1118 15. THE CASE .Cr(G, T) = (/JIf L ~ L 2 (p) for p a Mersenne or Fermat prime, then p > 7 by 15.2.24, and
CL 0 (Z) =SL. Then as K centralizes Z, and AutMI(L) is a 2-group, [K,£ 0 ] = 1,so L:::; N 1 (K) :::; Mc= !M(H) by 15.2.12.3, contrary to the choice of L.
Thus L is not L 2 (p). In the remaining cases m 3 (L) = 2, so L = 031 (I) by
A.3.18. Thus K :::; CL(Z), so m 3 (CL(Z)) = 2. Inspecting the list of groups
remaining in 1.1.5.3, we conclude L ~ J 4. But then 02 (CL(Z))/0 2 (0^2 (CL(Z)) ~
M22, A whereas Ca(Z) =Mc by 15.2.14.1, Kc= 0 3' (Mc) by part ( 4) of 15.2.12, and
Kc contains no such section by part (2) of the latter result. This contradiction
completes the proof of Theorem 15.2.15.
15.2.3. The final contradiction. For X:::; G, write A(X) for the subgroup
generated by all involutions of X.
LEMMA 15.2.25. (1) Case (1) or (4) of 15.2.1 holds, with M ~ 83, D10, or
Sz(2).(2) A(T):::; S.
PROOF. If (1) fails, then conclusion (ii) of 15.2.13.1 holds. In this case there
is Z1 of order 2 in Zs with C.M(Z1) ~ 83. In particular as Mn Mc= CM(V)T by
15.2.14.5, CM(Z 1 ) 1. Mc, contrary to Theorem 15.2.15. Therefore (1) holds. By
(1), S = !11(f'), so (2) holds. D
Recall UH= (VH) from Notation 15.2.9.
LEMMA 15.2.26. (1) r(G, V) > 1 and hence s(G, V) > 1.
(2) Wo(T, V):::; CT(V).(3) Wo(QH, V):::; CH(UH)·
PROOF. Let Ube a hyperplane of V. Suppose first that Mis not Sz(2). Then
Z =Zs is of rank 2 by examination of the cases in 15.2.25.1, and hence Zn U =/= 1.
Then as Ca(Z) =Mc by 15.2.14.1, Ca(Z) = Ca(ZnU). Similarly for g EM -Mc,
Ca(Z9) = Ca(Z9 n U) and ZZ^9 = V, so that
Ca(U):::; Ca(Z n U) n Ca(Z^9 n U)) = Ca(Z) n C 0 (Z^9 ) = Ca(V).
Thus r(G, V) > 1 in this case.
So assume M ~ Sz(2). In this case m(Zs) = 2 and m(Z) = 1. Here M
has two orbits on nonzero vectors of V of lengths 5 and 10, and hence two orbits
on hyperplanes of V, which are also of lengths 5 and 10. Notice by 15.1.9.5 that
Hypothesis E.6.1 is satisfied, so if U is T-invariant then E.6.13 says Ca(U) :::;
Na(V). If U is not T-invariant, then IUMI = 10, so as f' is cyclic, we may assume
that s normalizes U and hence centralizes a nontrivial 2-subspace of U, so that
Zs = Cv(s) :::; U. As V = (ZM), there exists g E M - Mc with Z9 1. U. By
Theorem 15.2.15, Ca(Z~ n U) :::; Mg, so as Mg = C 0 (Z9) by 15.2.14.1, with
Z^9 1. Z~ n U =/= 1 and Z~ of rank 2, we conclude that Ca(Z~ n U) = Ca(Z~). Thus
Ca(U) = Ca(V) as in the previous paragraph.
Therefore r(G, V) > 1 in either case. Since m(M, V) > 1 by 15.2.14.6, also
s(G, V) > 1, so that (1) holds. Furthermore a(M, V) = 1 by 15.2.14.6. Now
E.3.21.1 implies (2), and (2) implies (3). D
LEMMA 15.2.27. (1) 02,F*(Mc):::; Kc(M n Mc)·
(2) V:::; 02(Mc)·
(3) If Zs n V9 =/= l, then [V, V9] = 1.