Geometry of Sporadic Groups II. Representations and Amalgams A.A. Ivanov and S.V. Shpectorov July 14, 2016 Contents Preface v 1 Preliminaries 1 1.1 Geometries and diagrams . 1 1.2 Coverings of geometries . 3 1.3 Amalgams of groups . 4 1.4 Simple connectedness via universal completion . 6 1.5 Representations of geometries . 10 I Representations 15 2 General features 17 2.1 Terminology and notation . 17 2.2 Collinearity graph . 20 2.3 Geometrical hyperplanes . 21 2.4 Odd order subgroups . 24 2.5 Cayley graphs . 28 2.6 Higher ranks . 29 2.7 c-extensions . 30 2.8 Non-split extensions . 34 3 Classical geometries 37 3.1 Linear groups . 37 3.2 The Grassmanian . 39 1 3.3 Pe is uniserial . 41 3.4 G(S4(2)) . 43 3.5 Symplectic groups . 44 3.6 Orthogonal groups . 47 3.7 Brouwer's conjecture . 48 3.8 G(3 · S4(2)) . 53 3.9 G(Alt5) ............................. 55 n [ ]2 3.10 G(3 2 · S2n(2)) . 57 i ii CONTENTS 4 Mathieu groups and Held group 65 4.1 G(M23) ............................. 65 4.2 G(M22) ............................. 66 4.3 G(M24) ............................. 69 4.4 G(3 · M22)............................ 70 4.5 D(M22) ............................. 74 4.6 G(He).............................. 77 5 Conway groups 79 5.1 Leech lattice . 79 5.2 G(Co2).............................. 82 5.3 G(Co1).............................. 85 5.4 Abelianization . 86 23 5.5 G(3 · Co2)........................... 88 5.6 G(3 · U4(3)) . 92 6 Involution geometries 95 6.1 General methods . 95 6.2 I(Alt7) ............................. 98 6.3 I(M22) ............................. 100 6.4 I(U4(3)) . 102 6.5 I(Co2; 2B) ........................... 104 6.6 I(Co1; 2A) ........................... 107 7 Large sporadics 111 7.1 Existence of the representations . 111 7.2 A reduction via simple connectedness . 114 7.3 The structure of N(p) . 116 7.4 Identifying R1(p) . 122 7.5 R1(p) is normal in Rbpc . 126 7.6 Rbpc is isomorphic to Ge(p) . 131 7.7 Generation of Ge(p) \ Ge(q)................... 133 7.8 Reconstructing the rank 3 amalgam . 134 7.9 G(34371 · BM) . 138 II Amalgams 139 8 Method of group amalgams 141 8.1 General strategy . 141 8.2 Some cohomologies . 143 8.3 Goldschmidt's lemma . 147 8.4 Factor amalgams . 150 8.5 L3(2)-lemma . 151 8.6 Two parabolics are sufficient . 154 CONTENTS iii 9 Action on the derived graph 155 9.1 A graph theoretical setup . 155 9.2 Normal series of the vertex stabiliser . 158 9.3 Condition (∗i) . 161 9.4 Normal series of the point stabiliser . 164 9.5 Pushing up . 168 10 Shapes of amalgams 171 10.1 The setting . 171 10.2 Rank three case . 173 10.3 Rank four case . 177 10.4 Rank five case . 181 10.5 Rank six case . 183 10.6 The symplectic shape . 183 10.7 Summary . 184 11 Amalgams for P -geometries 187 11.1 M22-shape . 187 11.2 Aut M22-shape . 189 11.3 M23-shape . 190 11.4 Co2-shape . 192 11.5 J4-shape . 197 11.6 Truncated J4-shape . 203 11.7 BM-shape . 204 12 Amalgams for T -geometries 209 12.1 Alt7-shape . 209 12.2 S6(2)-shape . 210 12.3 M24-shape . 212 12.4 Truncated M24-shape . 215 12.5 The completion of Af . 218 12.6 Co1-shape . 221 12.7 M-shape . 226 12.8 S2n(2)-shape, n ≥ 4 . 226 Concluding Remarks 233 13 Further developments 235 13.1 Group-free characterisations . 235 13.2 Locally projective graphs . 238 Bibliography 240 Index 249 iv CONTENTS Preface This is the second volume of the two-volume series which contains the proof of the classification of the flag-transitive P - and T -geometries. A P -geometry (Petersen geometry) has diagram P ◦ ◦ · · · ◦ ◦ ◦; 2 2 2 2 1 where ◦ P ◦ denotes the geometry of 15 edges and 10 vertices of the 2 1 Petersen graph. A T -geometry (Tilde geometry) has diagram ∼ ◦ ◦ · · · ◦ ◦ ◦; 2 2 2 2 2 where ◦ ∼ ◦ denotes the 3-fold cover of the generalized quadrangle of 2 2 ∼ order (2; 2), associated with the non-split extension 3 · S4(2) = 3 · Sym6. The final result of the classification as announced in [ISh94b], is the following (we write G(G) for the P - or T -geometry admitting G as a flag- transitive automorphism group). Theorem 1 Let G be a flag-transitive P - or T -geometry and G be a flag- transitive automorphism group of G. Then G is isomorphic to a geometry H in Table I or Table II and G is isomorphic to a group H in the row corresponding to H. In the first volume [Iv99] and in [IMe99] for the case G(J4) the following has been established. Theorem 2 Let H be a geometry from Table I or II of rank at least 3 and H be a group in the row corresponding to H. Then (i) H exists and of correct type (i.e., P - or T -geometry); (ii) H is a flag-transitive automorphism group of H; ∼ (iii) either H is simply connected or H = G(M22) and the universal cover of H is G(3 · M22). v vi Geometries of sporadic groups II Table I. Flag-transitive P -geometries Rank Geometry H Flag-transitive automorphism groups H 2 G(Alt5) Alt5, Sym5 3 G(M22) M22, Aut M22 G(3 · M22) 3 · M22, 3 · Aut M22 4 G(M23) M23 G(Co2) Co2 23 23 G(3 · Co2) 3 · Co2 G(J4) J4 5 G(BM) BM G(34371 · BM) 34371 · BM If F is a geometry and F is a flag-transitive automorphism group of F then A(F; F) denotes the amalgam of maximal parabolics associated with the action of F on F. In these terms the main result of the present volume can be stated follows: Theorem 3 Let G be a flag-transitive P - or T -geometry of rank at least 3 and G be a flag-transitive automorphism group of G. Then for a geometry H and its automorphism group from Table I or Table II we have the following: A(G; G) =∼ A(H; H): In the above theorem we can assume that H is simply connected. Then by Theorem 1.4.5 H is the universal cover of G and H is the universal completion of A(G; G). Notice that Theorem 3 immediately implies that a geometry H from Table I or Table II does not have flag-transitive automorphism groups ex- cept those already in the tables. Particularly, the largest of the groups corresponding to H is the full automorphism group. Preface vii Table II. Flag-transitive T -geometries Rank Geometry H Flag-transitive automorphism groups H ∼ 2 G(3 · S4(2)) 3 · Alt6, 3 · S4(2) = 3 · Sym6 3 G(M24) M24 G(He) He 4 G(Co1) Co1 5 G(M) M n n [ ]2 [ ]2 n G(3 2 · S2n(2)) 3 2 · S2n(2) Now in order to deduce Theorem 1 from Theorems 2 and 3 it is sufficient to observe the following Proposition 4 The set of geometries in Tables I and II is closed under taking coverings commuting with the actions of the flag-transitive automor- phism groups given in these tables. Proof. Let H be a geometry from Table I or II and H be a flag- transitive automorphism group of H (also from the table). Suppose that σ : H! H is a proper covering which commutes with the action of H on H and let H be the action induced by H on H. Let N be the kernel of the homomorphism of H onto H (the subgroup of deck transformations with respect to σ). In order to identify N we look at the normal structure of H. If Q = O3(H) then either Q is trivial or it is an elementary abelian 3-group which is irreducible as a GF (3)-module for H=Q. Furthermore, H=Q is either a non-abelian simple group or such a group extended by an outer automorphism of order 2, finally H does not split over Q. Hence either jHj ≤ 2, or N = Q, or N = 1. If jHj ≤ 2 then clearly H cannot act flag-transitively on a P - or T -geometry. If N = Q 6= 1, then the elements of H are the orbits of Q on H with the natural incidence relation. We know from [Iv99] that under these circumstances H is a cover of H only if the former is G(3 · M22) and the latter is G(M22) (in the other cases σ is only a 1- or 2-covering). Thus N = 1 and H acts flag-transitively on both H and viii Geometries of sporadic groups II H. In this case the 2-part of the stabilizer in H of a point of H must be strictly larger than that of the stabilizer of a point of H. This is impossible since for all the pairs (H;H) from the tables the stabilizer in H of a point from H contains a Sylow 2-subgroup of H. 2 Below we outline our main strategy of proving Theorem 3. Let G be a P - or T -geometry of rank n ≥ 3, G be a flag-transitive automorphism group of G and A = A(G; G) = fGi j 1 ≤ i ≤ ng be the amalgam of maximal parabolics associated with the action of G on G (here Gi = G(xi) is the stabilizer in G of an element xi of type i in a maximal flag Φ = fx1; :::; xng in G).