Symplectic Amalgams

1 Introduction.- 1.1 Symplectic Amalgams.- 1.2 Goldchmidt G4-Amalgam Again.- 2 Preliminaries.- 2.1 Some Group Theory Results.- 2.2 Some Representation Theory Results.- 2.3 Sesquilinear Forms.- 2.4 Two Theorems of McLaughlin.- 2.5 Ultraspecial and Extraspecial Groups.- 2.6 Tensor Products and Group Actions on p-Groups.- 2.7 The Goldschmidt Amalgams.- 3 The Structure of SL2(q) and its Modules.- 3.1 Group Theoretic Properties of SL2(q).- 3.2 Modules for SL2(q).- 4 Elementary Properties of Symplectic Amalgams.- 4.1 The Coset Graph.- 4.2 Proof of Theorem 1.6.- 5 The Structure of Q?.- 6 The L?-Chief Factors in V?.- 7 Reduced Symplectic Amalgams.- 7.1 A Reduced Symplectic Subamalgam.- 7.2 Reduced Amalgams and Consequences of Theorem 6.1.- 8 The Largest Normal p?-Subgroup of L?/Q?.- 9 The Components of L?/Q?.- 9.1 The Action of L? on Compp(L?).- 9.2 Two or more Normal Components in L?/Q?.- 10 The Reduction to Quasisimple when $$
C_{U_/alpha } (U_/alpha /Z_/alpha ) /nleqslant Q_/beta
$$.- 11 A First Look at the Amalgams with |V?/Z(V?)| = q4.- 11.1 A Characteristic 3 Amalgam.- 11.2 The Proof of Theorem 11.1.- 12 The Story so Far.- 13 Groups of Lie Type.- 13.1 Weyl Groups and Parabolic Subgroups.- 13.2 Sylow p-subgroups of Lie Type Groups.- 13.3 Automorphisms and Centres.- 13.4 The Order of Abelian p-subgroups.- 13.5 Extremal Subgroups.- 14 Modules for Groups of Lie Type.- 14.1 Modules in Characteristic p.- 14.2 Module Results for Low Rank Groups of Lie Type.- 14.3 Modules for Lie Type Groups and (2, q)-Transvections.- 14.4 Natural Modules for Orthogonal Groups.- 14.5 Natural Modules for the Symplectic Groups.- 14.6 Natural Modules for G2(q).- 14.7 Some Spin Modules.- 14.8 Modules for Lie Type Groups in Non-defining Characteristic.- 14.9 Some Non-containments.-15 Sporadic Simple Groups and Their Modules.- 16 Alternating Groups and Their Modules.- 17 Rank One Groups.- 18 Lie Type Groups in Characteristic p and Rank ?.- 18.1 A Subamalgam of A.- 18.2 The Examples.- 18.3 L?/Q? a Symplectic Group and V?/Z(V?) a Spin Module.- 19 Lie Type Groups and Natural Modules.- 19.1 The Symplectic and Orthogonal Groups.- 19.2 Sp4(2) — A Special Case.- 19.3 Groups of Type G2(q).- 20 Lie Type Groups in Characteristic not p.- 21 Alternating Groups.- 21.1 Large Alternating Groups.- 21.2 Small Alternating Groups.- 22 Sporadic Simple Groups.- 23 The Proof of the Main Theorems.- 24 A Brief Survey of Amalgam Results.- 24.1 Amalgam Results.- 24.2 Pushing-up.- References.- Indexs.