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
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