16.5. IDENTIFYING J1, AND OBTAINING THE FINAL CONTRADICTION 1195Thus L ~ £ 2 (4). Then H = LK by 16.4.6, so R induces inner automorphisms
on L. Recall R is faithful on L, so R is elementary abelian; hence as IUcl = 2,
Uc =Tc is of order 2. Therefore CK(z) =Tc by 16.4.4.2, so that Gz = L x Tc ~
L 2 (4) x Z 2 , and hence G is of type J 1 in the sense of I.4.9. Then we conclude from
that result that G ~Ji. D
In the remainder of the chapter, assume G is not Ji; therefore by Proposition
16.5.1, L* is not a Bender group. To complete the proof of our main result Theorem
16.5.14, we must eliminate each remaining possibility for Lin (E2).
Recall from 16.4.3.2 that L' = [L', z], and that z induces an inner automorphism
on L' since KE b. 0 (K') by the symmetry in 16.4.11.2.
LEMMA 16.5.2. (1) If Rn Z(T) =f. 1 and g is chosen as in 16.4.2.4, then
g E Na(T).
(2) Assume u, z are involutions in R, Tc whose projections on L, LY are 2-
central, and that IT: TcTLI :::; 2. Then either
(a) Rn Z(Ti) =f. 1 for some Ti E Syb(H), and we may choose Ti so thatR ::;] Ti E Syl2(H n H'), or
(b) TcTl =:To E Syl2(HnH^1 ) for some l EL, with R ::;] To, ITol = ITl/2,
and there exists g E Nc(T 0 ) with KY= K'.
PROOF. Assume that Rn Z(T) =f. 1, and that g is chosen as in 16.4.2.4. NowT:::; Ca(R n Z(T)) :::; Nr(K') using 16.4.2.5, so TY = T as Nr(K') :::; TY by the
choice of g. Thus (1) holds..
Assume the hypotheses of (2). Then u centralizes Tl for some l E L, so
Tl :::; H' by 16.4.2.5. Also Tc :::; H' by 16.4.11.1, so To := TcTl acts on some
Ri E Syb(K' n H). But by 16.4.10, RE Syl2(K'), so by Sylow's Theorem there is
x EK' nH with Rf= R, and thus T2 := T/f acts on R. Let T2 :::; Ti E Syb(H). By
hypothesis I Ti : T21 :::; 2, so either R ::;] Ti, or T2 = Nr 1 (R) E Syl2(H n H'). In the
former case, Rn Z(Ti) =f. 1 and conclusion (a) of (2) holds, so we may assume the
latter. Thus To = T!f-
1
E Syb(H n H'). By 16.4.11.2 we have symmetry between
Kand K', so there is S E Syl 2 (K'L') with S Sylow in H n H'. Thus there is
h E HnH' with T/) = S, so ash acts on K' L', To is Sylow in K'L'. Let y E G with
KY= K'; then To and T;j are Sylow in K'L', so there is w E K'L' with T;jw = T 0 ,
and hence conclusion (b) of (2) holds with g := yw. D
We now begin the process of eliminating the possibilities for L remaining in
(E2). Let u denote an involution in U, and recall z E Tc n Z(T). Also Rn K = 1by 16.4.2.1, so that by 16.4.11.1,
R* ~ R~Tc.In particular as U:::; LK,
U~ U*:::; T{.
LEMMA 16.5.3. Lis not A5.
PROOF. Assume otherwise. Then U ~ U* :::; TJ, ~ Ds, and hence U ~ Z2,E 4 or D 8. Now in the notation of Definition F.4.41, X := I'i,u(L) :::; H' using
16.4.2.5. But if U ~ D 8 , then X = L, contrary to 16.4.2.2. Assume U ~ E 4.
Then X ~ 84 with 02 (X) = U; so as X acts on K' while U = Oi(R), X acts
on K' n 02 (XU) = U, and hence 3 E n(AutH'(U)). Then as Out(L') is a 3^1 -
group, 3 E n(NK'(U)), so that m 2 , 3 (H) > 2, contradicting G quasithin. Hence