12.2. GROUPS OVER F2, AND THE CASE V A TI-SET IN G 795
THEOREM 12.2.2. Assume Hypothesis 12.2.1. Then one of the following holds:
(1) G is a group of Lie type of Lie rank 2 over F 2 n, n > 1, but G ~ U 5 (2n)
only for n = 2.
(2) G ~ M22, M23, or J4.
(3) T :SM:= Nc(L), and there exists VE Irrf-(L, R 2 (LT), T). For each suchV, V :SI T, V E R2(LT), the pair L, V is in the Fundamental Setup (3.2.1), V
is a TI-set under M, and either V :SI M or Cv(L) = 1. In addition, one of the
following holds:4, or 5.(a) V is the natural module of rank n for L/02(L) ~ Ln(2), with n = 3,
(b) m(V) = 4 and V is indecomposable under L/0 2 (L) ~ L 3 (2).
(c) L/02(L) ~ L5(2) 1 and Vis an irreducible of rank 10.(d) V/Cv(L) is the natural module for L/CL(V) ~An, with 5:::::; n:::::; 8.
(e) m(V) = 4, and L/CL(V) ~ A1.
(f) V/Cv(L) is the natural module of rank 6 for L/02(L) ~ G2(2)' ~
(g) V is a faithful irreducible of rank 6 for L/0 2 (L) ~ A 6.
PROOF. By Theorem 10.0.1, T :S Nc(L). Hence (L, T) = LT and by 3.2.3,
there exists Vo E Irr +(L, R2(LT), T) and for each such V 0 , Land V := (V[) satisfy
the FSU. Therefore by Theorem 3.2.5, one of the following holds:
(i) V = Vo :SI M.
(ii) V =Vo :SI T, Cv(L) = 1, and Vis a TI-set under M.
(iii) Case (3) of Theorem 3.2.5 holds.
Case (iii) was eliminated in Theorem 7.0.1 and Theorem 12.1.11. Thus case (i)
or (ii) holds, so that V = V 0 E Irr +(L, R 2 (LT)) and V :SI T. As 02 (LT) centralizesR2(LT) and L/02(L) is quasisimple, 02(LT) :::::; CLr(V) :::::; 02(LT)02,z(L), so that
VE R2(LT). In case (i), V :SI M, and in case (ii), Cv(L) = 1, so in either case V
is a TI-set under M. Thus it remains only to show either that G is described in (1)
or (2), or that Land its action on V are as described in one of the cases (a)-(g) of
part (3) of Theorem 12.2.2.
The possibilities for the pair (L, V) when V is not an FF-module under the
action of AutcL(V) (L/CL(V)) are listed in 3.2.9. If the first case of 3.2.9 holds,
then Vis the D4(2n)-module for L 2 (2^2 n) with n > 1, so by Theorem 6.2.20, either
G ~ U 4 (2n), or n = 2 and G ~ U 5 (4). Thus conclusion (1) of Theorem 12.2.2 holds
in this case. The remaining cases of 3.2.9 were treated in Theorem 7.0.1, where it
was shown that G is isomorphic to J4, so that conclusion (2) of Theorem 12.2.2
holds.
Thus we have reduced to the case where Vis an FF-module for AutcL(v)(L),
where L := L/CL(V). Therefore Land its action on V := V/Cv(L) are listed in
3.2.8. In the first case of 3.2.8, L ~ L 2 (2n) and V is the natural module. Then
by Theorem 6.2.20, the only groups G arising are: the groups of Lie rank 2 and
characteristic 2 (arising in our Generic Case), so that conclusion (1) of 12.2.2 holds;
and M 22 and M 23 , so that conclusion (2) of 12.2.2 holds. Indeed the only case of
the FSU with L ~ L2(2n) left open by Theorem 6.2.20 is the case where n = 2 and
V is the A 5 -module; this case is one of the subcases of 3.2.8.5, and it appears as a
subcase of case (d) of conclusion (3) of 12.2.2. The cases with h > 1 in (2), (3), and
(4) of 3.2.8, were eliminated in Theorem 11.0.1. On the other hand when n = 1,