13.3. STARTING MID-SIZED GROUPS OVER F2, AND ELIMINATING U 3 (3) 893the Thompson A x B-Lemma, X is faithful on Cv 3 ( Q J), so Q J centralizes V3. Now
by A.4.8.4, V3 S WK, so 1-/:-V1 S CwK(K). However we saw that either WK E
Irr +(KT, R2(KT), T) and so is described in 13.3.2.3, or WK is the sum of natural
modules for K/02(K) ~ L3(2). Thus as CwK(K) -/:-1, WK is a 5-dimensional
module for K+ ~ A 6 • Therefore A4 ~Lt S k+ and V3/V1 is an L+-invariant line
in WK/V1, with [V3,02(L+)] = V1, whereas in the 5-dimensional module WK, thepreimage of such a line is centralized by 02(Lt). This contradiction establishes
(g).Now as L+ i K by (g), but m3(KL+) S 2, A.3.18 eliminates the possibilities
for K/02(K) of 3-rank 2 in 13.3.2.1. Thus m3(K) = 1, so that K+ ~As or L 3 (2).
As L+ i K = [K, L+], and Out(K+) is a 2-group, L+ is diagonally embedded in
LKLe, where Le:= CKL+(K/0 2 (K)) and LK = 02 (LK) is the projection of L+
on K. But if K/0 2 (K)'~ L3(2), then LK = [LK,T n K], contrary to the fact
that L+ is T-invariant. Thus K/0 2 (K) ~As, and from earlier discussion W_K is
the As-module. Then as L"j{ -/:-1, LKT = (T n K)02(KT), so as Ri = 02 (L+T)
is of index 2 in T, R 1 = (T n K)0 2 (KT). Since K satisfies Hypothesis 12.2.3by 13.3.2.4, we may apply 13.2.4.1 with K in the role of "L to conclude that
C(G, Baum(R1)) S Na(K). It follows as Ki M by (e) that Na(Baum(R1)) i M.
Therefore by 13.3.10, Lis an L 3 (2)-block or an A 6 -block, and in either case L+ has
exactly two noncentral 2-chief factors.
As WK is the As-module, EndK(WK) = F2, so [WK,Lc] = 0. Thus L+ hasat least one noncentral 2-chief factor on WK, as well as one on 02(L"j(); so as L+
has just two noncentral 2-chief factors, [02(J), L+] S WK. Hence as K = [K, L+],
K is an As-block. Then as Le centralizes K+ and WK, [K, Le]= 1 by Coprime
Action. By C.1.13.c, 02 (J) = Co 2 (J)(K) x WK, so as J E 'He, Le has a noncentral2-chief factor in Co 2 (J)(K), contradicting [02(J),L+] S WK. This contradiction
finally completes the proof of 13.3.14. DLEMMA 13.3.15. Assume Hypothesis 13.3.13. and that L ~As. Then
(1) h := (Qf^2 ) ::::! G2.(2) I 2 = XQ1, where X := L2,+ when L/02(L) ~ A5 and X := L2 otherwise.
(3) C1 2 (Vi) = 02(I2) and I2/02(I2) ~ 83, with 02 (12) = X.
(4) Cq 1 (Vi) S 02(h) s 02(G2).(5) m3(Ca(Vi)) S 1.
(6) Ca(V3) S Mv. Hence [V, Ca(V3)] S Vi.
PROOF. Part (1) holds by construction. As L ~ As, X/0 2 (X) is of order
3. Recall one consequence of Hypothesis 13.3.13 is that V2 is of rank 2. Then as
[Q 1 , Vi] -/:-1 by 13.3.14, XQ1 induces GL(Vi) on V2, with X transitive on vt.
Hence G2 = Ca(V2)Q1X, with Ca(V2)Q1 S G1, so
I2 = (Qf^2 ) = (Qf) S XQ1,
and X = [X, Qi] Sh so h = XQ1 and (2)-(4) hold. As X = 02 (h) <I G2 by
(1) and (3),