484 INTRODUCTION TO VOLUME II
and in particular is more complex, so that this centralizer is more difficult to deal
with.
We seek to translate these properties of linear groups, and in particular the
notion of "characteristic", into analogous notions for abstract groups. If G is a
finite group and p is a prime, G is defined to be of characteristic p-type if each
p-local subgroup Hof G satisfies
F*(H) = Op(H),
or equivalently CH(Op(H)) ::; Op(H). Every group of Lie type in characteristic p
is of characteristic p-type; indeed for large p, they are the only examples of p-rank
at least 2-though for small primes, there are groups of characteristic p-type which
are not of Lie type in characteristic p.
If a simple group G of p-rank at least 3 is "connected" at the prime p (as
discussed in the next section) but is not of characteristic p-type, then the centralizer
of some element of order p will behave like the centralizer of a semisimple element
in a group of Lie type-that is, it will have components, making it easier to analyze.
Thus the aim is to find a prime p such that G has a reasonably rich p-local structure,
but G is not of characteristic p-type. Recall also that one chooses p to be 2 whenever
possible. The original proof of the Classification partitioned the simple groups into
two classes: those of characteristic 2-type, and those not of characteristic 2-type;
furthermore different techniques were used to analyze the two classes.In the remainder of this subsection, we'll try to give some insight into how more
recent work (done since the original proof of the Classification) has suggested that
it is useful and natural to change the boundary of this even/odd partition. Wementioned earlier that in the GLS program, the notion of even type replaces the
notion of characteristic 2-type. However for the purpose of dealing with quasithingroups, our notion of even characteristic seems to be more natural than that of even
type. Notice that a group of characteristic 2-type is of even characteristic, since
the former hypothesis requires all 2-locals to be of characteristic 2, while the latterimposes this constraint only on locals containing the Sylow group T. Thus the class
of groups of even characteristic is larger than the class of groups of characteristic
2-type, since the 2-locals in the former class are more varied.
In a moment, we will discuss two classes of groups where this extra flexibil-
ity is useful. But before doing so, we'll say a word about the influence of thesegroups and others on our work. In December 1996, Helmut Bender gave a talk at
the conference in honor of Bernd Fischer's 60th birthday, in which he suggested
approaching classification problems like ours with a list of groups in mind, to serveas a guide to where difficulties are likely to occur. However, that list should include
not only the "examples"-the groups which appear in the conclusion of the the-
orem; it should also include "shadows"-groups not in the conclusion, but whose
local structure is very close to that of actual examples, since these configurations of
local subgroups will also arise in the analysis, and typically they can be eliminatedonly with real effort. Thus in our exposition, we try to emphasize not only how the
examples arise, but also where the shadows are finally eliminated. Our Index lists
occurrences in the proof of examples and shadows.
In particular we must deal with shadows of the following two classes which areQTKE-groups but not simple-since it is hard to recognize locally that the groups
are not simple.