680 6. REDUCING L2(2n) TO n = 2 AND V ORTHOGONAL
the role of V, Z :=:; VL. Further replacing V by VL if necessary, we may assume .v
is the natural module.
By Theorem 6.1.27, Ca(Zs) :::; M, so by 6.1.7.1, Ca(Zs) :::; Mv := NM(V);
hence by 6.1.9.5:
LEMMA 6.2.1. Na(Zs):::; Na(V):::; M.
Observe that Zs is the T-invariant 1-dimensional F4-subspace of V regarded
as a 2-dimensional F 4-space. Let Mv := Mv /CM(V).
LEMMA 6.2.2. (1) LT~ 85.
(2) Z is of order 2.
(3) Ca(Z) i M.
PROOF. Part (3) follows from 6.1.5. Recall Z :::; V, so if Lf' ~ A5, then
Zs= Cv(T) = Z, and (3) contradicts 6.2.1. Hence (1) holds and Z = Cv(T) is of
order 2 by (1), establishing (2). DLEMMA 6.2.3. If g E G with V:::; Na(VB) and VB:::; Na(V), then [V, VB]= 1.
PROOF. If [V, VB]# 1, then 6.1.11 says VB E VCG(Zs). But Ca(Zs):::; Na(V)
by 6.2.1, contradicting our assumption that 1 # [V, VB]. DLEMMA 6.2.4. Assume U:::; V with m(V/U) = 2 and H := Ca(U) i Na(V).
Choose notation so that Tu := Nr(U) E Syh(NM(U)), and let Q := Cr(V),
Lu:= 02 (NL(U)), UH:= (VH), fl:= H/U, and H* := H/CH(UH)· Then
(1) U = Cv(t) for some t ET inducing a field automorphism of order 2 on L.(2) F*(H) = 02(H), R := Q(t) E Syh(H), Na(R) :::; Na(J(R)) :::; M, Tu E
Syl2(Na(U)), and IT: Tul = 2.(3) Wo(R, V):::; Q.
(4) UH is elementary abelian, UH :::; Z(02(H)), and CH(UH) = 02(H), so
UH E R2(H).
(5) Lu/02(Lu) ~ Z3 with 02(Lu) =Lu nH.
(6) There is at most one K E C(H) of order divisible by 3, and if such a K
exists then either
(i) K = 031 (H) and m 3 (K) = 1, or
(ii) K/02(K) ~ (S)L3(q), and a subgroup of order 3 in Lu induces adiagonal automorphism on K / 02 ( K).
PROOF. Observe by 6.1.8 that as H = Ca(U), H n M =" NH(V), so that
our hypothesis H i Na(V) is equivalent to Hi M. As Ca(Zs) :::; M, case (3)
of 6.1.13 must hold, proving (1). Next by (1), IT : Tul = 2, and the remaining
statements of (2)-(4) follow from 6.1.14, except for the inclusion 02 (H):::::: CH(UH)in part (4). Part (5) follows from (1), and (6) follows from A.3.18 in view of (5).
Finally CH(UH):::; Na(V):::; M, and by Coprime Action, Y := 02 (CH(UH)):::;
CM(V) :::; CM(L/02(L)). Thus LT normalizes 02 (Y0 2 (L)) = Y. Therefore if
Y # 1 then H :::; No(Y) :::; M = !M(LT), contradicting our initial observation
that Hi M. Thus CH(UH) is a 2-group, completing the proof of (4), and hence
also the proof of 6.2.4. DDefine a 4-subgroup F of VB to be of central type if F is centralized by a Sylow
2-subgroup of LB; of field type if F is centralized by an element of Mfr inducing a