732 9. ELIMINATING n:tc2n) ON ITS ORTHOGONAL MODULE
n = 2, so the lemma holds. Finally if t-=/:-er, then t induces an automorphism on
B of order [Jo[, so that [fol = 2i. Then since B = CD(Vi), we calculate in Lo that
[CB(t)[ = 2m + 1. This is impossible as 2m -1-=/:-2m + 1. Thus the Proposition is
established. D
9.3. Reducing to n(H) == 1
In this subsection, we assume n(H) = 2, and eventually arrive at a contradic-
tion.
Set Gi := No(Vi). By 9.2.7.3, K/02(K) ~ L2(4), f3 [D, er], and B =
CD(Vi).
PROPOSITION 9.3.1. D acts on K and [K, Vil = 1.
·PROOF. Define Dr; := C15(er). Then D = [D, erlDr;, and hence D = BDr; for a
suitable preimage Dr; in D of Dr;. Thus Dr; is of order 3 and faithful on Vi. The
proof begins with a series of three reductions:
First, notice if Dr; :::; No(K), then D :::; No(K), and hence Vi :::; (zD") :::;
Ca(K), so that we are done. Thus we may assume Dr; i N(K); in particular, K
is not normal in Gi.
Second, suppose that K:::; Gi. Then KE £(Gi,T), so by 1.2.4, K:::; Ki E
C(Gi), and indeed K <Ki by the previous paragraph, so Ki is described in A.3.14.
Suppose m3(Ki) = 2. Then Ki :SI Gi by 1.2.2.b. As Dr; i K, comparing the
list in A.3.18 to that of A.3.14, we conclude Dr; induces diagonal automorphisms
on Ki/02(Ki) ~ L3(4) or U 3 (5), and so D normalizes K from the embedding
described in A.3.14. Thus in this case we are done by our first reduction, so we
may assume that m3(Ki) =). Then by A.3.14, Ki/02(Ki) is Ji, L2(25), or L2(P),
or Ki/02:2,(Ki) ~ SL2(P) for suitable p. We can reduce the fourth case to the
third case by noting that Ko:= NK 1 (T n 02 , 2 1, 2 (Ki))^00 is D-invariant. But in the
first three cases, D = CD(Kif02(Ki))B acts on K, contrary to the first reduction.
Therefore we may assume that Ki Gi. In particular, [K, Vil -=/:-1.
Third, we recall that K centralizes Z, so K :::; Gi if f' :::; L 0 (er) by 9.2.2.3,
contrary to the second reduction.
In view of our three reductions, we may assume D does not act on K, Ki Gi,
and f' i Lo(er). To complete the proof, we construct _an overgroup X of K, and
obtain a contradiction in X.
By the third reduction and 9.1.2.2, there is t ET with f = erf, where f is an
involution inducing a field automorphism on L 0. As er and f invert f3, t centralizes
B, so T2 := (t)02(DT) is B-invariant and Dr;T2/02(Dr;T 2 ) ~ 83. Set X := (D, H).
Suppose 02(X) = 1. Then K, Dr;T2, T satisfies Hypothesis F.1.1 in the roles of
"Li, L2, S", so the amalgam a := (KT, BT, DT) is a weak EN-pair of rank 2 by
F.1.9. Further T2 is maximal in Dr;T 2 , so the hypotheses ofF.1.12 are satisfied, and
hence a is one of the amalgams listed in that lemma. As Dr;T 2 /0 2 (Dr;T 2 ) ~ L 2 (2)
and K/02(K) ~ L2(4), a is of type U4(2), J2, or Aut(h), so that [Tl :::; 28. This
contradicts [V[ = 28 with V < T.
Thus 02 (X) -=/:- l, so X E 'H(T) ~ 'He by 1.1.4.6. By 1.2.4, K :::; Kx E
C(X), and Kx :SI X by (+) in 1.2.4, so X = KxTD. As D i Na(K), K <
Kx. Next Vi :::; Vx := (zx) E R2(X) by B.2.14. As [K, Vil -=/:-l, [Kx, Vxl -=/:-
- Set X := X/Cx(Vx). Then K -=/:- 1. Also K = [K, J(T)], or else K :::;
Na(J(T)02(K)) :::; M using 3.2.10.8. Thus J(T)* -=/:-l, so Vx is an FF-module