970 i3. MID-SIZE GROUPS OVER F2Hence (3) holds by application of the Burnside Fusion Lemma to the elements of
A and B. Next as M = Na(A), (4) follows from (3) and the identification a given
after the statement of Theorem 13.9.1, which says A is the orthogonal module for
M/A ~ ot(2) with zM the singular points and tM the nonsingular points. Now
(4) implies (5), and then as Z(T) = (z) has order 2, tis not 2-central by (5), so (6)
holds. DFrom now on let B be the group defined in 13.9.2.1, and set K :== Na(B). AsB :'.SI T by that result, K E 1i(T) ~He by 1.1.4.6.
In the following lemma, "diagonal involutions" in J(M) are those projecting
nontrivially on both factors of the decomposition. J ( M) = Mi x M2. The next two
lemmas follow from straightforward calculations.LEMMA 13.9.3. M has 6 classes of involutions Lli, 1 ::::; i ::::; 6, where
(1) Lli := zM consists of the diagonal involutions in A.
(2) Ll2 := tM = (An Ti)# u (An T2)#.
(3) Ll3 consists of the involutions in Mi - A and M2 - A.
(4) Ll4 consists of the diagonal involutions iii2 with ik E Mk n Ll3, k = 1, 2.
(5) Ll5 consists of the diagonal involutions ij with i E Mk nL'.13 and j E M3-k n
Ll2, k = 1,2.(6) Li 6 := sM consists of the involutions in M - J(M).
LEMMA 13.9.4. B n Li1 = {z }, IE n Li2I = 2, and IE n Liil = 4 for i = 3, 4, 5.
Further each set is an orbit under T.
LEMMA 13.9.5. Gz > T, so Gz f:. M.
PR09F. Assume that Gz = T. We will obtain a contradiction using Thompson
Transfer A.1.36 on s, based on an analysis of fusion which will eventually include
the explicit identification of 02 (Gt)·
First CM(s) ~ Z 2 x 8 4 is not a 2-group, sos~ z^0. Suppose that z is weakly
closed in B with respect to G. Then z^0 n M = Lli = zM by 13.9.4 and 13.9.3, and
Ca(z) = T ::::; M by hypothesis. Therefore by 7.3.1 in [Asc94], Mis the uniquefixed point of z on G/M. Hence by 7.4.2 in [Asc94), s^0 n M = sM. Therefore as
sM ~ M - J(M), s ~ 02 (G) by Thompson Transfer, contradicting G simple.
Therefore z is not weakly closed in B with respect to G. On the other hand,
CM(i) is not a 2-group for i E Ll2 U Ll3, so
z^0 n M ~ Li1 u L'.14 u L'.15 (*)
by 13.9.3. Also by 13.9.2.3, z^0 n B = zK. Thus by (*), and since the sets in
13.9.4 are T-orbits, zK is of order 5 or 9, so in particular, T < K. Set V := \zK)
and K := K/CK(V). Then 02 (K) = 1 by B.2.14, so that 02 (K) ::::; CK(V).
Also CK(V) is a 2-group as T = Gz, so CK(V) = 02(K). On" the other hand
if A ::::; 02(K), then J(T) = AB ::::; 02 (K), so K ::::; Na(J(T)) = T by 13.9.2.2,
contrary to T < K. Thus A 1:. CK(V), and hence by B.2.5, Vis an FF-module for
K. Then 3 E 7r(K) by Theorem B.5.6, while CK(z) =Tis a 2-group, so
IK: Tl= lzKI ~ 9
rather than 5. Hence the inclusion in(*) is an equality, and as B = (Li 4 nB, Li 5 nB),
V = B ~ E15. Then as B = Cr(B), we conclude that 02 (K) = B. Inspecting the
subgroups of GL 4 (2) with Sylow group D 8 and of order 72, we conclude Bis the