13.1. ELIMINATING LE .C;(G, T) WITH L/0 2 (L) NOT QUASISIMPLE 873
of order 42 with CL(u1)* ~ E4 for u1 E U1. Further either U = U 1 , or AutMJU) ~
La(2) x Z2 and U = U1 UU 2 , whereu I - U1 is of order 14,
lElrr+(L,U)where U1 is the unique T-invariant member of Irr+(L,U), and CL(u)* ~ 84 for
u E U2. In either case, (1) and (2) hold.
In case (2) of 13.1.10, U is the set of nonsingular vectors in the orthogonal space
U, so [U[ = 10, and CL·(u) ~Sa is nonabelian; thus (1) and (2) hold in this case.
Finally in case (4) of 13.1.10, Lis transitive on U = U - Cu(L) of order 14, so
CL* (u) ~ A4 is nonabelian, completing the proof of (1) and (2). Observe we alsoshowed in cases (1), (3), and (4) of 13.1.10 that (Tun L)+ contains a Q 8 -subgroup.
It remains to prove (3), so we assume Gui Mc, and derive a contradiction.
We next claim that Tu E Syb(Gu): For if not, then by (2), uB E Z for some
g E G. Then G~ = Ca(u^9 ) ::::; Mc= !M(Ca(Z)) by(+), and hence X::::; Mu=
Mc n Gu::::; Mr1• By (1), we may apply 13.1.11 to conclude that Mu is contained
in a unique conjugate of Mc, so that Mc = Mr
1- Then g E Mc as Mc E M, so u
is centralized by an Mc-conjugate of T, whereas u E U so [Mc : Mu [ is even. Hence
the claim is established.
Set Rx := 02(XT) and R := CRx(u). Observe that [L,Rx] ::::; 000 (L) =
CL(U), so L centralizes Rx. We claim that R = CRx (U). Since CRx (U) ::::; R,
it suffices to show that R centralizes U. If U E Irr +(L, U), then as L* centralizes
R*x, Rx centralizes U by A.1.41, so that the claim holds. Therefore we may assumethat U ~ Irr +(L, U), so case (3) of 13.1.10 holds. Then AutMc (U) S La(2) x Sa,
so we may assume AutR(U) = CAutMc (U) ~ Z2. But then Cu(R) is a T-invariant
member of Irr +(L, U), so as u E Cu(R), u is 2-central in Mc, contrary to u EU.
This completes the proof of the claim.
By the claim, R '.SJ XT, so as Mc= !M(XT) by(+),C(Gu, R) ::::; Mc n Gu= Mu.
Next, the hypotheses of A.4.2.7 are satisfied with Gu, Mu, Tu in the roles of "G,
M, T": For Nau(R) ::::; Mu and X '.SJ Mu, with Tu Sylow in Mu and Gu, and
R = 02(XT) = 02(XTu). Therefore R E B2(Gu) and R E Syb((RMu)) by
A.4.2.7. Thus the pushing up Hypothesis C.2.3 is satisfied with Gu, Mu in the role
of "H, MH". However, before we apply pushing up results from section C.2, we
will establish a number of further preliminary results.
We claim next that 02,F(Gu) S Mu: Set Gu := Gu/02(Gu) and recall p is
the odd prime in ?r(X). Let Rr denote a supercritical subgroup of Or(Gu)· As ·Mu
is irreducible on X/0 2 ,IJ?(X) by (2), we may apply A.1.32 with Gu, Rn X in the
roles of "G, R, P". If r #-p, then by part (1) of that result, X centralizes Rn
and hence QP(0 2 ,p(Gu)) normalizes X. If r = p, then by part (2) of A.1.32, either
X A = Rp, A or Zp '"" = Rp A = Z(X) A and X A = '"" p l+^2. In the former case, Op(Gu) A normalizes^0
the characteristic subgroup Rp = X, so the claim holds. In the latter case, since
th~ supercritical subgroup Rp contains all elements of order p in C Op (Gu) ( Rp),
we conclude that Op(Gu) is cyclic. Then as Mu is irreducible on X/02,1J?(X), X
centralizes Op(Gu), completing the proof of the claim. We have also shown that X