x PREFACE

signalizer functor theorem, which was later refined and extended by Goldschmidt,

Bender, Glauberman and McBride [Go2,Go3,Be4,Gll,McB2] . With further re-

finements introduced by Aschbacher, who also developed the theory of standard

components and tightly embedded subgroups [A4,A6], this approach culminated

in two major theorems which provide prototypes for the methods and results in

this volume. First, Aschbacher characterized the finite simple groups of Lie type

in odd characteristic [A9]. Second, for groups of characteristic 2 type containing a

sufficiently large elementary abelian p-group in a 2-local subgroup, Gorenstein and

Lyons [GL1] and Aschbacher [A13,AGL1] analyzed the local structure (this was

followed by recognition theorems of Gilman and Griess, Finkelstein and Frohardt

and Solomon [GiGr,FinFrl,FinS2,FinS3]).

This volume provides a unified treatment which grew out of much of the ma-

terial from those two works. We achieve two stages and part of the third stage in

the planned five stages of the Generic Case, analyzing the structure of centralizers

of elements of prime order in a generic simple group. Our analysis is thus the lineal

descendant of Gorenstein and Walter's work.

Characteristically for the Signalizer Functor Method, we are led to a dichotomy,

both forks of which will be resolved in later volumes. The non-triviality of a certain

signalizer functor leads to the existence of an almost strongly p-embedded subgroup.

When p = 2, i.e. in the cases leading to the alternating groups and to the groups of

Lie type in odd characteristic, we in fact have a contradiction via the 2-uniqueness

theorems established in Volume 4. However, when p is odd, further analysis will

appear in a later volume to arrive at a contradiction on this fork of the dichotomy.

On the other side, the triviality of the functor leads us to a "neighborhood" of

semisimple subgroups which have large intersections and are normal in centralizers

of subgroups of order p or p2. These are the data which will permit the identification

of G as either an alternating group or a group of Lie type. Again this identification

will be achieved in a subsequent volume of this series.

Working within the Classification Grid, we do not achieve proofs of such inde-

pendent results as the [/-Conjecture and the jB-conjecture, which had been proved

in the 1970's (see [Wa2,Wa3] for example, and [G4] for an account of these mile-

stones). Instead the conclusions of these conjectures are verified not for all local

subgroups but only for enough subgroups to enable us to construct our neighbor-

hood of semisimple subgroups.

We continue the notational conventions established in volume 2 of this series

[IG\- We refer to the chapters of the current book as [IIIi], [III2], [III3], [III4], [III5],

and [Illg]. As in the previous volume, the last chapter [IIIQ] collects the necessary

3C-group lemmas for the main chapters [III3, III4, III5] and thus logically precedes

them. In a handful of places results from [Ills] are used in [III2], or results within

these two chapters are used before they are proved, but this is just a consequence

of the way we have chosen to organize the many lemmas, and the reader may easily

verify that there is no circularity.1

The material in this volume, perhaps more than any other part of the classifi-

cation theorem, was close to Danny Gorenstein's heart and deeply pondered in his

1

Namely, in the proofs of Lemmas 2.3 and 2.4 of [III2], reference is made to Lemmas 2.4 and

3.5 of [Hie], and in Section 4 of [III2], we use Lemmas 2.2, 3.6 and 4.17 of [Ille]- Similarly Lemmas

7.1 of [III2] and 5.39, 7.8, 7.20 and 9.15 of [Hie] are each quoted once within their chapters before

they are proved.