1170 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC
The definition of even type is given on p.55 of [GLS94]. We will not need toassume that m 2 ( G) ~ 3 as in part (3) of that definition. Part (2) of that definition
says that if K is a component of the centralizer of an involution, then K/Z(K) is
in the set C 2 of simple groups listed in Definition 12.1 on page 100 in [GLS94];
and also that Z ( K) satisfies further restrictions given in the final sentences of that
definition. We will not reproduce that full list here, since as G is a QTK-group,
K/0 2 (K) also appears in Theorem C (A.2.3). Instead, intersecting the list of
possibilities from Theorem C (A.2.3) with the list of possibilities in Definition 12.1
on page 100 of [GLS94], it follows that when G is a QTK-group, G is of even type
if and only if:
(El) O(Ca(t)) = 1 for each involution t E G.
(E2) If L is a component of Ca(t) for some involution t E G, then one of the
following holds:
(i) L/0 2 (£) is of Lie type and characteristic 2 appearing in case (3) or (4) of
Theorem C; but Lis not SL 2 (q), q = 5, 7, 9 or As/Z2. Further if L/02(L) ~ £3(4),
then (02(L)) = 1.
(ii) L ~ L3(3) or L2(p), pa Fermat or Mersenne prime.
(iii) L/02(L) is Mn, M12, M22, M23, M24, J2, J4, HS, or Ru.
Observe that from Theorem C, in case (i) either L/0 2 (£) is of Lie rank 1, and so
of Lie type A 1 == £ 2 ,^2 B 2 = Sz, or^2 A 2 = U 3 ; or L/0 2 (£) is of Lie rank 2 and of
Lie type A2, B2, G2,^2 F4, or^3 D4; or Lis £4(2) or £5(2).
In the remainder of this introductory section assume that L is a component of
the centralizer of some involution of G, and set L := L/Z(L). Thus by Hypothesis
16.1.1, L is one of the quasisimple groups listed above. To provide a more self-
contained treatment, in this introductory section we collect some facts about L
which we use frequently.
First, inspecting the list of Schur multipliers in I.1.3 for the groups Lin (E2),
and recalling , that O(L) = 1 by (El), we conclude:
LEMMA 16.1.2. (1) If L is not simple, then Z(L) = 02 (£) and L ~ Sz(8),
£3(4), G2(4), M12, M22, J2, HS, or Ru.
(2) Either IZ(L)I ::::; 2, or L ~ Sz(8) or L3(4) with Z(L) ~ E4, or L ~ M22
with Z(L) ~ Z4.
Occasionally we need more specialized information about the quasithin groups
appearing in 16.1.2, which can be obtained from knowledge of the covering groups
L of L. Such facts are collected in I.2.2.
In the next two lemmas, we list the involutory automorphisms of L and their
centralizers in L. Notice that we write L rather than L in those cases where
Z(L) = 1by16.1.2.1.
We begin with the groups of Lie type and characteristic 2 in case (i) of (E2),that is, in case (3) or (4) of Theorem C. Recall that the involutions in classical
groups of characteristic 2 are determined up to conjugacy by their Suzuki type: In
orthogonal and symplectic groups, the types are denoted ak, bk, ck, as discussed in
Definition E.2.6; in linear and unitary groups, the types are denoted jk, as discussed
in Aschbacher-Seitz [AS76a]. In each case, k is the dimension of the commutator
space for the involution on the natural module for the classical group.