CHAPTER 16
Quasithin groups of even type but not even
characteristicThe original proof of the Classification of the finite simple groups (CFSG) re-
quires the classification of simple QTK-groups G of characteristic 2-type. (RecallG is of characteristic 2-type if F*(M) = 02 (M) for all 2-local subgroups M of
G.) Mason produced a preprint [Mas] which goes a long way toward such a clas-sification, but that preprint is incomplete. Our Main Theorem fills this gap in the
"first generation" proof of CFSG, since we determine all simple groups in the larger
class of QTK-groups of even characteristic. (Recall G is of even characteristic if
F*(M) = 02(M) only for those 2-locals M containing a Sylow 2-subgroup T of G.)
The "revisionism" project (see [GLS94]) of Gorenstein-Lyons-Solomon (GLS)aims to produce a "second-generation" proof of CFSG. In GLS, the notion of char-
acteristic 2-type from the first-generation proof is replaced by the notion of even
type (see p. 55 in [GLS94]). In a group of even type, centralizers of involutions
are allowed to contain certain components (primarily of Lie type in characteristic
2). In particular, if the centralizer of a 2-central involution has a component, then
G is not of even characteristic, and so does not satisfy the hypothesis of our Main
Theorem.
To bridge the gap between these two notions of "characteristic 2" , this final
chapter of our work classifies the simple QTK-groups of even type. More precisely,
our main result Theorem 16.5.14 (the Even Type Theorem) shows that Ji is the
only simple QTK-group which is of even type but not of even characteristic. Thus
the simple QTK-groups of even type are the groups in our Main Theorem, of even
type, along with Ji.
To prove Theorem 16.5.14, we will utilize a small subset of the machinery onstandard components from the first generation proof of CFSG. In sections I.7 and
I.8 of Volume I, we give proofs of all but one of the results we use; that result is
Lemma 3.4 from [Asc75], which is a fairly easy consequen~e of Theorem ZD on
page 21 in [GLS99].
We are grateful to Richard Lyons and Ronald Solomon for their careful reading
of this chapter, and suggestions resulting in a number of improvements.
16.1. Even type groups, and components in centralizers
In this chapter, we assume the following hypothesis:
HYPOTHESIS 16.1.l. G is a quasithin simple group, all of whose proper sub-
groups are JC-groups, but G is not of even characteristic. On the other hand, G is
of even type in the sense of GLS.
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