16.2. NORMALITY AND OTHER PROPERTIES OF COMPONENTS 1173PROOF. Let R := K/02(K). Since Lis a component of Ca(t) and i centralizes
L by hypothesis, Lis also a component of Cca(i)(t). We apply I.3.2 with Ca(i)in the role of "H": As O(Ca(i)) = 1 by (El), 02',E(Ca(i)) = E(Ca(i)) and the
2-components in that result are components. Then either
(i) K = KiKi for some component Ki i-Ki of Ca(i), L/02(L) ~ R 1 , and
L = CK(t)=, or
(ii) K is at-invariant component of Ca(i), and Lis a component of CK(t).In case (i), since G is quasithin, the possibilities in (2) are obtained by intersecting
the list of A.3.8.3 with that of (E2). Therefore we may assume that case (ii) holds,
and K > L since otherwise conclusion (1) holds. The simple group R is described
in (E2). The cases in (3) arise by inspecting 16.1.4 and 16.1.5 for involutionsi E Aut(K) such that CK(i) has a component. We use 16.1.2.1 to conclude that
02 (K) = 1 or 02 (L) = 1 when appropriate; in case (i) of (3), Z(L) = 1 from the
structure of the covering group K of R ~Ru in I.2.2.7a. DLEMMA 16.1.8. Assume t is an involution in G, L is a component of Ca(t), i
is an involution in Ca((t,L)), and SE Syb(Ca(i)) with IS: Cs(t)i :S 2. Then L
is a component of Ca ( i).
PROOF. Assume otherwise, and set K := (LE(Oa(i))); then K is described
in case (2) or (3) of 16.1.7, and it remains to derive a contradiction. As S E
Syb(Ca(i)) and K is subnormal in Ca(i), SK := Sn K E Syb(K). Further[SK: CsAt)i :SIS: Cs(t)i :S 2. However in case (2) of 16.1.7,
ISx: CsK(t)i ~ IKi/02(K1)l2 > 2,
so case (3) must hold. But in each subcase of (3), [SK: CsK(t)i > 2, a contradiction
establishing the lemma. D
16.2. Normality and other properties of components
Let P denote the set of pairs (z, L) such that z is a 2-central involution in G
and Lis a component of Ca(z).
LEMMA 16.2.1. P "f 0.
PROOF. Let T E Syb(G). By Hypothesis 16.1.1, G is not of even charac-
teristic, so there is M E M(T) such that 02 (F(M)) i- 1 and there is 1 i-z E
D 1 (Z(T)) n 02 (M). Then 02 (F(M))) :S 02 (F(CM(z))), so that F(CM(z)) #
02 (CM(z)). Then as M = Na(02(M)) since M E M, F*(Ca(z)) i-02(Ca(z))
by 1.1.3.2. On the other hand by Hypothesis 16.1.1, G is of even type, so by (El),
O(Ca(z)) = 1. Therefore E(Ca(z)) i-1, so there is a component L of Ca(z), and
then (z,L) E P. D
In view of 16.2.1, we assume for the remainder of the chapter:
NOTATION 16.2.2. TE Syl 2 (G), z is an involution in Z(T), (z, L) E P, Gz :=