0.2. CONTEXT AND HISTORY 483groups which again satisfy (QT) and (K), but instead of condition (E) they impose
a more technical condition (seep. 55 of [GLS94], and 16.1.1 in this work):
(E') G is of even type.The condition (E') allows certain components^2 in the centralizers of involutions t
(including involutions in Z(T), which are not allowed under our hypothesis of evencharacteristic); but these components can only come from a restricted list. To be
precise, a quasithin group G is of even type if:(E'l) O(Ga(t)) = 1 for each involution t E G, and
(E'2) If Lis a component of Ca(t) for some involution t E G, then one of the
following holds:
(i) L/02(L) is of Lie type _and in characteristic 2, but L is not SL 2 (q),
q = 5, 7, 9 or As/Z2, and if L/02(L) ~ L3(4) then 02(L) is elementary abelian.
(ii) L ~ L3(3) or L2(p), pa Fermat or Mersenne prime.(iii) L/02(L) is a Mathieu group, J 2 , J4, HS, or Ru.
In order to supply a bridge between our Main Theorem and the GLS program,
we also establish (as Theorem 16.5.14):
THEOREM 0.1.2 (Even Type Theorem). The Janka group J 1 is the only simplegroup of even type satisfying (QT) and (K) but which is not of even characteristic.
Since the groups appearing as conclusions to our Main Theorem are in fact of
even type, the quasithin simple groups of even type consist of J 1 together with that
list of groups.0.2. Context and History
In this section we discuss the role of quasithin groups in the Classification,
focusing on motivation for our basic hypotheses. We also recall some of the history
of the quasithin problem. Occasionally we abbreviate 'Classification of the Finite
Simple Groups" by CFSG.
0.2.1. Case division according to notions of even or odd "character-
istic". The Classification of the Finite Simple Groups proceeds by analyzing the
p-local subgroups of an abstract finite simple group G for various primes p. Further
for various reasons, which we touch upon later, the 2-local subgroups are preferred.
On the other hand the generic example of a simple group is a group G of
Lie type over a field of some prime characteristic p, which is the characteristic of
that group of Lie type. Such a group G can be realized as a linear group acting
on some space V over a finite field of characteristic p, and the local structure of
G is visible from this representation. For example if g E G is a p^1 -element (i.e.,
(lgl,P) = 1) then g is semisimple (i.e., diagonalizable over some extension field), so
its centralizer Ca(g) is.well-behaved in that it is essentially the direct product of
quasisimple groups of Lie type in characteristic p corresponding to the eigenspaces
of g. There are standard methods for exploiting the structure of these components.
On the other hand, if g is a p-element, then g is unipotent (i.e., all its eigenvalues
are 1), so Ga(g) has no components; instead its structure is dominated by the
unipotent subgroup
F*(Ga(g)) = Op(Ca(g))
(^2) See section 31 of [Asc86a] for the definition of a component of a finite group (namely
quasisimple subnormal subgroup), and corresponding properties.