532 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1£X = Nx(Qx) since Nx(Qx) ~ Nx(Q) = L. Thus Z(L) = P = Z(LX), contrary
to 2.4.4. This contradiction completes the proof that Cc(v)::::; GQ.
In the remainder of the proof, X again denotes an arbitrary member of1i(L) not
normalizing L; thus 1 =/=- 02 (X) = P::::; Z(L) by earlier remarks. Now Cc(Z(L)) ::::;
Cc(v) ::::; GQ = Nc(L) by the previous paragraph, so L E C(Nc(Z(L))). As
H ::::; Nc(Z(L)), S E Syl 2 (N 0 (Z(L))) by 2.3.7.2, so L ::::1 Nc(Z(L)) by 1.2.1.3.
Therefore N 0 (Z(L)) = N 0 (L) = GQ using (1). Also Bis transitive on Z(L)# and
Cc(v) ::::; GQ = Nc(Z(L)), so Z(L) is a TI-subgroup of G by I.6.1.1, completing theproof of (3). Then as 1 =/=- 02 (X) ::::; Z(L), X ::::; Nc(Z(L)) = GQ by (3), contrary
to assumption. This contradiction completes the proof of the lemma. DWe next repeat some arguments from sections 3 and 4 of [Asc78a], which forcethe 2-local structure of G to be essentially that of an extension of £ 3 (2n) or Sp 4 (2n);
this information is used later in transfer arguments to eliminate these shadows.In fact, by 2.4.8 and 2.4.11, the hypotheses of section 3 or 4 in [Asc78a] are
satisfied, in cases (a) or (b) of 2.4.8.4, respectively. Thus we could now appeal
to Theorems 2 and 3 of [Asc78a]. However those results are not quite strong
enough for our present purposes, and in any event we wish to keep our treatment
as self-contained as possible, as discussed in the Introduction to Volume I under
Background References. Thus we reproduce those arguments from [Asc78a] nec-
essary to complete our proof.LEMMA 2.4.12. (1) H is the split extension of L by a cyclic subgroup F of S
inducing field automorphisms on L / Q. Thus S is the split extension of R by F.
(2) If f is an involution in F, then all involutions in f R are fused to f under
R, CL(!) is an L2(q^112 )-block (or 84 or Z2 x 84 if q^112 = 2), and either
(a) Z(L) = 1, and CR(!) is special of orderq^312 ; in this case we say CR(!)
is of type L3(q^1 l^2 ).
(b) Z(L) ~ Eq, with JCz(L)(f)J = q^112 and JCR(f)J = q^2 ; in this case we
say CR(!) is of type Sp 4 (q^112 ).
PROOF. Recall H = LS, while by parts (3) and (5) of 2.4.8, S is the split
extension of RE Syl2(L) by Ns(B) =: F. Thus F n L = F n R = 1, so that (1)
holds.
Suppose f is an involution in F. As f induces a field automorphism on L :=
L/Q, q = r^2 , Cr,(!) ~ L 2 (r), and CQ/Z(L)(f) is the natural module for Cr,(!).
Indeed if Z(L) =/=- 1, then Z(L) ~ Eq by 2.4.8.4, so from I.1.6, Q is the largest
indecomposable extension of a submodule centralized by L by a natural L-module;
hence m(Z(L)) = 2m(Cz(L)(f)). Thus in any event m(Q) = 2m(CQ(J)), so Q is
transitive on the involutions in fQ. Then by a Frattini Argument, Cr,(!)= CL(!),
so CL(f) is as claimed in (2). Further by Exercise 2.8 in [Asc94], R is transitive
on involutions in f R, completing the proof of (2). D
DEFINITION 2.4.13. Relaxing somewhat the usual definition in the literature,
we define a Suzuki 2-group to be a 2-group I admitting a cyclic group of automor-
phisms transitive on its involutions, with [I, I] = Z(I).
LEMMA 2.4.14. Assume t ET - S with t^2 ER. Then (t)R splits over R, R is
transitive on the involutions in tR, and choosing t to be an involution, one of the
following holds: