0.2. CONTEXT AND HISTORY 485
Two non-simple configurations. Let L be a simple group of Lie type in char-
acteristic 2, and assume either
(a) G = L(t) is L extended by an involutory outer automorphism t, or
(b) G = (L x Lt)(t), for some involution t; i.e., G is the wreath product of L
by Z2.
Then G is in fact of even characteristic, but rarely of characteristic 2-type, since
Ga(t) usually has a component. However the components of Ga(t) are of Lie type
in characteristic 2, so G is also usually of even type. During the proof of the CFSG,
groups with the 2-local structure of those in (a) and (b) often arise. Under the
original approach, lengthy and difficult computations were required, to reduce to
a situation where transfer could be applied to show the group was not simple. In
the opinion of GLS (and we agree), the proof should be restructured to avoid these
difficulties. ·
This is achieved in GLS by replacing the old partition into characteristic 2-
type/not characteristic 2-type by the partition into even type/odd type, while we
achieve it for quasithin groups with the partition into even characteristic/not even
characteristic. Locals like those in the two classes of nonsimple groups above are
allowed under both the even characteristic hypothesis and the even type hypothesis,
but were not allowed under the older characteristic 2-type hypothesis. Thus under
the old approach, such groups would be treated in the. "odd" case by focusing on
the "semisimple" element t-rather artificially, as its order is not coprime to the
characteristic of its components-and usually at great expense in effort. Under
the new approach, such groups arise in the "even" case, where the focus is not on
Ga(t).
In the generic situation when G is "large" (see the next subsection for a dis-
cussion of size), GLS are able to avoid considering such centralizers by passing to
centralizers of elements of odd prime order, which can therefore be naturally re-
garded as semisimple. However, quasi thin groups G are "small", and in particular
the p-rank of G is too small to pass to p-locals for odd p; so we avoid difficulties
when G is of even characteristic by using unipotent methods applied to overgroups
of T, rather than semisimple methods applied to Ca(t). The case where G is of
even type but not of even characteristic is discussed later in section 0.4 of this In-
troduction. There we will again encounter local subgroups resembling those in our
two classes, when they appear as shadows in the proof of the Even Type Theorem.
0.2.2. Case division according to size. After the case division into char-
acteristic 2-type/not characteristic 2-type or even type/odd type described above,
both generations of the CFSG proceed by also partitioning the simple groups ac-
cording to notions of size. Here the underlying idea is that above some critical size,
there should be standard "generic" (i.e., size-independent) methods of analysis; but
that "small" groups will probably have to be treated separately.
In the even/ odd division of the previous subsection, we indicated that the
generic examples for the even part of the partition should be the groups of Lie type
in characteristic 2. For these groups the appropriate measure of size is the Lie rank
of the group, and as we mentioned in section 0.1, e(G) is a good approximation of
the Lie rank for G of Lie type and characteristic 2. From this point of view, the
quasithin groups are the small groups of even characteristic, so our critical value
defining the partition into large and small groups occurs at e( G) = 2.
