1102 15. THE CASE .Cf(G, T) =^0PROOF. By 15.1.24.2, Lis not an L 3 (2)-block, while in all other cases of 15.1.22,
m 3 (L) = 2; then we obtain (1) from A.3.18.
Let G]' := Gi/C 01 (L/0 2 (L)). By (1), Y :S L, so Y = 0
31(L n M). By
· 15.1.17, J(S) = J(T) and No(J(S)):::; M. By 15.1.9.1, J(T) centralizes V, so by a
Frattini Argument, Y = Cy(V)Y 0 , where Yo:= 02 (Ny(J(S))). Now Cy(V) :S Mc
by 15.1.5.2, but we saw Y 1:. Mz, so Yo i=- 1. However if L ~ Mu, then as
Yo ::=; 02 (N 01 (J(S))), Yo centralizes Las case (6) of 15.1.22 holds, contradicting
1 i=-Yo :::; L. Hence (2) is established. ·Next recall by 15.1.24.1, that we are in case (4) or (6) of 15.1.7, so that IT:
SI = 2 from our construction in 15.1.16; thus (6) holds.
We turn to the proof of (3). Lett ET - S; then V{ =Vi since we are in case(4) or (6) of 15.1.7. By 15.1.24.3, Vi = [Vi, Y], and so Vi = [Vi, yt]. Further a
Sylow 3-group of L, and hence also of Y, is of exponent 3, so there is X of order 3 in
yt faithful on Vi. However if Vi centralizes L, then as Z1:::; Vi, L = 031 (Co(Vi)),
while X 1:. Las Xis faithful on V 1. This is a contradiction as L = 031 (Na(V 1 )) by
A.3.18, so (3) is established.
Part (4) follows from the action of Mon Vin cases (4) and (6) of 15.1.7; use
15.1.23 in case (6).
Finally suppose L is an A 7 -block. Represent LS on Q := {1, ... , 7}, and
adopt the notation of section B.3. By 15.1.22, Y*S* contains the stabilizer P* ofthe partition {{1, 2}, {3, 4}, {5, 6}, {7}}. Let P be the preimage of P* and Y1 :=
02 (P). By 15.1.24.4, V 1 :::; Cs(Y 1 ), and from the representation of LS on U(L),
Cs(Y1):::; Cs(L)(u,s), where u := e(), e := {1, ... ,6}, ands*:= (1,2)(3,4)(5,6).
Therefore as s* does not induce a transvection on U(L), we conclude from (4)that V1 :S CLs(U(L)) = 02(LS). So as V1 :S Cs(L)(u, s), Vi :S Cs(L)(u), so V1
centralizes K := L7 with K/02(K) ~ A5. As Z1 :::; Vi, K = 0
31
(Co(Vi)) by(1), and then it follows from A.3.18 that 031 (No(Vi)) = K :::; Ca(Vi). This is a
contradiction, as the subgroup X of order 3 defined earlier acts nontrivially on Vi.
Hence the proof of (5) and of 15.1.25 is complete. DLEMMA 15.1.26. F*(L) = 02 (L) and L/0 2 (L) ~ L4(2) or L 5 (2).
PROOF. Assume otherwise. In cases (2), (4), (5), and (7) of15.l.22, Cs(0^31 (Mn
L)) = Cs(L). Thus by 15.1.24.4, Vi :::; Cs(Y) :::; Cs(L), contrary to 15.1.25.3. Fur-
ther cases (1) and (6) of 15.1.22 were eliminated in parts (5) and (2) of 15.1.25,
leaving only case (3) of 15.1.22, where the lemma holds. DBy 15.1.26, F*(LS) = 02 (LS). Let U := (Zf) and UL:= [U,L]. By 15.1.24.1,
case (4) or (6) of 15.1.7 holds, where by construction in 15.1.16, Z is a full diagonalsubgroup of Z1 E9 Z2, so Ca 1 (Z) = Ca 1 (Z2) and S = Cr(Z1) = Cr(Z2). Thus
U E R2(LS) by B.2.14. Set (LS)* := LS/CLs(U), and recall CLs(U) = 02(LS)
since L/02(L) is simple and U E R2(LS). From 15.1.20.2, L = [L, J(S)], so that U
is an FF-module for L*S*; then we conclude from Theorem B.5.1 and B.4.2 using
I.1.6 that:LEMMA 15.1.27. One of the following holds:
(1) UL is the orthogonal module or its 7-dimensional cover for L* ~ L4(2).
(2) UL is a JO-dimensional irreducible for L* ~ L5(2).