1128 15. THE CASE .Cr(G, T) = f/J
PROOF. Observe that 15.3.19, 15.3.20, and 15.3.21 have eliminated all other
possibilities for L* from the list of C.2.7.3 provided by 15.3.16. Then as L/02(L)
is quasisimple by 15.3.15.2, and the Schur multiplier of L* is a 2-group by I.1.3,
02,z(L) = 02(L) so that L/02(L) ~ L* is simple. D
LEMMA 15.3.23. Lj0 2 (L) is not L3(2).
PROOF. Assume Lj0 2 (L) ~ L3(2), and set UL := [WL,L]. By 15.3.12.2,
Y+ = YLYe with IYLl3 = 3 = IYel and Y+/02(Y+) ~ Eg. By 15.3.16, R acts on
L, and by C.2.7.3, ML = SLYL is a minimal parabolic of Land R = 02(Y+S) =
02(LR)02(YL).
By C.2.7.3, (LR, R) is described in Theorem C.1.34. In particular L has k := 1,
2, 3, or 6 noncentral 2-chief factors. If k = 6, then case (4) of C.1.34 holds so that
m(0 1 (Z(S))) = 3, contrary to 15.3.8. Thus 1:::; k:::; 3.
From C.1.34, either UL is the direct sum of s:::; 2 isomorphic natural modules,
or UL is a 4-dimensional indecomposable. Thus either [UL, Ye] = 1, or s = 2 and
UL= [UL, Ye].
Assume first that [V, YL] .= 1. Then as YL =/=- 1 and Y is faithful on Vin case
(1) of 15.3.7, case (2) of 15.3.7 holds with Y/0 2 (Y) ~ 3i+^2 , and YL = 02 (Cy(V)).We saw Y+/0 2 (Y+) ~ E 9 , so Y+ < Y, and hence case (2) of Hypothesis 15.3.10
holds. Then Na(Vi) :::; M by 15.3.11.3, Y+ centralizes V1, and V2 = [Vi, Y+] =
[Vi, Ye]:::; CwL(YL)· As Na(Vi):::; M but L 1:. M, L centralizes neither V1 nor V2.
From the previous paragraph, UL is either a sum of isomorphic natural modules,
or a 4-dimensional indecomposable with a natural quotient, while ML is a minimal
parabolic of L with R = 02 (MLR). Thus V:::; Z(R) while the fixed points of the
unipotent radical Ron any extension in B.4.8 of UL with trivial quotient lie in UL,
so we conclude that
V::; ULCwL (L).
By the previous paragraph, either [UL, Ye] = 1, ors= 2 and UL = [UL, Ye]. In
the first case,
Vi= [Vi,Yc]:::; [ULCwL(L),Yc] = [CwL(L),Yc]:::; CwL(L),
contrary to an earlier remark. In the second case,
Vi= Cv(Ye):::; CuL(Ye)CwL(LYe) = CwL(LYe),
contrary to the same remark.
Therefore [V, YL] =/=-1. Now YL = [YL, SL] while [Ye, SL] ::; 02(Ye), so from
the action of Son Y+, YL and Ye are normal in Y+S, and {YL, Ye} is the set Y of
S-invariant subgroups of Y+ with Sylow 3-group of order 3. In particular, S acts
on YL and hence on L.
Suppose that case (2) of Hypothesis 15.3.10 holds. Then by 15.3.14.2, Y/0 2 (Y)
~ 31+^2 with Cy(V) > 02(Y). As Y = {YL, Ye} while [V, YL] =/=-1, it follows that
Ye = 02 (Cy(V)), so Na(Ye) :::; M as M = !M(Na(Ye)) by 15.3.7.2. But L
normalizes 02 (Ye0 2 (L)) =Ye, contradicting L 1:. M.
Thus we are in case (1) of Hypothesis 15.3.10, so Y = Y+, and hence from earlier
discussion, Y = YeYL and Y/02(Y) ~ E 9 • Therefore we may take Ye= 02 (M 1 )
and YL = 02 (M2), since {YL, Ye} = Y. Thus Yf, =Ye, and V 2 = [V, YL]· AsVi = [V, YL] is S-invariant, YLSL is the parabolic of L stabilizing the line Vi in
Z(02(L)). Hence case (5) of C.1.34 does not hold, as no such line exists in that
case.