12.9. THE FINAL TREATMENT OF Ln(2), n = 4, 5, ON THE NATURAL MODULE S59M24 in the latter. Now we can repeat our argument in the first paragraph of the
proof for the case i = 2: In each case K 2 = 031 (Na(K 2 )) by A.3.18. Also we saw
K2 ::::; Ca(Vi), while G2 contains a subgroup X of order 3 fixed-point-free on Vi,
so X ::::; 0
31
(Na(K2)) = K2 ::::; G1, a contradiction. This completes the proof of
12.9.3. DFor the remainder of the section, let K 1 be defined as in 12.9.3.
LEMMA 12.9.4. (V^01 ) is abelian.PROOF. Assume otherwise; then we have the hypotheses of the latter part of
section 12.8, so we can appeal to the results there .. Adopt the notation of the
second subsection of section 12.8; in particular take H := G 1 , U := UH = (VH),
and H* := H/CH(U).
As n 2 4, we conclude from 12.8.8.5 that L 1 is not normal in H, so that
L1 < K1 and in particular K1 f:_ M. Hence Ki/0 2 (K 1 ) is described in (2) or (3)
of 12.9.3. By 12.8.12.4, fr and its action on U are described in Theorem G.11.2.
As [V, L1] -:/-1, L1 -:/-1, so K 1 is a nontrivial normal subgroup of fr, and is also a
quotient of Ki.
If Ki~ SL2(7)/E49 then either K1 = L 1 ~ L3(2) or K1 ·~Ki- However by
inspection of the list in Theorem G.11.2, fr has no such normal subgroup. Thus
one of the remaining cases holds, where Ki is quasisimple, and hence Ki/Z(K 1 ) ~
Kif Z(Ki). Comparing the list in (2) and (3) of 12.9.3 to the normal subgroups of
groups listed in Theorem G.11.2, we conclude n = 4 and one of conclusions (4), (5),
or (8) of Theorem G.11.2 holds. Conclusion (8) does not occur, as there fr ~ S 7 ,
so that there is no T-invariant subgroup L 1 with Li/0 2 (L 1 ) ~ L 3 (2). Conclusion
( 4) does not hold by 12.8.13.6.
Thus conclusion (5) of Theorem G.11.2 holds; that iti U is the 6-dimensional
natural module for K 1 = F*(fr) ~ As. Let D := zg, ZD := Zn Z(h), and
VD= (Zfj^1 ). By 12.8.13.5, K 1 E C1(G, T), K 1 acts nontrivially on the submodule
VD of Zu E R2(KT), and K1 = [K1,D].
As K1 E C1(G, T), K1 ::::; KE Cj(G, T) by 1.2.9.2. Then either K 1 = K, or
K/02(K) ~ L5(2), M24, or J4 by A.3.12. Thus K1 =KE Cj(G, T) by 12.9.2.
As F*(fr) ~As, fr~ As or Ss, so as T normalizes Li with Li/02(L1) ~ L3(2),
we conclude fr~ As. Thus L 1 is a maximal parabolic of fr corresponding to an
end node. Next set Lo := 02 (0L(Vi)) = 02 (0£ 1 (Vi). Then L 0 T* is the minimal
parabolic of Li centralizing \/2. As V 2 is a singular 1-space in the orthogonal space
U, L 0 T is one of the two permuting minimal parabolics in the maximal parabolic
Po := Oir(f2) corresponding to the middle node of the Dynkin diagram for fr; in
particular Po normalizes L 0. Similarly L 1 is the maximal parabolic of fr normalizing
the totally singular 3-subspace V of U, and so corresponds to an end node of the
diagram for fr, with Lo = Pon L 1. Finally Lot' is the minimal parabolic of L
centralizing Vi, with I2'i' the other minimal parabolic in the maximal parabolic
Nr,(V2) for the middle node, so that h normalizes Lo.
By 12.8.13.2, D ::::; Cr(V) = 02(LT), and hence iJ ::::; 02(L1T). By 12.8.12.2,
- o • • • A J_ A
D :'::) H2 := HnG 2 , so as L 0 T::::; H2, D :'::) L 0 T. By 12.8.10.4, [V2 ,DJ= [W,D]::::;