13.1. ELIMINATING LE .L:;(G, T) WITH L/02(L) NOT QUASISIMPLE 875
(i) K =Ko, with k isomorphic to L 3 (p) or (S)L 3 (2n) where 2n = 1 mod p,
or
(ii) Ko= KKt fort E Tu - Nr,,,, (K), and K is isomorphic to Sz(2n) or L 2 (2n)
with 2n = 1 mod p, L2(r) for a suitable odd primer, or J 1 with p = 7.
Recall we saw earlier that Hypothesis C.2.3 holds. We are now ready to apply
pushing up results from section C.2.
Suppose first that F*(K) = 02 (K). If R i. Na,,,, (K) then K < K 0 , and by
C.2.4.2, Rn KE Syl2(K), so K is a x 0 -block of Gu by C.2.4.1. Then from our list
of possibilities for k, k ~ L2 ( 2n). On the other hand if R :::; N 0 ,,,, ( K), then K is
described in C.2.7.3. We compare the list of C.2.7.3 with our list of possibilities for
kin (i) and (ii): If k ~ (S)L3(2n), then case (g) of C.2.7.3 occurs, so Kn Mc is
a maximal parabolic of S L3 ( 2n), impossible as X :::l K n Mc. The only remaining
possibility in both lists is case (a) of C.2.7.3 with Ka x-block, so again K ~ L 2 (2n)
and K <Ko.
Thus in any case, Ko= KKt fort E Tu - Nr,,,, (K), and [K, Kt] = 1 by C.1.9.
Let P be the set of subgroups Po of P oforder p such that [Co 2 (x) (P 0 ), P] =f-1, and
set XK := X n Kand PK := P n K. Then X = XKX_k and P ={PK, Pk}. But
Mc = Na(X), so NMJP) permutes P, contrary to the fact that NL(P) induces
either SL 2 (p) or SL2(5) on P, and thus has no orbit of order 2 on the set of
subgroups of P of order p.
Therefore F*(K) =/-02(K), so as O(Gu) = 1 and K/0 2 (K) is quasisimple, K
is quasisimple. Then as X:::; Ko, and F*(X) = 02(X), we conclude by comparing
the list in 1.1.5.3 with our list of possibilities fork in (i) and (ii) that K 0 /Z(K 0 ) ~
(S)L3(2n), L2(2n) X L2(2n), or Sz(2n) X Sz(2n) for some n, or L2(r) x L2(r) for r.
a Fermat or Mersenne prime. In the latter three cases the components commute,
so as in the previous paragraph we conclude that N Mc (P) permutes the subgroups
P n K and P n Kt, for the same contradiction. Furthermore a similar argument
works in the first case: Namely X lies in a Borel subgroup of K, so that 02 (X)
is the full unipotent group A, which is special of order 23 n with center of order
2n. Therefore NMJP) acts on Cp(Z(A)) ~ Zp, for the same contradiction. This
finally completes the proof of 13.1.12. D
LEMMA 13.1.13. U is a TI-set in G.
PROOF. Suppose 1 =f-u E Un U9 for some g E G. Then by 13.1.12.3, X^9 :::;
CMg(u):::; Mc for XE X, and by 13.1.12.1, CMg(u) is irreducible onX9 /0 2 ,w(X^9 ),
so 13.1.11 says Mc = Mg. Thus g E No(Mc) = Mc as Mc E M, so U = U^9 ,
completing the proof. D
Recall the weak-closure parameters w(G, U) and r(G, U) from Defintions E.3.23
and E.3.3.
LEMMA 13.1.14. {1) Wi(T, U) centralizes U for i = 0, 1, so Na(Wo(T, U)) :::;
Mc.
(2) w(G, U) > 1 < r(G, U).
{3) If H E 1i(T) with n(H) = 1, then H :::; Mc.