812 12. LARGER GROUPS OVER F2 IN Cj(G,T)Let Y := KL2T, U := (ZY), and Y* := Y/Cy(U). As Le = L1L2, Ll <
CK(Z), and Z = Cz(L)(Z n V) by 12.3.7.1,
[Z,Le] = [ZnV,L2] = (e1,1,e(2)).
Then CL 2 ([Z, L2]) = 02(L2), and CK([Z, K]) = 02(K) by 12.3.12, so Cy(U) =
02 (Y). Thus Y* ~ 85 x 83 since M1M2/02(M1M2) ~ 83 x 83, and U E R2(Y).
Also K = [K, J(T)] by 12.3.12, and L2 = [L2, J(T)] using 12.3.7.1, so Y* = J(Y)*.
Therefore by Theorem B.5.6,[U, Y] = [U, K] EB [U, L2] = [Z, K] EB [Z, L2],
so in particular K:::; Ca([Z, L 2 ]) :::; Ca(e(2)) :::; M by 12.3.8, contrary to Ki. M.
This contradiction completes the proof of Theorem 12.3.1.12.4. Some further reductions
We begin section 12.4 with a technical lemma 12.4.1, which we use in particular
to prove the main result 12.4.2 of the section.
As we will be assuming Hypothesis 12.2.3, as usual we adopt the conventionsof Notation 12.2.5, including Z ~ fh(Z(T)).
LEMMA 12.4.1. Assume Hypothesis 12.2.3. In addition assume:(i) Ca(Z) :::; M, and
(ii) s(G, V) > 1.
Then there exists g E G with 1 #-[V, Vg] :S V n Vg.
PROOF. Assume the lemma is false. Let HE H*(T, M), K := 02 (H), VH :=
(ZH), and H* := H/CH(VH)· As Ca(Z) :S M by (i), 12.3.2 says either Hissolvable, or [VH, K] is the sum of at most two A 5 -modules for K* ~ A5· Then
a(H*, VH) = 1, by E.4.1 in the former case, or by an easy direct computation in
the latter.Observe that the triple G1 := LT, G2 := H, V satisfies Hypothesis F.7.6.
Form the coset geometry r as in that section, with parameter b := b(I', V). If
Wo(T, V) :::; 02(H), then by F.7.14, b is even. Hence by F.7.11.2, there exists
g E G with 1 #- [V, Vg] :::; V n Vg, contrary to our assumption that the lemma ·
fails. Therefore Wo(T, V) i. 02 (H). So there is A := Vg with A* #-1. Now as
s(G, V) > 1 by (ii), A* E A2(H*, VH) by E.3.10, contradicting a(H*, VH) = 1. D
The main result of this section is Theorem 12.4.2. It eliminates two of the fourcases in 12.2.2.3 where Cv(L) #- 1 (cases (b) and (f)), leaving only A 6 and A 8
in case (d). In particular when L is L3(2) or G2(2)', the result reduces V to the
natural module. The analogous reduction will be carried out later for L 4 (2) and
L5(2) in Theorems 12.6.34 and 12.5.1. Theorem 12.4.2 also moves in the direction(begun in 12.2.13) of showing that Ca(V n Z) i. M.
THEOREM 12.4.2. Assume Hypothesis 12.2.3. Then(1) If L/02(L) ~ L3(2) or G2(2)', then Cv(L) = 1.
(2) If L/02(L) ~ L5(2) and dim(V) = 10, then Ca(Z n V) i. M.
Until the proof of Theorem 12.4.2 is complete, assume G, L, V affords a coun-
terexample. Let Zv := Cv(L).
When L/0 2 (L) ~ L 3 (2) or G 2 (2)', Zv #-1 as we are in a counterexample to
Theorem 12.4.2. Hence by Theorem 12.2.2.3, Vis an indecomposable for L/0 2 (L),