Preface

At every stage the proofs of the major classification results concerning finite

simple groups have been inductive in nature. Each major theorem rests on a body

of results concerning the subgroups, automorphisms, coverings and representations

of the simple groups which arise in the conclusion of that theorem or which are

permitted by the hypotheses to arise as significant sections of a minimal counterex-

ample. It should come as no surprise then that the proof of the entire Classification

Theorem rests on a large body of such results concerning all of the finite simple

groups. The purpose of this volume is to establish fundamental facts about the

finite simple groupsâ€”those of Lie type, the alternating groups and the sporadic

groupsâ€”as well as to set up a framework for viewing these groups in which all of

the results needed in the ensuing volumes of this series can be derived. More de-

tailed specific results are postponed to the volumes in which they are first required.

The appropriate type of group to be analyzed here is an almost simple group X,

that is, one for which F*(X) is quasisimple.

As in the case of the preceding volume in this series, on general group theory, we

cannot and have not attempted an encyclopedic treatment. Our utilitarian focus on

the proof of the Classification Theorem has dictated our choice of topics, so that our

attention is devoted almost exclusively to local subgroup structure, automorphisms

and covering groups. In particular the vast and lively field of ordinary and modular

representations of the finite simple groups is barely touched. We do not even

broach the topic, important for us, of failure-of-factorization modules, quadratic

modules and related issues; indeed U. Meierfrankenfeld and G. Stroth are preparing

a monograph on this subject which we intend to add to our Background Results.

Furthermore, the extensive theory of the maximal subgroups of the finite simple

groups developed since 1980 is hardly represented here, since we need only a few

classical results concerning low rank groups of Lie type and a smattering of results

for sporadic simple groups. This reflects the fact that (luckily) extensive knowledge

of the subgroup structure of finite simple groups is necessary only for the smallest

of these groups.

The finite groups of Lie type get most of our attention, for their theory is

both richer than that of the alternating groups and better developed than that of

the sporadic groups. Our Background References for properties of these groups

are R. W. Carter's Simple Groups of Lie Type [Cal], R. Steinberg's Yale Lecture

Notes [Stl] and Steinberg's Memoir Endomorphisms of Linear Algebraic Groups

[St2]. In addition the Gorenstein-Lyons Memoir [GL1] is occasionally useful in

this connection. As the underpinning of our development we use the classification

theory of semisimple linear algebraic groups. Accordingly our opening chapter

reviews the main features of that theory which are important for our subsequent

computations, but omits proofs. This omission we make comfortably because of

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