12.7. THE TREATMENT OF A. 6 ON A 6-DIMENSIONAL MODULE 847
Next the preimage Ur is isomorphic to Es and contains V 1 , so by 12.7.10.5,
CH(Ur) ::; Mz = L1T. Then CH(Ur) = CL 1 r(Ur) ::; T, and hence CH* (Ur) is a
2-group by Coprime Action, so that H is not A1. Therefore H ~ L 3 (2). Then
arguing as in the proof of 12.7.7, U = U 1 EE! U 2 is the sum of two nonisomorphic
natural modules for H*. Therefore as V 1 = Cu-(LiT*), V 1 ::; Ui for some i, soU = (V 1 H) ::; Ui < U. This contradiction finally completes the proof of Theorem
12.7.14.
12. 7.4. The final contradiction. Because of Theorem 12.7.14, we can as-
sume in the remainder of the section that:LEMMA 12.7.21. V::; 02(Gz)·
LEMMA 12.7.22. (1) If g E G with v n V9-=/=-1, then [V, V9J = 1.
(2) Either W1 := Wi(T, V) centralizes V, or W 1 = R 2 and r(G, V) = 3.(3) Ca(Ci(T, V))::; M.
(4) If r(G, V) > 3, then Ca(C2(T, V))::; M.
(5) If Cv(V9) -=/=-1, then (V, V9) is a 2-group.
PROOF. Under the hypotheses of (1), we may take z E Vg by 12.7.9.3 and
12.7.4.1. Then by 12.7.9.2, wemaytakeg E Gz. Now by 12.7.21, Vg::; 02 (Gz)::; T,
so by 12.7.11, [V, V9J = 1. That is, (1) holds.
We next prove (2), (3), and (4). Let A:= V^9 n M::; T be a w-o:ffender. Thus
A-=/=- 1 and w := m(V^9 /A). By 12.7.11, w > 0. If w > 1, then W 1 centralizes V bydefinition, so that (2) holds, and then Ca(C 1 (T, V))::; M by E.3.34.2, so that (3)
holds. That result also shows that ( 4) holds if w > 2.
Next as 1-=/=-[A, VJ::; [V^9 , VJ, V n Vg = 1 by (1). If B::; A with m(Vg /B) <
r(G, V) =: r, then Cv(B)::; Na(V^9 ), so [Cv(B), AJ::; V n V^9 = 1; thus Cv(B) =
Cv(A), so that A E Ar-w(T, V) Then by 12.7.3, r -w ~ 2; and in case of equality,
A= R2 = Ww(T, V). Thus if r - w = 2, then Vi.= Cv(R2) ::; Cw(T, V), so thatCa(Cw(T, V)) ::; Gt::; M by 12.7.8.
By 12.7.10.2, r;:::: 3. Assume first that r > 3. Then by the previous paragraph:
first w > 1; and then either w > 2-or w = 2 and r = 4, so that (4) holds. Thus
the lemma holds when r > 3 by paragraph two, so we may assume that r = 3.Then (4) is vacuous, and (2) and (3) hold by paragraph two when w > 1, so we ·
may assume that w = 1. Then r - w = 2, so that (2) and (3) hold by paragraph
three. This completes the proof of (2), (3) and (4).
Assume the hypotheses of (5). By 12.7.4.1, we may assume Vg centralizes
v :=tor z. We observe V::; 02(Gv): if v = t, this follows from 12.7.8, and if v = z
it follows from 12.7.21. Hence (V, V9) is a 2-group, proving (5). D
If Gz ::; M, then by 12.7.4.1 and 12.7.8, we may apply Theorem 12.2.13 to
conclude that G ~ M24; but then V 1:. 02 (Gz), contrary to 12.7.21. Therefore
Gz 1:. M, so we can choose HE H*(T, M) with H::; Gz. By 3.3.2.4, we may apply
the results of section B.6 to H.
LEMMA 12.7.23. (1) n(H) = 2.
(2) 02 (H/02(H)) ~ L2(4) or L3(4).
(3) L1T = HnM.
PROOF. Let KH := 02 (H). By 12.7.10.2, s(G, V) > 1, and by 12.7.11,