842 12. LARGER GROUPS OVER F2 IN L'.j(G, T)
Qz = P :::;! Gz, and thenasT/P 95: Ds is Sylow in Gz/P::; ot(2) 95: Ss, we conclude
from the list of maximal subgroups of Ss that either Y = Gz or Gz/ P 95: A1. From
the structure of Gt 95: PI' L 3 ( 4) / E 4 , we see that Gt,z is of order 3 · 29 , so that Y is
transitive on t^0 n P of order 14. Thus if Gz/ Pis A 7 , Gt,z contains an A5-section,
contradicting Gz,t a {2, 3}-group.
Therefore Gz = Y. We have also seen that z is not weakly closed in P with
respect to z, so that Theorem 44.4 of [Asc94] applies. Since Mv/V 95: 86, G ~
£ 5 (2), and since Gt f;_ M, G ~ M 24. Therefore as G is simple, Theorem 44.4 in
[Asc94] shows G 95: He. D- 7.3. The case V f;_ 02 (Gz), including the identification of M24· Be-
cause of Theorem 12.7.7, we can assume in the remainder of this section that
LEMMA 12.7.8. Gt ::; M.
LEMMA 12.7.9. (1) M controls fusion of its involutions.
(2) Gv is transitive on {V9 : v E V9} for each v EV.
(3) V is the unique conjugate of V containing t.
PROOF. By 12.7.8 and 12.7.4.2, tis not 2-central in G, so t ~ z^0. Thus (1)
follows from 12.7.4.1. Then (1) and A.l.7.1 imply (2), and (2) and 12.7.8 imply (3)
usi.ng A.1.7.2. D
LEMMA 12.7.10. (1) m(Mv, V) = 2.
(2) r(G, V) > 2. Hence s(G, V) = 2.
(3) If L < Mv then there are two classes Oj, j = 1, 2, of involutions in Mv
not in L. Further Zj E Oj, where (Ij) = Z(L/f'), and m(Cv(Ij)) = 3. Finally 22acts on a conjugate of Vt, but 21 does not.
(4) If U ::; V with m(V /U) = 3, then one of the following holds:
(i) CM(U) = CM(V).(ii) Up to conjugation in L, U is a hyperplane of V2 and CM(U) = CM(V)R2.
(iii) U = Cv(I) for some involution IE Mv - L, and CM(U) = (i)CM(V).
(5) If U ::; V with m(V/U) = 3 and Ca(U) f;_ M, then U = Cv(I) for some
IE 01 ..
PROOF. First Lis transitive on the set 0 of involutions in L, and by 12.7.2.4,
V2 = Cv(I) for IE 0 n R 2. Assume L < Mv. Then Mv 95: 86 by 12.7.2.1, so there
are two classes Oj, j = 1, 2, of involutions in Mv-L, and we can choose notation so
that Ij E Oj, where Ij is defined in (3). As Ij inverts X, m([V, Ij]) = 3, completing
the proof of (1). If we represent Mv on the set n of 6 cosets of H := N.Mv(Vt),
then each involution I E H -L induces a transposition on n. Consequently the
members of the other class 01 have cycle type 23 on D. This completes the proof
of (3), and part (4) also follows since CM(U) is a 2-group for each U ::; V with
m(V/U) < 4.
Next let U::; V. If U is a hyperplane of V, then 1 #-Un Vt, so Ca(U) ::; M by
12.7.8. Thus r(G, V) > 1. Assume U::; V with Ca(U) f;_ Mand k := m(V/U) < 4.
By E.6.12, CM(U) > CM(V). Hence U is centralized by some involution IE Mv
by (4). Thus if k = 2, we can take U = V2 by the previous paragraph; however ·
V2 = V 1 Vt, so Ca(U) ::; M by 12.7.8. We conclude k = 3, so r(G, V) > 2, and
hence s(G, V) = 2 using (1), so (2) holds. Indeed this argument shows U f;_ V2,
as each hyperplane of V2 intersects Vt nontrivially. Thus U = Cv (I) and I ~ L