11.4. ELIMINATING THE REMAINING SHADOWS 775LEMMA 11.3.8. There exist F q-structures on P and V, preserved by Y and L,
respectively, which agree on P n V = V 2 •
PROOF. LetEv:= EndFqL(V), Epnv := EndFqL 2 x(P n V), and Ep := EndFqY(P).
Then Ew ~ Fq for each WE {P, V,Pn V}. In particular we may regard Epnv asthe restriction of Ew to P n V for W := P, V, so the lemma holds. D
Let X1 := Cx(Li/02(L1)) and X2 := Xn£2. Let W 1 be an X-complement to
V n Pin V, and W2 an X-complement to P n R 1 in P. Finally let W 3 := [W 1 , W 2 ].
Then W3 is X-invariant, and (W 1 , W 2 ) = W 1 W 2 W 3 is a special group of order q^3
with center W3. By 11.3.8, the F q-structures on P and V restrict to X-invariantF q-structures on Wi, which agree on W 3. Thus we may regard Wi as an F qX-
module.LEMMA 11.3.9. The map c: W1 x W 2 -+ W 3 defined by c(w, w') := [w, w'] is
X-invariant and F q-bilinear.PROOF. Since x acts on wi, c is X-invariant and F2-bilinear. Pick generators
Wi for wi as an F q-space with [w1, w2] = W3. Using the F q-structure on P, we may
write X2 = {x(>.): >. E Ff} so that x(>.)w2 = >.w2. Next [V,X2]::::; [V,L2]::::; Vi=
P n V from the action of Lon V, so as W 1 is X-invariant, [W 1 ,X 2 ] = 1. As W 1centralizes X2, it acts on the >.-eigenspace of x(>.) on P; then as W 2 is contained
in that eigenspace, so is W3 = [W1, W2]-and hence x(>.)w3 = Aw3. Thus
Aw3 = x(>.)w3 = [x(>.)w1, x(>.)w2] = [w1, >.w2],and hence c is linear in its second variaqle. Similarly X1 centralizes W 2 , since W 2
covers a Sylow 2-group of Lif 02 (£ 1 ), so W 1 W 3 is an eigenspace for each member
of Xf on V, and the same argument shows c is linear in its first variable. DWe are now in a position to obtain a contradiction, and hence finally eliminate
the shadow of £ 4 (q). Let y be a generator for X n K = D. Then y has twoeigenspaces on P: P n V = V 2 and (Wf^2 ). Let ).. be the eigenvalue on the second
space; then as y is of determinant 1 on P, y has eigenvalue ).. -^1 on P n V. Similarly
y is of determinant 1 on V, so the eigenvalue for yon W 1 is >.^2. Then by 11.3.9,
the eigenvalue for y on W 3 is the product >.^2 ).. = >.^3 of its eigenvalues on W 1 and
W 2. This is impossible, as W 3 = [W1, W 2 ] ::::; [V, P] = P n V and the eigenvalue for
y on P n V is >.-^1. This contradiction completes the proof of Proposition 11.3.2.
11.4. Eliminating the remaining shadows
Recall from earlier discussion that the shadows other than Sp 6 (q) and nt(q).2
with L ~ SL3(q), have been eliminated. In these remaining shadows, the central-
izer of a 2-central involution is not quasithin, and we essentially eliminate those
configurations in 11.4.4 in this section.
LEMMA 11.4.1. (1) Cv(L) = 1. In particular, Lis transitive on V#.
(2) If L1 <KE C(Ca(Vi), then K is described in case (1) or (2) of 11.1.2.