958 i3. MID-SIZE GROUPS OVER F2
LEMMA 13.8.23. Assume m(U;) = 1, and K is nontrivial on W. Then
(1) u'Y induces transvections on wand UH, DH is a hyperplane of UH, CH:=Cun(U'Y) is a hyperplane of DH, and CH= Cun(U;)..
(2) u; < v;:
(3) Either Ai f:. W and [Cw-(U;), v;J = 1, or Ai :S: W and [Cun (U;), v;J = 1.
(4) u; < CH·(CE(U;)) for at least one of E := W or UH.PROOF. By 13.8.22, H* is faithful on Wand case (2) of 13.8.8 holds, sou; E
Q(H*, UH)· Therefore as m(U;) = 1 it follows that m(UH/Cun(Ury)) :S: 2; then
since K is nontrivial on W, equality holds and u; induces transvections on both
Wand UH. Therefore by 13.8.10, DH is a hyperplane of UH.
Let CH := Cun(Ury)· By F.9.13.7, [Dry,DH] = 1, so as m(U;) = 1 and
[DH, Ury] :S: Ai by F.9.13.6, CH is a hyperplane of DH. Therefore CH= Cun(Ury)
as both subgroups are of codimension 2 in UH. Hence (1) holds.
Part (2) follows from 13.8.18.4. Next [CH, Vry] :S: Ai by 13.7.3.7. Further by
( 1), u; is not a strong FF* -offender on UH or W. Assume Ai :S: W. Then W :S: D Ji
by 13.8.22.1, so fJH = CuH(U;) =CH by (1). Thus if [C(JH(U;), v;J =f. 1, DH>WGH, and hence W :S: CH as !DH : CHI = 2 by (1). However this contradicts
[W, Ury] =f. 1. So suppose instead Ai 1:. W. Then by F.9.13.6, [DH n W, Ury] :S:
Ai n w = 1, so DH n w :s: CH, and hence
[n--;r=1w, Vry] :s: [CH, Vry] n w :s: Ai n w = LSince Cw-(U;) :S: DH -------n W, this establishes (3).
Finally (2) and (3) imply (4). DLEMMA 13.8.24. K* is not isomorphic to A7.
PROOF. Assume K* ~ A7. We adopt the notational conventions of section
B.3, and represent H on f2 := {1, ... , 7}, so that T has orbits {1, 2, 3, 4}, {5, 6},
and {7}. Let /3 := '"'(g/;i for gb as defined earlier, and let 6 E {/3,1}. By (1) and (2)
of F.9.13, VsY :S: 02(LJ.T) for some y EH.
Suppose first that case (1) of 13.8.8 holds, and pick 6 as in that case. Then Vo or
Us induces a nontrivial group of transvections on UH, so in ·particular KT ~ 87.
But as case (2b) of 13.8.5 holds, L/0 2 (L) ~ A.6 so ILil3 = 32 , and hence LJ.T ~
84 x 83 is the stabilizer of the partition { {1, 2, 3, 4}, {5, 6, 7}} of n. Thus 02 (LJ.T)
contains no transvections, whereas we showed VsY :S: 02 (LJ.T) and Us :S: V';,.
Therefore case (2) of 13.8.8 holds. Define a as in that case; thus V~ :S: 02 (LJ.T*)
and u; E Q(H*,UH)·
Pick W maximal in UH, so that UH is an irreducible H* -module. It will suffice
to show m(U;) = 1 and [W, K] =f. 1: for then 13.8.23.4 supplies a contradiction,
since for each faithful F2H-module Eon which some h EH* induces a transvec-
tion (that is, with [E, H] the A7-module), (h) =CH (CE(h*)).
As LiT = T Li, LiT stabilizes either {1, 2, 3, 4} or a partition of type 23 , 1.
Assume the first case holds. Then the stabilizer 8 in H of {1, 2, 3, 4} is solvable,
so 8 :S: M by 13.8.13. Thus 8 = H n M; hence by 13.7.3.9, Li = fJ(8) is of 3-rank
2, so that L/02(L) ~ A5. Next either L 0 = ((5, 6, 7)) and Li,+ = 02 (K5, 6 , 7 ), or
vice versa. As Li,+ is inverted in T n L :S: CT(L 0 ), H ~ 87. As q(H, UH) :S: 2