1174 16. QUASITHIN GROUPS OF EVEN TYPE BUT NOT EVEN CHARACTERISTIC
PROOF. Let CT(t) ::; S E Syl2(CG(t)), so that CT(t) ::; Cs(z). Since T E
Syh(G),
IS: Cs(z)/::; IS: CT(t)I::; IT: CT(t)I::; 2,
and hence the lemma follows from 16.1.8 with z, tin the roles of "t, i". D
Of course the component Lis subnormal in Gz; the main result in this section
is 16.2.4 below, showing that in fact L is normal in Gz.
Our eventual goal will be to show that Lis standard in G, as defined in the next
section. As Ronald Solomon has observed, rather than proving that L is normal in
Gz, we might instead prove that Lis "terminal" in the sense of [GLS99J (ie. for
each t E CG(L), Lis a component of CG(t)), and then appeal to Corollary PU4 in
chapter 3 of [GLS99J to prove that Lis standard. Instead we show directly that
L is standard, later in 16.3.2. This allows us to keep our treatment self-contained,
and avoid an appeal to a fairly deep result such as Corollary PU4 of [GLS99], with
a minimal amount. of extra effort. ·
THEOREM 16.2.4. L ::::1 Gz.
Until the proof of Theorem 16.2.4 is complete, assume (z, L) is a counterexam-
ple. Set Lo:= (LGz) and H := NG(Lo). By A.3.8, IT: NT(L)I = 2 and Lo= LLu
for u E T - NT(L), so that T ::; H and [L, Lu] = 1. The possibilities for L are
obtained by intersecting the lists of A.3.8.3 and (E2); 16.1.2.1 allows only one case
with 02(L) =f-1:
LEMMA 16.2.5. Either L ~ L2(2n), Sz(2n), or L2(P) withp odd, or L/02(L) ~
Sz(8) with 02(L) =f-1.
In the remainder of this section we will eliminate the possibilities in the list of
16.2.5.LEMMA 16.2.6. (1) L is a component of CG(t) for each involution t E (z)Lu.
(2) If L/02(L) ~ Sz(8) and 02(L) =f-1, then L is a component of CG(s) for
each involutions E 02(L).
PROOF. Let t be an involution in (z)Lu. From our list in 16.2.5, either
(I) L is simple and has one conjugacy class of involutions, or(II) L/02(L) ~ Sz(8), and 02(L) =f-1.
If (I) holds, then conjugating in L, we may take t E Z(NT(L)); then as IT :
NT(L)/ = 2, Lis a component of CG(t) by 16.2.3, establishing (1) in this case.
Therefore we may assume that (II) holds. Let s be an involution in 02 (L);
then the same argument also establishes (2), since 02 (L) = Z(L) ::; Z(NT(L)) as
Out(L/Z(L)) is of odd order. Thus it remains to establish (1) in case (ii).
By (2), Lis a component of CG(s). Thus we can apply 16.1.7 to s, tin the roles
of "t, i". Ass EL::; 02 (CG(t)), s acts on each component of CG(t) by A.3.8.1, sothat case (2) of 16.1.7 does not occur. Also the only subcase of case (3) of 16.1.7 in
which L/Z(L) ~ Sz(8) is subcase (i), and in that subcase Lis simple, whereas here
02(L) =f-1. Thus case (1) of 16.1.7 holds, completing the proof that conclusion (1)
holds. D