R˜ is normal in C. H l C 0 :. Let t g H l C2 _ C0 . As C0 l C2 is of index Moreover, R 0 s ² V two in C2 , t normalizes C0 l C2 . Also t permutes r and r 9 and we conclude that R˜ s R 0 R 0t is normalized by R 0 , C0 l C2 , and t. Thus both ˜ Since C1 s C0 s R 0 Ž C0 l C2 . and C2 s Ž C0 l C2 .² t : normalize R. ˜ ² C1 l C2 , C1 l H :, R˜ is indeed normal in C. ˜ ˜ is a completion of the amalgam Note that CrR

˜ ˜ C RrR, ˜ ˜C ž C RrR, 1

2

12

˜ ˜ WRrR ˜ ˜ . RrR,

/

˜ we can apply Corollary 5.5 and conclude that CrR ˜ ˜( C ( As O 2 ŽW . F R, 1q 24 ˜ ˜ Co1. Thus C ; 2q ? Co1 and since C is a quotient of C, we obtain C˜ ( C. In view of the preceding proposition our nearest goal is to find such an amalgam inside of G. The first part, namely finding the subgroups, is already accomplished. Indeed the groups C0 , C1 , C2 , C12 s H and W had been defined for X, in particular for G and for M. It remains to show that the pairwise intersections are the same when regarded as subgroups of G and M, respectively. The fact that the pairwise intersections between C1 , C2 , H, and W are correct follows immediately from the definitions of these groups. Also H F Ma and C2 F Md . Since r is perpendicular to Qa in Q d we conclude from Lemma 2.2 that C0 intersects C2 and H correctly. Moreover, NC 1Ž r . F NMs Ž r . F NMs Ž² sd :. F Md and so C0 and C1 intersect correctly. It remains to check the intersection C0 l W. As C0 F Mr and W F Mm this is accomplished by LEMMA 6.3.

NMrŽ m . s Mr l Mm .

Proof. Let F s rm. Then F is a non-degenerated 2-space of ‘‘plus’’type with respect to the Md invariant quadratic form on Q d . Hence Ž . Ž . NM dŽ r , m . ; 3 8 ? Vy 6 3 ? 2 and Frr is of type 3C in Mr rr ( M 24 Žcompare wAtx.. This shows that NM Ž m . s NM Ž r , m . F NM Ž m . F Mm . r d d COROLLARY 6.4. Let C˜ be the subgroup of G generated by C0 , C1 , C2 , H, 1q 24 and W. Then C˜ ; 2q ? Co1 and C˜ is the normalizer of the connected component of J containing s . We now proceed finding the remaining terms E4 and E8 Žcf. Section 3. of the 2-local geometry of M. Of the 3-local subgroups considered so far

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405

only the normalizers of Fischer type subgroups contain a conjugate of E4 . ŽThis follows from the fact E4 centralizes all subgroups of odd order in M which are normalized by E4 .. This is not enough to reconstruct N as a subgroup of G and we are forced to first locate a further 3-local subgroup of G containing E4 . By Lemma 4.2Žg. there exists a 2-space F in A all of whose 1-spaces are in B and so of Fischer type. Moreover NH Ž F .rCH Ž F . ( GL2 Ž3. and there exists L1 , L2 in L with F F L1 l L2 . Choose F so that r F F and let d be a further Fischer type subgroup of F. We are trying to locate subgroups of NG Ž F . and for this we will produce a quadric d9 with F F Q d9. Let z9 be an involution in H so that PaU z s PaU z9, but Pa z / Pa z9. Then by Lemma 3.3Žb., z9 s z r for some r g Ma . Let A9 s Ar and V9 s V r. Since si s CA9Ž L i . has 12-conjugates under H l H r ; 2 2 ? Mat 12 , si g V9. Thus  s 1 , s 2 4 is an edge in V9 and there exists a unique quadric d9 adjacent to a , s 1 , and s 2 . In Q d9 we see that Q d9 l Pa s s 1 s 2 w Q d9 l Pa , z9x and w Q d9 l Pa , z9x has order 3 5. As w Qa , z9x s AQa and CAQ Ž s 1 s 2 . has order 3 5 we conclude that a CAQ aŽ s 1 s 2 . s w Q d9 l Pa , z9x. Hence F F Q d9. Since all 1-spaces in F are of Fischer type, F is a non-degenerate Ž . Ž . 2-space of ‘‘minus’’-type in Q d9 and CM d9Ž F . ; 3 8 ? Vq 6 3 . Since C M d9 F F Mr we conclude wAtx that Frr is of type 3 A in Mrrr , which means that Ž . CMrŽ F . ; 3 2 ? PVq 8 3 . Let g be a point incident to d9 such that Q g is perpendicular to F in Q d9. Then Q g is centralized by a Sylow 3-subgroup of CM d9Ž F .. Hence Q g FrF is 3-central in CM d9Ž F .rF and so also 3-central in CMrŽ F .. Thus CMrŽ F . l NMrŽ Q g . is a maximal subgroup of CMrŽ F . different from CM d9Ž F .. Hence CMrŽ F . s CM d9Ž F . , CMr Ž F . l NMr Ž Q g . F² NM d9Ž d . , NM g Ž d .: F Md .

¦

;

Put T s CMrŽ F .. We conclude that T s CMdŽ F . and so NH Ž F . normalŽ .. Ž . izes T. Put MF s TNH Ž F .. Then MF ; Ž3 2 = PVq 8 3 ? GL2 3 and in particular, MF maps isomorphically onto the full normalizer of F in M. ˜ As z centralizes F, Note that CM FŽ z . s NH Ž F .CT Ž z . : HCMrŽ z . : C. 3Ž . qŽ . z g O T s T 9 ( PV 8 3 . As NH Ž F . induces the full group of outer automorphisms on T 9 and by wAtx, T 9 has a unique class of involutions invariant under all automorphisms, z is 2-central in T 9. In particular, there exists a pure Conway foursgroup E in T 9 with z g E F O 2 Ž CT 9Ž z .. F ˜ Let t be an involution in E distinct from z. Then O 2 Ž CMrŽ z .. s R 0 F R. t s z g for some g g T 9 F Mr l MF . Put C˜t s C˜g . Then by conjugation of the corresponding statements for z we get CMrŽ t . F C˜t and CM FŽ t . F C˜t .

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LEMMA 6.5. CC˜Ž E . F C˜t . Proof. Put CE s CC˜Ž E .. Then CE ; 2 2q 11q22 ? Mat 24 . Moreover CE l Mr s NC EŽ r . and so modulo O 2 Ž CE ., CE l Mr has shape 3 ? SymŽ6.. Similarly modulo O 2 Ž CE . the intersection CE l MF is of shape 3 2 ? GL2 Ž3.. By wAtx no proper subgroup of Mat 24 has two such subgroups and thus CE s ² CE l Mr , CE l MF : O 2 Ž CE .. Since r has fixed points on any composition factor for CE on O 2 Ž CE . this implies CE s ² CE l Mr , CE l MF : F C˜t . Let E8 be a pure Conway type eight subgroup of T 9 such that E8 F O 2 Ž CT 9Ž x .. for all 1 / x g E8 and E F E8 . Put E4 s E and for i s 4, 8 put CE i s F1 / x g E i C˜x . Then by Lemma 6.5, CE i s CC Ž Ei .. Moreover NT Ž Ei . normalizes CE i and induces on Ei its full automorphism group. Put ˜ s CE 8 NT Ž E8 .. Then x maps the amalgam Ž C, ˜ N, ˜ L ˜. N˜ s CE4 NT Ž E4 . and L ˜ be the isomorphically onto the amalgam Ž C, N, L. as in Section 3. Let M ˜ N, ˜ and L. ˜ Then by Lemma 3.1, x maps M˜ group generated by C, isomorphically onto M. Thus to complete the proof of Lemma 1.6 it remains to show that G s M. For this note first that Mr is generated by its ˜ Moreover, M1 and M8 are both generated by intersection with C˜ and N. ˜ Finally M1 and M8 generate G and so their intersections with Mr and C. G s M and Lemma 1.6 is proved.

REFERENCES wAsx wAtx wBCNx wBvLx wBFx wDMx wIv1x wIv2x

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