11.5. THE FINAL CONTRADICTION 781acts on RiX, and hence by a Frattini Argument can be taken to normalize X. Then
[X, Bo] ::::; XnBoT = 1 using (1), contrary to an earlier observation. Thus (3) holds
when n is even.· So assume instead n is odd. Then X is a 3' -group of odd order coprime to III
by (1). Therefore as ma(Y) ::::; 1 by (2), (3) follows from an examination of the
list of A.3.15, unless possibly X/Cx(I/0 2 (!)) induces a nontrivial group of field
automorphisms on I/0 2 (!) ~ L 2 (2k) or L 3 (2k). In that event, k = 2am for someodd m divisible by IX : Cx(I/02(I))I, and X/02(X) induces a faithful group of
field automorphisms on the subgroup Ii of I with Ii/0 2 (Ii) ~ L 2 (2m) or L 3 (2m).
Further a Borel subgroup of Ii acts on T, unless possibly some t ET induces a graph
automorphism on Ii/02(Ii) ~ L 3 (2m), in which case a subgroup of order 2m - 1
acts on T. Then arguing as in the previous paragraph, [X, Bo] = 1, contradicting
the fact that X / C x (I/ 02 (I)) induces nontrivial field automorphisms on Ii/ 02 (I 1 ).This contradiction completes the proof of (3).
Assume the hypotheses of (4). Applying (3) and appealing to 1.2.1.1, we con-
clude X centralizes Y^00 /0 2 (Y), and hence so does [P, X] = P. Thus P centralizes
E(Y/02(Y)). As if>(P)::::; 02(G1), P = [P,X] centralizes F(Y/0 2 (Y)) by A.1.26.
Thus P centralizes F*(Y/02(Y)), establishing (4).
If L1 = K, then 02(LT) ::::; 02 (KT), while if L1 < K, then by 11.4.1.3,[02(LT), X] ::::; [0 2 (L1T), X] ::::; 02(KT). Thus (5) is established.
Finally if 02 (H1) ::::; K, then L1 < K so K is described case (1) or (2) of 11.1.2.
In particular in each case, m2(Ri/CR 1 (K/02(K)) ::::; 1 as Ri = 02(L1T). But
CR 1 (K/0 2 (K)) :S 02(H1), since 02 (H 1 ) :SK. Thus (6) holds. D
PROPOSITION 11.5.7. (VG^1 ) is abelian.PROOF. Assume that (VG^1 ) is nonabelian. Set U := ("'3G^1 ). By 11.5.5 applied
to G1 in the role of "H1", fJ::::; Z(02(G1)) and <l?(U) ::::; 1/i. Let Y := Ca(V1) n
Ca(K/02(K)).
We first treat the case L ~ SL 3 (q). Then Va = V so that U is nonabelian
by assumption. Let x, y E NL( X) with V = V1 EB V{ EB Vt As U = (VG^1 ) is
nonabelian, Vi Z(U), so U f. 1. From the proof of 11.5.5.1, Hypothesis G.2.1 is
satisfied, so by G.2.5, L::::; I:= (U, ux, UY)= LU, U = 02(L1) = R1,S := 02(I) = Cu(V)Cu,,(V)Cuv(V),
US/S = 02(L1)S/S, S has an £-series
1 =: So ::::; S1 ::::; S2 ::::; Sa :S S4 := S
such that S1 := v, S2 := u n ux n UY, (and setting wi :=Si/Si-I) L centralizes
W 2 , W 3 is the direct sum of r copies of the dual V* of V, and W4 is the direct sum
of s copies of V. As I= (UL), M 1 acts on I and hence on S, as does L since L::::; I.
We claim that S::::; 02(G1). Set E := uxnUY. By 11.5.5.2, if>(E) :S V{nV 1 Y =
1. From the discussion above, W4 = [W 4 , £ 1 ] and for each irreducible J in Wa, the
image of E in J is the X-invariant complement to [J, L1] in J. Hence S = [S, L1]E.
Further X acts onE and S/S2 = [S/S2,X], so E = [E,X]S2 and S = S,X.
Now as if>(E) = 1, [E, X] centralizes Y/0 2 (Y) by 11.5.6.4, so as S 2 ::::; U::::; 02(G1),
E = [E, X]S 2 centralizes Y/0 2 (Y). Then as [£1, Y] ::::; [K, Y] ::::; 02 (Y), S =
[S,L 1 ]E centralizes Y/02(Y). Also we saw that S::::; 02(LT), so [S,X] centralizes
K/0 2 (K) by 11.5.6.5, and hence so does S = [S,X](S n U). Thus S centralizes