Symplectic Amalgams

The aim of this book is the classification of symplectic amalgams - structures which are intimately related to the finite simple groups. In all there sixteen infinite families of symplectic amalgams together with 62 more exotic examples. The classification touches on many important aspects of modern group theory: * p-local analysis * the amalgam method * representation theory over finite fields; and * properties of the finite simple groups. The account is for the most part self-contained and the wealth of detail makes this book an excellent introduction to these recent developments for graduate students, as well as a valuable resource and reference for specialists in the area.

Table of Contents:
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.