496 INTRODUCTION TO VOLUME IIThis fundamental lemma can be used to show first that L :::l Ca(z)-which
is very close to showing that L commutes with none of its conjugates. Then the
fundamental lemma also shows that L :::l Ca(i) for each involution i E Cc(L), after
which it is a short step to showing that Cc(L) is tightly embedded in G, and Lis
standard in G.
At this point, we could quote some of the theory of standard subgroups and
tightly embedded subgroups (developed in [Asc75] and [Asc76]) to simplify the
remainder of the proof. But since GLS do not use this machinery, we content
ourselves with using only elementary lemmas from that theory which are easy to
prove; the lemmas and their proofs are reproduced in sections I. 7 and I.8. In
particular, we use I.8.2 to see that our hypothesis that G is not of even characteristic
shows that for some £9 distinct from L, an involution of Ca(L^9 ) normalizes L; this
provides the starting point for our analysis. Then, making heavy use of the factthat z is 2-central, and that the component.£ is highly restricted by the even type
hypothesis, we eliminate all configurations except Na(L) ~ Z 2 x £2(4). Then
we identify G = J 1 via the structure of Cc(z) as noted above. Along the way,
we encounter various shadows coming from groups which are not perfect, like thegroups in the examples in subsection 0.2.1. In most such cases it is possible to
apply transfer to contradict G = 02 ( G), given the fact that the Sylow 2-group T
of G normalizes L.This shows the advantages of introducing the notion of a group of "even char-
acteristic", and hence of the the partition of the quasithin groups of even type into
those of even characteristic, and those of even type which are not of even char-
acteristic: The first subclass we studied via unipotent methods, and the latter by
semisimple methods at the prime 2. If instead we had used unipotent methods to
treat only the more restricted subclass of groups of characteristic 2-type, then our
semisimple analysis at the prime 2 would have had to deal with the shadows of thenonsimple configurations in subsection 0.2.1, in which involution centralizers Cc(z)
with components do not contain a Sylow 2-group T of G. When z is not 2-central
the road to obtaining T, so that one can show G is not simple via transfer, is much
longer and very bumpy.As a final remark, we recall that for the generic groups of even type, GLS are
able switch to semisimple analysis of elements of odd prime order, and so are able
to avoid dealing with shadows of the nonsimple examples of subsection 0.2.1. Thus
they do not need the concept of groups of "even characteristic" in their generic
analysis.