11.3. ELIMINATING THE SHADOW OF L 4 (q) 773As m([B, VH]) = 4n, VH is the 6n-dimensional maximal central extension of the
natural module for K* appearing in I.2.3.1. Then from the structure of that module
as orthogonal 3-space over F 2 n, [VH, a] n [VH, b] = 1 for a* -/=-b* in B*; whereas
from the action of Rg on va, for elements a, b E Vg in distinct cosets of Vi, we
[a, VH] = [a, Rg] = V2 = [b, Rg] = [b, VH]· This contradiction completes the proof
of the lemma. DWe now obtain successive restrictions forcing various 2-locals to closely resemble
those in the shadow L4(q).
Set K := 02 (H), let D be a Hall 2'-subgroup of HnM, and further set Do:=
03 (D)fh(0 3 (D)). Recall Xis a Cartan subgroup of L acting on TL.
LEMMA 11.3.7. (1) KE C(H) and K* ~ K/02(K) ~ L2(2n).(2) Do:::; L.
(3) Cz(L) = Cv(L) = 1 = CvH(K).
(4) V =A and V* E Syb(K*).
(5) Vi= [V1,Do] and we may take Do:::; X.PROOF. Part (1) follows from 11.3.6 and 11.3.5.2, and then (2) follows from
11.0.4. Next L ~ SL 3 (q) or Sp4(q) by 11.3.4.2, so that m = s(G, V) = n by 11.2.2.
Thus A E An(H, VH) by 11.3.4.1, so it follows that A E Syb(K) is of rank n
and CvH(A) = CvH(a) for all a EA#. Then by G.1.6, VH/CvH(K) is a direct
sum of s copies of the natural module for K*.
Let VL := [(ZL), L] = [Z, L], so that V :::; (zL) = VLCz(L) using B.2.14.
Similarly VH = [VH, K]Cz(K). As [Z, H] -/=- 1 by Hypothesis 11.3.1.1, L = [L, J(T)]
by Theorem 3.1.8.3; therefore by Theorem B.5.1.1, either VL = V and V is the
natural module for L, or L ~ SL 3 (q) and VL is the sum of two isomorphic naturalmodules for L. As Do :::; L and TD 0 = DoT, we may take Do :::; X, and either
Cz(D 0 ) = Cz(L), or conjugating in NL(V2) if necessary, we may assume [V1, Do] =
1 = [Z,D 0 ]. On the other hand as VH/CvH(K) is a sum of natural modules for
K* and VH = [VH, K]Cz(K), Cz(Do) = Cz(K). In particular, [Z, Do] -/=-1, so
Cz(D 0 ) = Cz(L). Then as K 1. M = !M(LT), 1 = Cz(K) = Cz(Do) = Cz(L),
so (3) follows,^3 [VH,K] = VH, Vi= [V 1 n Z,Do] :::; VH, and the proof of (5) is
complete. By 11.2.2.4, Vi:::; CvH(Wo) :=::: CvH(A).
Next Ca(V2) :::; Cc(V1) :::; M using 11.3.4.4. Thus as m(A/CA(VH)) = n,
11.2.2.3 says VH:::; Cc(CA(VH)):::; Nc(A). Now VH centralizes CA(VH) of corank
n in A, so VH/CvH(A) is contained in the group A of all Fq-transvections on A
with axis CA(VH)· From the action of A* on VH, CvH(A) is of rank sn, so as
m 2 (A) = 2n, n for L ~ SL 3 (q), Sp4(q), conjugating in L^9 if necessary, either:
(i) s = 1, m(VH/CvH(A)) = n, and [A, VH] = V{, or
(ii) s = 2, L ~ SL3(q), AutvH(A) = AutR~(A), and Vi= [A, VH]·
In case (i), V 19 = [A, VH] = CvH (A), so V{ = V1 using our earlier observation that
Vi:::; CvH(A); hence A= V by 11.3.4.4. Similarly in case (ii), from the action of Rg
on A, for each u E VH - Vi, [u, A] = vr for some x E Lg :::; Nc(A). Further VH
is the sum of two natural modules for K* and V1 = [V1,Do]:::; CvH(A), so V1 is a
1-dimensional F q-subspace of CvH (A). Therefore Vi = [A, u] for some u E VH -Vi,
so again Vi= V{" for some x E Nc(A), and hence we may assume Vi= V{, so
A= V by 11.3.4.4. This completes the proof of (4). D