15.3. THE ELIMINATION OF Mr/CMf(V(Mf)) = Ss wr Z2 1129Set Q := [02(L), L] as in C.1.34, and observe that [Z, L] :::; UL :::; Q. We
will complete the proof by showing that fort E T - S, W := QQt is normalized
by LS, and hence also by T as IT : SI = 2. Then as Y = YLYe :::; (YL, t),
(L, T):::; Na(W):::; M = !M(YT) by 15.3.7, contrary to the choice of L.
Assume that k = 3, so that case (3) of C.1.34 holds. Then as YL stabilizes
the line V 2 in the natural module Z(Q), Q = [Q, YL], so Q :::; 02 (YL)· Fur-
ther Ye centralizes the natural module Z(Q) since EndL;o 2 (LJ(Z(Q) = F 2. AsQ/Z(Q) is the direct sum of two natural modules, either Ye centralizes Q, or
Q = [Q, Ye]. In the latter case Q = 02 (Ye) n 02 (YL), so Q is t-invariant, whereas
Ye and YL have three and two nontrivial 2-chief factors, on Q/Z(Q), respectively.
Therefore [Q, Ye] = 1, so Ye = 02 (Ye) centralizes L by Coprime Action. ThenQt :::; 02(YL)t = 02(Ye) :::; Cs(L), so that W = QQt _::::i LS, which suffices as
mentioned above.Suppose finally that k = 1 or 2, so that case (1) or (2) of C.1.34 holds. In
each case Q = [Q, YL]Cq(PL), for PL E Syls(YL), and as Ye centralizes the line V 2
stabilized by YL in a natural submodule in Q, Ye centralizes L from the structure
of Aut(L). Thus Q = 02(YL)CQ(P) :::; 02(YL)Cs(P), for PE Syls(Y), and by a
Frattini Argument we may assume t ET-S normalizes P, and hence also Cs(P).Therefore Qt :::; 02 (Ye)Cs(P) :::; 02 (LS), so [Qt, LS] :::; [0 2 (LS), L] = Q, and
hence W = QQt ::;! LS, which again suffices as mentioned earlier. DLEMMA 15.3.24. L/02(L) is not L4(2) or L5(2).
PROOF. Assume L/02(L) ~ Ln(2) for n := 4 or 5. Then Y+ is solvable and
S-invariant of 3-rank 2, Y+ ::::; L by 15.3.17, and Sn LE Syl2(L) as SE Syb(I).
Thus LS E 1-f+, so we may take I = LS, and hence U 1 = (ZL). As Sylow 3-subgroups of L are isomorphic to E 9 , Y+/02(Y+) ~ E 9 rather than 3i+^2. Then
Y+S / R ~ S 3 x S 3 from the action of S on Y, so S is trivial on the Dynkin diagram
of L/02(L).
Suppose first· that n = 4. Then Y+S is the maximal parabolic of LS over S
determined by the end nodes, so Y+SL = L n Mas L i. M. This parabolic has
unipotent radical RL/02(L) ~ E24·
Set UL := [WL, L]. By 15.3.16, L = [L, J(R)], so there are FF-offenders on
UL with respect to R, and in particular UL is an FF-module for L/02(L). As
1 of [ Z, Y+] :::; UL, UL/ Cu L ( L) is not the orthogonal module, so we conclude from
Theorem B.5.1 that UL is either the sum of a natural module and its dual, or the sum
of at most n -1 isomorphic natural modules. Now by B.2.14, ULZ = ULCuLz(L),
and we let ZL denote the projection of Z on UL with respect to this decomposition.Then Z:::; ZLCwL(L), so that CL(ZL) = CL(Z).
Assume first that UL is a sum of isomorphic natural modules. Then WL =
ULCwL(L) by I.1.6.6. Also 1 of 02 (CY+(ZL)) = 02 (CY+(Z)) ::::; 02 (CY+(V)), so
case (2) of 15.3.7 holds. Hence Y+ < Y, so that case (2) of Hypothesis 15.3.10
holds, and thus Nc(Vi)::::; M by 15.3.11.3. But now Vi::::; CwL(Y+) = CwL(L), soL:::; Cc(Vi) :::; M, contrary to the choice of L.
Therefore UL is the sum of a natural module and its dual. Since Y+SL = LnM
is the maximal parabolic over SL determined by the end nodes, each Ls ( 2 )-parabolic
P over SL satisfies Pi. M and [ZL, 02 (Y+ n P)] of 1. Then applying 15.3.11.12 to
Pin the role of "L", no nontrivial characteristic subgroup of S is normal in PS.
Thus (0^2 (P)S, S) is an MS-pair, and hence is described in C.1.34. But P has two