618 4. PUSHING UP IN QTKE-GROUPS
Na(R2) = NH(R 2 ) acts on the parabolic K 2 of K, since we saw after 4.3.6 that
K :::;I H, so (1) holds.
Next using A.4.2.4, R 2 is Sylow in Syb(CH(K2/02(K2)), Now K2 :::l Gz nH,
so by 0.1.2.4, R 2 E !3 2 (NKzTnH(R2)). Therefore (2) follows from 0.1.2.3. By
(2) and 0.2.1, 02 (KzT) ::::; R 2 , so by (1) K 2 = 02 (K202(KzT)). Then 4.3.13
completes the proof of (3). D
Set Go:= LiR2Y and G 0 := Go/Ca 0 (Li/02(L1)). By 4.3.14.3, 02(KzR2)::::;
R2. As Y acts on R2, 02(KzR2) E Syb(Ca 0 (Kz/02(Kz))), so Na 0 (R2)* =
Na 0 (R2) by a Frattini Argument. Thus K2 :::l Na 0 (R2); so in view of 4.3.13 and
4.3.14:
LEMMA 4.3.15. R2 =f 1.
Now K'Z/Z(K'Z) is a group appearing in Theorem C (A.2.3), satisfying the
restrictions on prime divisors of 2n - 1 in 4.3.12.2.
Inspecting the automorphism groups of those groups for a proper 2-local sub-
group Nx'Z(R2) with a normal subgroup K2 such that K2/02(K2) ~ L2(2n), we
conclude:
LEMMA L1.3.16. One of the following holds:
(1) Kz/02(Kz) ~ L2(2^2 in) for some i 2:: 1.
(2) Kz/02(Kz) ~ (S)Us(2n).
(3) n = 2 and Kz/02(Kz) ~ Ls(5) or Ji.
(4) n = 2, Kz/02(Kz) ~ Ls(4) or Us(5), and Y induces outer automorphisms
on Kz/02(Kz).We are now in a position to complete the proof of Theorem 4.3.2.
Assume that one of cases (1)-(3) of 4.3.16 holds and let p be a prime divisor
of 2n - 1. As Y* centralizes K2 / 02 ( K2) and hence K2, but the groups in those
cases do not admit an automorphism of order p centralizing K2, we conclude that
Y* = 1. By 4.3.11, 02(KS) = [02(KS), Y], so as R2/02(KS) = [R2/02(KS), Y],
also R2 = [R2, Y]. Then since Y* = 1, R2 = 1, contrary to 4.3.15.
Thus case (4) of 4.3.16 holds. Choose X of order 5 in K2· Recall that K has
three noncentral 2-chief factors, U and two copies of the dual of U on Q/U. Thus
K 2 has four noncentral 2-chief factors, and each is a natural module for K2R2/ R2·
Therefore X has four nontrivial chief factors on R2· As Gz E H(T) and Kz :::l Gz,
F*(Kz) = 02 (Kz), so at least one of those chief factors is in 02(Kz).
Suppose that Kz/0 2 (Kz) ~ Us(5). Then X = Z(P) for some PE Syls(Kz),
and P ~ 5i+^2. Thus from the representation theory of extraspecial groups, X
has five nontrivial chief factors on any faithful P-chief factor in 02 (Kz). But
02 (Kz) ::::; R 2 by 4.3.14.3, and we saw that X has just four nontrivial chief factors
on R2, with at least one in 02(Kz).
Therefore Kz/02(Kz) ~ Ls(4). Therefore Kz/02(Kz) ~ Ls(4). Let X be
a subgroup of order 3 in 02 ,z(K). Then X is faithful on Zs, so X :::; Gz but
X 1::. Kz, and hence XKz/02(Kz) ~ PGLs(4) by A.3.18. Further X centralizes
K2/02(K2), and from the structure of [0 2 (K), K] in 0.1.34.3, there are four non-
trivial K 2 -chief factors in 02 (K), all natural modules for K2/0 2 (K2) ~ L 2 (4), and
CR 2 (X)/CR 2 (K2X) is a natural module for K2/02(K2). It follows from B.4.14
that each nontrivial KzX-chief factor W in 02 (Kz) is the adjoint module for
Kz/02(Kz), and Cw(X)/Cw(XK2) is an indecomposable of F 4 -dimension 4 for