i2.5. ELIMINATING L5(2) ON THE 10-DIMENSIONAL MODULE 8i7We mention that there is L E .Cj(G,T) with L ~ L 5 (2)/E 210 in the non-
quasithin groups Sp10(2), Dt 0 (2), !"21 2 (2), and 0;1;(2). These shadows cause little
trouble, as they are essentially eliminated immediately in 12.5.3 below.The proof involves a series of reductions. As usual we adopt the conventions of
Notation 12.2.5. Observe that af? Tacts on V, T induces inner automorphisms on
L, so Mv = L.
We next discuss the parabolic subgroups of Lover 'f, and their action on the
module v. Let r be the natural 5-dimensional module for L with a basis for r
denoted by {1, ... , 5}, and let rk := (1, ... , k). Choose notation so that Tacts on
rk for each k. We regard V as the exterior square A^2 (r), so that V has basis i /\ j
for 1 Si < j S 5. Then T acts on the subspaces Vk of dimension k defined by
Vi := A^2 (I'2) = (1/\2), V3 := A^2 (I'3) = (1/\2, 1/\3, 2 /\ 3),
Vi:= ri Ar= (1Ai:1 <is 5),
V6 := A^2 (I'4) = (i/\j: 1Si<j<5), Vi:= I'2/\I' = (1/\i,2/\j: i > 1,3 s j s 5).For i = 1,3,4,6, set Gi := Na(Vi), Mi := NLr(Vi), Li := NL(Vi)^00 , and
Ri := 02(LiT).
Notice that Li E .C(G, T) for each i.
LEMMA 12.5.2. {1) Mi = NL(I'2), Mi/02(Mi) ~ L3(2) x L2(2), Li/Ri ~
L3(2), and 0 <Vi <Vi< V is a chief series for Mi.(2) M3 = NL(I'3), M3/02(M3) ~ L2(2) x L3(2), L3/R3 ~ L3(2), and V3 is a
natural module for L3 / R3.
{3) M4 = L4 = NL(I'i), and V4 is a natural module for L4/R4 ~ L4(2).
(4) M6 = L6 = NL(I'4), v6 is the orthogonal module for L5/R6 ~ L4(2) ~
nt(2), and V/V 6 is a natural module isomorphic to R5.
PROOF. These are easy calculations. D
Observe that from 12.5.2.1, Mi = PLi, where Pis the minimal parabolic of
LT over T which is in Mi, but not in LiT. Further 02 (P) = 0
31(P) :s:) Mi with
[0^2 (P),Li] s 02 (Mi). Similarly 02 (P) S L 3 n L 6 , Pis a minimal parabolic of
LiT for i = 3, 6, and Mi n Mi is the product of P with the minimal parabolic
Pi := Mi n LiT.
Recall from chapter 1 the definition of Bp(X) for XE .C(G,T) with X/02(X)
not quasisimple.
LEMMA 12.5.3. For each i = 1, 3, 4, 6, Li S Ki E C(Nc(Vi)) with Ki :s:) Nc(Vi),
and one of the following holds: ·
{1) Li= Ki.
{2) i = 1, Ki/02(Ki) ~ L5(2), M24, or J4, and 02 (P) S Ki.
{3) i = 1, Ki= B1(Ki)Li and Ki/02(Ki) ~ SL2(7)/E49.
PROOF. The existence and normality of Ki follows from 1.2.4 and the fact that
T normalizes Li· By 12.5.2, Li/02(Li) ~ L 3 (2) if i = 1, 3 and L4(2) if i = 4, 6.
We first treat the case i = 1. We may assume that Li <Ki, so that Kif02(Ki)
is described in the sublist of A.3.12 where B/02(B) ~ L3(2). If Ki/02(Ki) ~
L5(2), M24,. or J4, then Ki = 0 3' (Gi) by A.3.18, so 0 2 (P) S Ki and hence
(2) holds. Thus we may assume that Ki/0 2 (Ki) is not one of these groups, nor