964 13. MID-SIZE GROUPS OVER F2
LEMMA 13.8.29. {1) K2 E C(G2) with K2/02(K2) ~ L3(2).(2) G2 = K2L2,+T and Ch= k2 x j2 ~ L3(2) x 83.
(3) Uo = Vs n V.J-= UsUiI and U 0 /V 2 is the tensor product of the natural
modules V/Vi and Us/Vi for j2 and K2.
2(4) V 0 = VsV.J-V.J-, and Vo/Uo is the tensor product of V/Vi or V/Vi EB F2
with the dual of Us /Vi.
(5) Vs/Us is the 6-dimensional orthogonal module for H* ~ L4(2).PROOF. As Us is the natural module for H ~ L4(2), Ns(Vi) = Cs·(°V2) is
the parabolic subgroup L 3 (2)/E 8 of H* stabilizing the point V2, so K2 E C(HnG2)
with K 2 /0 2 (K 2 ) ~ L 3 (2). As h :'.Si G2, and I2 acts transitively on vt, G2 =
I 2 (H n G 2 ) with H n G 2 = K2T and [K 2 , j2] = 1. Thus (1) and (2) hold.
Next Us/Vi is the natural module for K2 ~ L3(2). Thus as K2 :'.Si. Ch, Uo/Vi
is the direct sum of h-conjugates of Us/Vi. Further Us= (V3K^2 ) with V/Vi the
natural module for j 2 , so as h :'.Si. G 2 , U 0 /V 2 is the direct sum of conjugates of
V/Vi. Thus U 0 /Vi = UsUiI/Vi is the tensor product of V/Vi and Us/Vi. Further
Uk= (V/K^2 )::; Vs, so Uo = UsUiI::; Vs and so Uo = ug::; Vs n V_J-.
Let Vs:= Vs/Us. Then Vs= (Vs) with the maximal parabolic L]_T of H
centralizing the point V, and (VK^2 ) ~ UiI/Vi ~ Us/Vi as a K2-module, so we
conclude from B.4.13 that (5) holds. In particular K2 is irreducible on Vs /Uo, so
either Uo =Vs n V.J-or Vs= V.J-. In the latter case, both LT= (L2,+, L1T) and
H act on Vs, contrary to H 1. M = !M(LT). This completes the proof of (3).
By (5), Vs/U 0 is isomorphic to the dual of U 0 /Us as a K 2 -module, and by (3),
Vs< Vo. Thus (4) holds. D
LEMMA 13.8.30. L 0 has at least 9 noncentral 2-chief factors.PROOF. Recall· V < UL = (Uf{) ::; 02(LT) = Q by (7) and (2) of 13.8.4.
Let W be a normal subgroup of L maximal subject to being proper in UL, and set
·[h := UL/W. As Us/Vi is a 2-dimensional irreducible for Lo :'.Si. L, and UL= (UJi-),
f~h = (Uf{) = [UL,Lo] is a faithful irreducible for L+ := L/02(L) ~ A5, and so
may be regarded as an F 4-module on which Lt ~ Z 3 acts by scalar multiplication.
In particular from the 2-modular character table for A5, dimF 4 (UL) = 3 or 9, so to
complete the proof, it suffices to show dimF 4 (UL) > 3.
From 13.8.29.3, B2 := (Uf?) ~ (Uf?)/V is ofF4-dimension 2. Let S 3 := (S~^1 );from 13.8.29.5, S3/UL ~ (Uf^1 )/UsV is of F4-dimension 2, so dimF 4 (S 3 ) = 3.
Finally by 13.8.29.4, L 2 ,+ does not act on (Uf^1 ) /U 0 , so UL > S 3 , completing the
~~ D
LEMMA 13.8.31. {1) A1 i. Us.
{2) Case {2) of 13.8.8 holds.
{3) [Hc,K] 1. Vs; and if K has a unique noncentral chief factor on He/Vs,
it is not a 4-dimensional module for H* ~ L 4 (2).PROOF. In case (1) of 13.8.8, A 1 ::; Us by 13.8.27.3, so to prove (2), it will
suffice to establish (1).
Assume (1) fails, so that A 1 ::; Us. Then as H is transitive on Uf;, there
is k E H with A~ = V2 = Vl; and since [Vi, Qs] = Vi by 13.7.3.6, we may
assume A~ = Vf. Then as G1 = H by 13.8.27.2, 1k = 11g, so that by 13.8.29.3,