486 INTRODUCTION TO VOLUME IIThis leaves the question of why the boundary of the partition according to
size occurs when e( G) = 2, rather than e( G) = 1 or 3 or something else. Theanswer is that when one passes to p-locals for odd primes p, e( G) ~ 3 is needed
in order to use signalizer functors. (See e.g. chapter 15 of [Asc86a]). Namely·
such methods can only be applied to subgroups E which are elementary abelianp-groups of rank at least 3, and E needs to be in a 2-local because of connectedness
theorems for the prime 2 (which will be discussed briefly in the next section). Using
both signalizer functors and connectedness theorems for the prime 2, one can showthat the centralizer of some element of E looks like the centralizer of a semisimple
element in a group of Lie type and characteristic 2. Then this information is usedto recognize G as a group of Lie type.^3
Thus, in both programs, the two partitions of the simple groups indicated
above, into groups of "even" and "odd" characteristic, and into large and small
groups, give rise to a partition of the proof of the Classification into four parts.
Since groups of even characteristic include those of characteristic 2-type, our Main
Theorem determines the groups in one of the four parts-the small even part-in
the first generation program.
To integrate our result into the GLS second-generation proof, we need to rec-
oncile our notion of "even characteristic" with the GLS notion of "even type". Theformer notion is more natural in the context of the unipotent methods of this work,
but the latter fits better with the GLS semisimple methods. Our Even Type Theo-
rem provides the transition between the two notions, and is relatively easy to prove.We will say a little more about that result in section 0.4 of this introduction. The
Main Theorem, together with the Even Type Theorem, determine the groups in
the small even part of the second generation program.0.2.3. Some history of the quasithin problem. We close this section with
a few historical remarks about quasithin groups, and more generally small groups
of even characteristic.
The methods used in attacking the problem go back to Thompson in the N-
group paper [Tho68J; in an N-group, all local subgroups are assumed to be solvable.
In particular, Thompson introduced the parameter e(G), and used weak closure
arguments, uniqueness theorems, and work of Tutte [Tut47] and Sims [Sim67].
We discuss some of these techniques in the next section; a more extended discussion
appears in the Introduction to Volume I.
Groups G of characteristic 2-type with e(G) small were subsequently studied
by various authors. Note that e( G) = 0 means that all 2-locals are 2-groups, which
is impossible in a nonabelian simple group of even order by an elementary argument
going back to Frobenius; cf. the Frobenius Normal p-Oomplement Theorem 39.4 in
[Asc86a]. In [Jan72], Janko defined G to be thin if e(G) = 1, and used Thompson's
methods to determine all thin groups of characteristic 2-type in which all 2-locals are
solvable. His student Fred Smith extended that classification from thin to quasithin
groups in [Smi75]. The general thin group problem was solved by Aschbacher in
[Asc78b]. Mason went a long way toward a complete treatment of the general
quasithin case in [Mas], which unfortunately has never been published. See however
his discussion of that work in [Mas80].
(^3) In both the original proof of OFSG and in the GLS project, the case e( G) = 3 requires
special treatment.