954 i3. MID-SIZE GROUPS OVER F2PROOF. Assume otherwise. By 13.8.16, we may take H = KLiT. By 13.8.17,
K* ~ L 2 (2n), (8)L3(2n), 8p 4 (2n), or G 2 (2n). By 13.8.5, case (2) of 13.8.8 holds,
so in particular u; E Q ( H*, UH).
Let B 0 be the Borel subgroup of K* containing T 0 := T* n K*, and let B :=
02 (B 0 ). As K is defined over F 2 n with n > 1, and LiT = TLi, Li acts on B; so
by 13.8.13, B :=:; M. Then using 13.7.3.9, Li= B(BLi), and BLi = BaLi, where
Ba:= 02 (CBL 1 (L/02(L))):::; CM(V).Let X :=Li if L/02(L) ~ A5, and X :=Li,+ if L/02(L) ~ A5. If L/02(L) ~ A6
then Be = 03 (B), while if L/02(L) ~ A5, then Ba = 03 (B)Lo. In either case,
[BX : Ba02(B) I = 3.
Next X/0 2 (X) is inverted by some t ET n L, and [Bo, t] :=:; 02(Bc). Nowfrom the structure of Aut(K*), one of the following holds:
(i) Cr• (0^3 (B)02(B)/02(B*)) = 02(Bo)·
(ii) n = 2 or 6, and K* is not U 3 (2n).
(iii) K* ~ (8)Us(8).
In case (i) as [t, 03 (B)] :=:; 02 (B), t E 02 (B), a contradiction as t in-
verts X /0 2 (X). In case (ii) if t induces an outer automorphism on K, then
[[B,t]/02([B,t])[ > 3 unless K* is (8)£ 3 (4) or L2(4). Therefore we conclude
that either:
(a) X 1:. K, X* :=:;CH· (K*) so that X :S:J KLiT = H, and X* is inverted in
CH·(K*), or(b) K ~ L2(4), (8)Us(8), or (8)L 3 (4), and t induces an outer automorphism
on K*.
Assume first that (a) holds. Then as H is an SQTK-group, m 3 (K) = 1, so
that K* ~ L 2 (2n), L 3 (2n) for n > 1 odd, or U 3 (2n) for n even. Further X :S:J H
and Vs = [Vs, X], so UH = [UH, X]. Hence as X is inverted in CH ( K*), each
noncentral chief factor for Hon UH is the sum of a pair of isomorphic K* -modules.
Then case (ii) of F.9.18.4 holds, so that each i E Irr+ (K, UH, T) is a T-invariantFF-module for KT. Therefore JH := (JH) is the sum of two X-conjugates of J,
and K* is not (8)U 3 (2n).
Suppose K* is L 2 (2n). Observe that if n is even, then m 3 (XB) > 1, so we
conclude from 13.7.3.9 that L/0 2 (£) ~ A 6. We saw earlier that Ba = 03 (B)L 0 ,with [BX : 02(B)BcJ = 3; then since X 1:. K as case (a) holds, we conclude that
B = Ba centralizes V. On the other hand if n is odd, then B is a 3'-group, so
again B =Ba centralizes V.
Next as KT has no strong FF-modules by B.4.2, applying F.9.18.6 to JH in
the role of "W", we conclude [UH, K] = JH. As i/C1(K) is an FF-module, by
B.4.2 it is either the natural L 2 (2n)-module or the A 5 -module. In the first case as
B centralizes V, Vs :=:; C-uH(BT 0 ) = C-uH(K), a contradiction since UH = (Vl)
and K i=-l. Thus i is the A5-module, so that J(H) ~ 85 by B.4.2.5; hence
H* ~ 85 x 8s and UH is the tensor product of the 85-module and 83-module.
Since case (2) of 13.8.8 holds, there is an H-conjugate a of 'Y such that v; :=:;
02(LiT*) = To :::; K*. Then as v; is quadratic on UH, rv;r = 2, contrary to
13.8.18.4.
This leaves the case K* ~ L 3 (2n), n > 1 odd. This time the FF-module i is