924 13. MID-SIZE GROUPS OVER F2By (iii), m([UH, VJ)= 2, so as m(V) = 1, q(KV*,UH)::::; 2. Therefore B.4.2
and B.4.5 describe K and the possible noncentral 2-chief factors W for KV on UH.
As m(KV*,UH) = 2, K is not one of the sporadic groups in B.4.5; cf. chapter H
of Volume I, and recall that the 12-dimensional module for J2 is the restriction of
the natural module for G 2 (4). If K ~ A 7 , then by (iv), V induces a transposition
on k, contradicting (ii).
In the remaining cases in B.4.2 and B.4.5, k is of Lie type over F2"' for some
m. By (i), either Vis generated by a long-root involution, or K ~ Sp 4 (2m)'. Then
by (iv), m = 1, and if k is A 6 , then Vik. Turthermore if K* ~ A5, then as
H E He, for some choice of W, W is the faithful 6-dimensional module for K*.
But then as V i k, m([UH, VJ) ;::: 3, contrary to (iii). Thus K* is not A5, so in
particular Z(K*) = 1, and hence K* ~ K is simple. Similarly by (iii), Wis neverthe 10-dimensional module for K* ~ £5(2).
The cases remaining appear in B.4.2. Turther KV ~ Lz(2), 3 ::::; l ::::; 5, 86,
or G 2 (2)', (keeping in mind that Vik iff K ~ A 6 ) and Wis either the natural
module for K*, or the 6-dimensional orthogonal module for K* ~ £ 4 (2). As V
centralizes Y =f. 1 by (ii), K is not £ 3 (2). Therefore m3(K) = 2, so by A.3.18,£ 1 ::::; 031 (H) = K, completing the proof of (1). Turther in each case m3(C.K(V)) =
1, so as L1 ::::; C.K(V), we conclude that m 3 (L 1 ) = 1, and hence (4) holds. Next
W = Ui/U2 for suitable submodules Ui of UH, and by (iii),
[W, V]::::; Vw := (V3 n U1)U2/U2,
and £ 1 is irreducible on V3, so Vw = [Vw, L1] is of rank 2 and Vw = [W, V]. Thiseliminates the possibility that Wis a natural module for £4(2) or £5(2), since there
V is a long-root involution, so V induces a transvectio~ on W. Hence W is the
natural module for KV ~ 86 , A 8 , or G 2 (2)', establishing (2). Turthermore by
(iii), W is the unique noncentral chief factor for K on UH. As V3 = [V3, L1] ::::;
[UH, K], UH= (V 3 H) =[UH, K]. This completes the proof of (3).
Finally we verify (5) and (6). As £ 1 ::::; Kand Tacts on £ 1 , Tacts on K, so
K :'::! H by 1.2.1.3. By (iii), [UH, VJ= V 3 is of rank 2 and is LiT*-invariant, andin each of the cases in (2), P := NK•T•([UH, VJ) is a minimal parabolic of KT*,
so IP* : T*I = 3. Thus P* = LiT*. Turther [V, QH] ::::; v n QH = V3, so that
UH= [QH, K]. This completes the proof of (5). From the action of P* on U, wedetermine that (6) holds in each case. Thus the proof of the lemma is complete. D
By 13.5.21.6, the hypotheses of 13.5.14 are satisfied. Choose l as in 13.5.14, and
set L_ :=(UH, Uk). By 13.5.14, UH::::; £ 1 , and L_ = 02 (L_0 2 (LT)) is described
in G.2.4. Turther if n = 5 then L_ = L by 13.5.14.3. As in G.2.4, let S := 02 (£_ ),
82 = V(UH n Uk) and lets denote the number of chief factors for L_ on S/82, asin G.2.4.6. We maintain this notation throughout the remainder of the section.
LEMMA 13.5.22. (1) ISi = 24 Cs+l)IS2: v1.
(2) L1 has 2s + 2 noncentral 2-chief factors.
(3) IUHI = 22 s+5IS2: Vl-
(4) UH::::; £1.