11.1. THE SUBGROUPS Na(V;) FOR T-INVARIANT SUBSPACES V; OF V 76ithat Na(Vi) E 7-le by 1.1.4.6. Notice when L ~ SL 3 (q) that V3 = V, while in the
other cases, from the action of NGL(V)(L) on V, NM(V3) = NM(Vi).We begin by considering the embedding of Li in a C-component Ki of NG (Vi). i
LEMMA 11.1.1. Assume either{i) i = 1, 1 =f Vo::=; Vi, H := Na(Vo), and To:= Nr(Vo) E Syb(H), or
{ii) i = 2 and H := Na(Vz).
Then Li ::=;KE C(H) with K :::) H, and one of the following holds:
{1) Li= K.
(2) K/02(K) ~ (S)L3(q), Sp4(q), G2(q),^2 F4(q),^3 D4(q), or^3 D4(qif^3 ).
{3) n = 2 and K/02(K) is isomorphic to A7, A7, L 2 (p) for a prime p with
p = ±1 mod 5 and p = ±3 mod 8, L2(25), (S)LH5), M22, M22, M23, Ji, J2, J4,HS, Ru, SL2(5)/Po for a suitable nilpotent group Po of odd order, or SL2(p)/Ep2
for a prime p satisfying the congruences above.PROOF. If i = 1, V 0 and To are defined in (i); if i = 2, set Vo :=Vi and T 0 := T.
Thus in either case H = Nc(Vo), and· To E Syl 2 (H) acts on Li, so Li E .C(H, T 0 ).
Thus by 1.2.4, Li is contained in a unique KE C(H), and the embedding Li ::=; K
appears on the list of A.3.12. As To acts on Li, To also acts on K, so K :::) H by
1.2.1.3. The possibilities for K are determined by restricting the list of A.3.12 to
Li/0 2 (Li) ~ L2(q). The groups in (2) are the gro'ups of Lie type, characteristic 2,
and Lie rank 2 in Theorem C (A.2.3). When n = 2, we use the list in A.3.14, and
get the further examples
0
in (3). DWe next determine the possible embeddings of Li in Na(Vi) for i = 1 and 2.
Recall that Xis a Hall 2'-subgroup of NL(TL), so X ::=; NL(Vi).
PROPOSITION 11.1.2. For i = 1, 2, Li :::; Ki E C(Na(Vi)) with Ki :::) Nc(Vi)
and Ki E 7-le. Furthermore for K := Ki either Li = ·K, or i = 1, q = 4, and one
of the following holds: ·
(1) K/02, 2 1(K) ~ SL 2 (p) where p = 5, or p?: 11 is prime.
{2) K/0 2 (K) ~ L 2 (p) for a suitable prime p?: 11, and L/02(L) is not SL3(4).
{3) KX/02(KX) ~ PGL3(4). Further if Ko denotes the member of .C(G, T) nK distinctfromK andLi, and!:= (Ko,L2), then! E .Cj(G,T), and interchanging
the roles of Land I if necessary, L/02(L) ~ G2(4) and l/02(I) ~ Sp4(4).
PROOF. By 11.1.1, Li::=; Ki :::) Na(Vi). Recall Na(Vi) E 7-le, so Ki E 7-le by
1.1.3.1. So we may assume Li < Ki =: K, and K appears in case (2) or (3) of
11.1.1, but not among the conclusions of 11.1.2. In particular K 1:. M.
Let Gi := Ca(Vi); observe that X ::=; Na(Vi), and set (GiX)* := GiX/02(K).
As Nc(Vi) E 7-le, Gi E 7-le by 1.1.3.1. Set Xi := Cx(Li/02(Li)). By 11.0.2.2,
IXil=q-1.
Suppose first that i = 1. By inspection of the possiblities for K, namely in (2)
and (3) of 11.1.1 but not in 11.1.2, K/0 2 (K) is_ quasisimple and either
(i) mp(K) = 2 for some prime p dividing q - 1, or
(ii) q = 4 and K/02(K) ~ L2(P) for a prime p?: 11, L2(25), L3(5), or Ji.