Preface

This volume contains the proofs of Theorems C2 and C3, as stated in the first

volume of this series [I2].

Theorems C2 and 63 constitute the classification of finite simple groups G of

special odd type. This condition requires that no 2-component K of an involution

centralizer of G is of generic type in the sense that KjOyiK) G S2? but G is not of

restricted even type. The latter condition is rather technical, but primarily it entails

that either G has 2-rank 2 or, for some 2-component K of an involution centralizer

of G, K/Z*(K) = L2(q) for some odd q or K/0^{K) is a member of a small

finite set of additional quasisimple groups. In fact, we prove a strengthened version

of Theorem C2, which we call Theorem Q^, in which the ban on 2-components of

generic type is relaxed. Theorem C2 permits us to classify all X-proper simple

groups G having a 2-Thin Configuration in the sense of [IIIi; 2.1]. This extension

is an essential ingredient in the proof of Theorem 67, the classification of groups

of generic type (begun in the previous volume and to be completed in the next

volume). The simple groups arising as conclusions to our theorems are the finite

simple groups of Lie type in odd characteristic of BN rank 1 or 2 (with some

exceptions) together with L±(q), (q odd, q ^ 1 (mod 8)), An, n G {7,9,10,11},

and the five sporadic groups Mu , M12, Mc, Ly and O'N.

The special odd condition represents our measure of smallness for simple groups

which are not of even type. Other measures have been used in the past, namely

2-rank, normal 2-rank or sectional 2-rank. Indeed, our list of conclusions differs

little from the conclusions of the Sectional 2-Rank 4 Theorem of Gorenstein and

Harada [GH1]. The Gorenstein-Harada Memoir depends on a long list of prior

results. Some of these likewise form part of our Background Results, most no-

tably the Feit-Thompson Theorem [FT1] on the solvability of groups of odd order,

and the body of results yielding recognition theorems for the split (B, iV)-pairs

of rank 1. Others have been incorporated into our proof of Theorem 62, notably

the classification of finite simple groups of 2-rank 2 by Gorenstein-Walter [GW1],

Alperin-Brauer-Gorenstein [ABG1] and Lyons [LI] and much of the classification

of finite simple groups with an abelian Sylow 2-subgroup by Walter [Wal], together

with involution centralizer recognition theorems for finite simple groups of Lie type

in odd characteristic of £W-rank 2 by Brauer [Br5], Fong and W. J. Wong [FW1],

[Fol].

The classification of finite simple groups of 2-rank at most 2 by Brauer and

Suzuki [BrSul], Feit and Thompson, Gorenstein and Walter, Alperin and Brauer

and Gorenstein, and Lyons was a major accomplishment of the 1960's. During the

late 1960's, the Signalizer Functor Method was developed, primarily by Gorenstein

and Walter, and with particular emphasis on the prime 2. The importance of

2-connectivity for this method (cf. [Is; Section 22]) again focussed attention on

ix