510 1. STRUCTURE AND INTERSECTION PROPERTIES OF 2-LOCALS
R = P-or R ~ Zp, P ~ p1+^2 , and R = Z(P). In particular by A.1.21, P
centralizes QP(F(fl)).
Suppose for the moment that Op(fl)-:/-1, and chooser= p. If mp(R) = 2 then
P = R :s) fl, so X = 02 (R) :s) H by 1.3.2.5, and the lemma holds. Therefore
we may assume mp(R) = 1. Then as R is supercritical, it contains all elements of
order r in Cor(H) (R), so Op(fl) is cyclic.
Thus in any case, we may assume that Op(fl) is cyclic. In particular Pi. F(fl)
as Pis noncyclic. Hence as Aut(Op(fl)) is cyclic and P = [P, NT(P)], P centralizes
Op(fl); therefore as P centralizes QP(F(fl)), P centralizes F(fl).
By 1.3.2.3, NT(P) is irreducible on P /ip(F), so as Op(fl) is cyclic, PnOp(fl) ::;
ip(F); therefore as P centralizes F(fl), we conclude Cp(E(fl)) ::; ip(F). Thus
there is a component Li of fl with [Li, P] -:/-1. By A.3.3.4, there is LE C(H) with
L =Li. Set K := (LT), so that K :s) H by 1.2.1.3. As 1-:/-[£, P], [L, P] i. 02(L),
so L ::; [L, P] ::; [K, P] by A.3.3. 7. Then as T acts on P, K = (LT) = [K, P].
We claim P::; K. Suppose first that L < K = LLt. Then ip(NT(P)) ::; NT(L)
as jLTI = 2. Notice that the groups listed in 1.2.1.3 have Out(L) abelian. But by
1.3.2.4,
P = [P, ip(NT(P))] = [P, NT(P) n NT(L)],
so P induces inner automorphisms on L and then also on K. Then by 1.2.2.a,
P::; QP' (H) = K, establishing the claim in this case.
Next suppose that L = K. This time we examine Out(L) for the groups L
appearing in Theorem C, to see in each case there are no noncyclic p-subgroups Uwhose normalizer is irreducible on U/ip(U)-as would be the case for the image of
Pin Out(L), if P did not induce inner automorphisms on L. Thus P::; LCFI(L).
Then as NT(P) is irreducible on P/ip(P), either P::; L = K as claimed, or PnL::;
ip(P). However as Cp(L) ::; ip(F), mp(L) = 2; so in the case where P n L::; ip(P),
there exists x of order pin CpL(L) -L, and hence mp(L(x)) > 2, contradicting H
an SQTK-group. This completes the proof of the claim.
Thus P::; K by the claim. Then by 1.3.2.2, X = (P^02 (X)) :=:; K.
We next establish the lemma in the case L < K = LLt. Here mp(L) = 1 by
1.2.1.3, so
p = (P n L) x (P n L)t ~ Ep2,
and by 1.2.1.3, L is L 2 (2n), Sz(2n), L 2 (q) for some odd prime q, or Ji. If
L is L 2 (q) for some odd prime q, then conclusion (1) of the lemma holds, so
we may assume we are in one of the other cases. Then as PT = TP, P lies
in the Borel subgroup NJ<(T n K) of K if L is a Bender group, and similarly
P ::; NK(T n K) when L is Ji. In the first case P lies in a Cartan subgroup,
so 2n = 1 mod p, and in the second, p = 3 or 7. Further as Pacts on T n K,
[NTnK(P),P]::; (TnK) nP = 1. Therefore AutT(P) is isomorphic to a subgroupof Out(K), so as NAutT(K)(Autp(K)) is irreducible on Autp(K)/ip(Autp(K)) by
1.3.2.4, JOut(K)l2 > 2. Then as jOut(K)I = 2 J0ut(L)J^2 , Out(L) is of even order,
which reduces us to L ~ L 2 (2n), n even-so that conclusion (1) of the lemma holds.
It remains to treat the case K = L :s) H, where we must show that one ofconclusions (2)-(4) holds. Thus P::; L by the claim.
Suppose first that p > 3. Then the possibilities for L and P with PT = T P
are determined in A.3.15. Suppose case A.3.15.3 holds. Then p plays the role of"r" in that result, and it follows that the signs 5 and E there conincide. Thus
L ~ (S)Lg(q) with q = 5 mod 4; further CL(Z(T))^00 ~ SL 2 (p) plays the role of