i0.2. WEAK CLOSURE PARAMETERS AND CONTROL OF CENTRALIZERS 7438i,i x 8i,2 with 8i,j 9'! Ds, and T acts transitively as D 8 on the four members of
.6. := { 8i,j : i, j}. As 8 is the direct product of the subgroups in .6., by the Krull-
Schmidt Theorem A.1.15, Na(8) permutes r := {DZ(8): DE .6.}. Let K be the
kernel of Na(8) on rand Na(8)r := Na(8)/K. Then D 8 9'! Tr~ N 0 (8)r ~ 84.
Observe that for F E r, A(F) = {VF, AF} is of order 2, where VF := V n F.
Thus 02 (K) acts on each VF. Then as V ::::) T, K = 02 (K)(K n T) acts on
(VF : FE r) = V. Hence K ~ Na(V) = M, so as we are assuming Na(8) 1:. M,
there is x E Na(8) with x inducing a 3-cycle on r. Therefore Na(8)r 9'! 84. Let
KR be the preimage in Na(8) of 02 (Na(8)r) and R := T n KR· By a Frattini
Argument, Na(8) = K(Na(8) n Na(R)), so we may take x E Na(R). But R
normalizes just two members V and A := (AF : F E r) of A(8) = A(T), so xacts on V and A. Therefore Na(8) = KT(x) ~ Na(V) = M, contrary to our
assumption.
Thus in the remainder of the proof, we assume that C z (Lo) # 1. We maychoose H E 1-l*(T, M) with H ~ Na(8). Let E := fh(Z(8)), VH := (ZR), and
H* := H/CH(VH)· As usual VH E R'2(H) by B.2.14. Now Z ~ E and hence
VH ~ E. As Cz(Lo) # 1, CH(VH) ~ Ca(Cz(Lo)) ~ M = !M(LoT), so H* # 1.
Observe applying E.2.3.3 to LoT that for ti ET n Li - 8, ti induces a transvection
on E with center Vi EV;. If ti E CH(VH), then
So := ToS = (ti, t2, S) ~ Cr(VH ).
But we saw earlier that Na(B+) ~ H for each S+ ::::) T with T 0 ~ B+, so by a
Frattini Argument, H = NH(Cr(VH))CH(VH) ~ M, contrary to our assumption.
Thus ti # 1, so as VH ::::; E, ·ti induces a transvection on VH with center Vi·
Then comparing the possibilities in E.2.3 to the list of groups in G.6.4 containing
Frtransvections, we conclude that either H* 9'! Ot(2) with m([VH, HJ) = 4, or
H* is one of Ss or Ss wr Z2. The latter cases are out, as then NM(S) is not a
3'-group, contrary to 10.1.3 and the fact that NL 0 (S) is a 3'-group. So [VH, HJ is
the orthogonal module for H 9'! Ot(2). Let Y := 02 (CH(v2)); then Y 9'! Z3, and
YnM::::; 02(H).
Let X := Ca(v2). Then Ti = Cr(v2), IT: Til = 2, and L ::::; X. As T 1:. X
but Na(Ti) ~ M, Ti E Sylz(X). Thus by 1.2.4, L ~IE C(X). Suppose first that
L =I. Then L ::::! X by 1.2.1.3, so X acts on [02(L),L] =Vi. As Y = [Y,Ti]
while Y n M ~ 02 (H), we conclude from the structure of Aut(L/02(L)) that Y ~
02 (Cx(L/0 2 (L))). Further EndFz(L/Oz(L))(Vi) 9'! F2, so that Y must centralize
Vi. However, Y does not centralize vi E Vi. This contradiction shows that L <I.
Suppose that Vi ~ 02 (1). Then since the As-block L has a unique nontrivial
2-chief factor Vi, and Vi is projective, W := (Vl) = Vi EB Cw(L) ::::; Z(02(I))
and I has a unique nontrivial 2-chief factor. In particular W E R2 (I) and setting ·
i := I/Cr(W), [W,a] =[Vi,&] for each involution a EL, so q(i, W)::::; 2. Also we
conclude from A.3.14 that I/02(I) 9'! A1, A1, Ji, £2(25), or L2(P) with p = ±1
mod 5 and p = ±3 mod 8. Then as q(i, W) ::::; 2, we conclude from B.4.2 and B.4.5
that I /0 2 (!) ~ A 7. Since the unique nontrivial £-chief factor Vi is the As-module,
we conclude that Wis the A 7 -module, so I is an A 7 -block. However I= 031 (X)
by A.3.18, so 1 # 031 (CL 2 (v2)) ~ 02 (Cr(L)), contradicting 02 (Cr(L)) = 1.
Therefore Vi 1:. 02 (1), so as L is irreducible on Vi, Vi n 02(I) = 1. Set
j := I/0 2 (!). Then Lis a Ti-invariant As-block in j, a situation that does not
occur in A.3.14. This contradiction completes the proof. D