834 12. LARGER GROUPS OVER F2 IN .Cj(G, T)
PROOF. Recall Mv acts as A 8 or 8s on the set 0 of eight points. Thus if I is
an involution in Mv, the Mv-conjugacy class ofI is determined by the number n(I)
cycles of I of length 2 on 0. Thus n(I) = 1, 2, 3, 4, and we check in the respective
cases that I is of Suzuki type (cf. Definition E.2.6) bi, c2, b3, az on the orthogonal
6-space V. In particular,
Cv (I) -=/:-Cv (3) for involutions ] -=/:-I. ( *)
We first prove (1) and (2), so we assume that either A E A3(Mv, V) or B E
Az(Mv, V). Then m(A) > 3 by (*), so that m(A) = 4 = m2(Lf'). Similarly
m(B) ~ 3.
Now the possibilities for A of rank 4 are described in cases (ii)-(iv) in the
proof of 12.6.22. If A :::; L, then A is in case (ii); thus A is conjugate to R1 =
J(L n T), whereas R 1 ~ A 3 (Mv, V). Therefore A 1:. L, so we are in case (iii) or
(iv), and hence we may take I= (1, 2) E A. Let W := Ov(I) and X := CMv (I).
Then (I) = CMv(W) and AutA(W) E A 3 (Aut_x(W), W). However Wis the coreof the 6-dimensional permutation module for 86 , and we compute directly that
a(8 6 , W) :::; 2. This contradiction completes the proof of (1).
Next Z(T) is generated byf := (1, 2) (3, 4) (5, 6) (7, 8);
and
J := Ov(f) =I EB (v),
where v := ei,3,5,7 and I:= (e1,2, e3,4, e5,6, e1,s). Let Io := [V, ~ and Y := CMv (f).
Then I is isomorphic to the 3-dimensional quotient of the permutation module forAuty-(I) ~ 84 or A4 on {e1,2, es,4, e5,6, e1,s}, Auty(I) :::; NaL(I)(Io), and Oy-(I) =
CMv (I) is
x := ((1,2),(3,4),(5,6),(7,8)),
when T 1:. L, or X n L, when T :::; L. In either case, Cy-(I)/ (f) acts faithfully as a
group of transvections on J with axis I.Since f is 2-central in M, we may take B :::; CMv(f) = Y, and then either B
centralizes I or AuttJ(I) E A 2 (Auty-(I), I).
Suppose first that B centralizes I. Then B :::; Cy-(I) :::; X, so as m(B) ~ 3, we
check that Cv(B) =I. If m(B) = 3 then m(B n L) ~ 2, and for each b E f3# n L,
m(Ov(b)) = 4 > m(I), so (b) E £(B, 1). Thus 1£(B, 1)1 > 2, and hence (2) holds
with fJ = 1. On the other hand if m(B) = 4 then B = X and Ov(B) =I. In thiscase we take D := (J) ford:= (1, 2). Then for each of the three other transpositions
d' in B, m(Cv( (d, d') )) = 4 > m(I), so that (d, d') E &(B, D), and again (2) holds.
So suppose instead that AuttJ(I) E A 2 (Auty-(I),I)). We saw that Auty-(I) is
a subgroup of P := NaL(I) (Io)~ 84 containing A4, so from the action of GL(I) on
I, Az(P,I) = {0 2 (P)}, and hence AuttJ(I) = 02 (P) is of rank 2. Hence it is easy
to calculate that (f) = C_x(B), so as m(B) ~ 3 and OfJ(I) :::; C_x(B), we conclude
that CtJ(I) = (f'; and m(B) = 3. In particular each member of f3# is regular on
0, of rank 3, and for each b E f3#, Cv(b) is of rank 4, so that (b) E &(B, 1). Hence
1£(B, 1)[ = 7, so that (2) holds with fJ = 1.
It remains to prove (3) and (4), so we may assume that for some y E G, A:=
VY n T with [A, VJ-=/:- 1 and k := m(VY /A):::; 1. Hence [VY, V]-=/:-1, so by 12.6.27,
v n VY = 1. Then [V n Na(VY), A] = 1, so in particular v 1:. Na(VY). On theother hand for each Ao:::; A with m(VY /Ao):::; 2, Ca(Ao):::; Na(VY) by 12.6.20.3.
Thus for each A 0 :::; A with m(A/Ao) < 3-k, Cv(A 0 ) :::; V n Na(VY) :::; Cv(A), so