i2.9. THE FINAL TREATMENT OF Ln(2), n = 4, 5, ON THE NATURAL MODULE 863Assume CA(V) =/= l. Conjugating in Na(V9), we may assume that CA(V) =
V{. Thus V::::; Gf::::; Na(U9), where U := (V^01 ). Hence [V, A]::::; U9::::; Ca(V9) as
U is abelian by 12.9.4.
We now prove (2), so we may assume A < V9. Then [V, A] is cyclic: for
otherwise V^9 ::::; Ca([V, A]) ::::; Na(V) ::::; M by 12.9.7, contrary to A < V9. As[V, A] is cyclic, A induces a group of transvections on V with center [V, A]; so as
Cv(B) =I= Cv(A) for each noncyclic subgroup B of A, IAI = 2. But now CA(V)
is noncyclic, contrary to paragraph one. This completes the proof of (2).
If Wo ::::; CT(V) then Na(Wo) ::::; M by E.3.34.2. Thus we may assume A= V9,and it remains to derive a contradiction. Suppose first that A acts nontrivially on
Vn-i· Then Vn-i i Cv(A) = I and hence m(CA(Vii-i)) ::::; 1 by paragraph one.Let M~-i := Mn-i/CM(Vn-i)), and observe M~-i 9'! Ln-i(2). Then
m2(M~_i) 2 m(A*) 2: n - 1,
so we conclude n = 5 and A= J(T). But now Cv 4 (A) =Vi< Cv 4 (B) for Ba
4-subgroup of A with B* inducing transvections on V4 with a fixed axis, contrary
to an observation in the first paragraph.Therefore A centralizes Vn-i, so A::::; Rn-i· Then as m(CA(V))::::; 1,
m(A) 2: m(A) - 1 = n - 1 = m(Rn-i),
so that ACT(V) = Rn-i· Thus Li= [Li, A] and Ai:= CA(V) is of order 2, so by
paragraph two we may assume Ai= V(, V::::; Na(U^9 ), and Vn-i = [A, VJ ::::; U9.Let Q := 02(Gi). For y E Q, [U,y]::::; Vi by 12.8.4.2, so m(U/Cu(y))::::; 1 and
hence as n 2 4, Cvn 1 (y^9 ) is noncyclic. Thus y^9 E Ca(Cvn 1 (y^9 ))::::; Na(V) by
12.9.7, so [Q9, V]::::; Q9 n V = Vn-i::::; U^9. If [Kf, VJ::::; 02(Kf), then V::::; Na(A).
by 12.9.5.1, contrary to I < V. Thus Kf = [Kf, V], so Kf centralizes Q9 /U9.
Then [Ki, Q] ::::; U ::::; CQ(U) as U is abelian by 12.9.4. Therefore [Ki, 02(Ki)] ::::;
Ca(U) ::::; Ca(V).Let P := 02 (KiT) and choose X of order 7 or 5 in Li for n = 4 or 5, re-
spectively. Recall KiT E 1-lz by 12.9.5.1, so that [V, P] = 1 by 12.8.4.2. Further
V =Vix [V,X], and by Coprime Action, P = Cp(X)[P,X] = Cp(X)[P,Ki] =
Cp(X)[0 2 (Ki), Ki]. Now Pacts on V, and [02(Ki), Ki] centralizes V by the pre-vious paragraph; then C p ( X) acts on [V, X], and hence P acts on [V, X]. Therefore
as Xis irreducible on [V,X] and normalizes P, P centralizes [V,X], so as P::::; Gi,
P centralizes V. As 02 (Ki) ::::; P and Vi ::::; V, this is contrary to 12.9.6.1, so the
proof of 12.9.9 is complete. DWe are now in a position to complete the proof of Theorem 12.9.l.
By 12.9.5.2, Ki 9'! A1, £4(2), or £5(2). In particular, there is an overgroup H
of Tin KiT not contained in M with H/02(H) 9'! 83. By 12.9.9.1, Na(Wo)::::; M,
so by E.3.15, W 0 i 02 (H). Thus there is A := V^9 ::::; T with A i 02(H). If
Vi::::; A, then by 12.8.3.3 and 12.9.4, A E v^01 nH ~ 02 (H), contrary to our choiceof A. Thus Vi n A = l.
Now H ¢ 1-lz since H does not contain Li, but we define some notation similar
to that in Notation 12.8.2: Let UH:= (Vf) and QH := 02(H). Then UH::::; (V^01 ),so UH is abelian by 12.9.4. Indeed Hypothesis G.2.1 is satisfied with H, Vi, 1 in
the roles of "G, V, L", so by G.2.2.1, UH E Z(QH). By 12.9.7, V2 < UH. As
H/QH 9'! 83 and Ai QH, B :=An QH is of index 2 in A. Then [UH, BJ::::; Vi, so