916 13. MID-SIZE GROUPS OVER F2
PROOF. As in the proof of 13.4.5, this follows from 13.3.2, once we observe that
by 13.3.2, we may apply various results to K in the role of "L": Theorem 13.3.16
says K/02(K) is not U3(3). Then since Hypothesis 13.5.1 excludes G ~ Sp5(2),
we conclude from Theorem 13.4.1 that CvK(K) = 1 when K/0 2 ,z(K) ~ A5. D
13.5.1. SetHng up the case division on (V~^1 ) for A 5 and A 6.
REMARK 13.5.3. In the remainder of this section (and indeed in the remainder
of the chapter), in addition to assuming Hypothesis 13.5.1, we also assume L/02(L)
is not L3(2); that is, we restrict attention to the cases where L/0 2 (L) ~As, A5,
or A5.
Then by 13.5.2.3, Cv(L) = 1 and Vis the natural module for L/CL(V) ~An,
n = 5 or 6.
As usual we adopt the notational conventions of section B.3 and Notation
13.2.1. We view V as the quotient of the core of the permutation module for
L/CL(V) on D := {1, ... , n}, modulo (en). Recall from Notation 12.2.5.2 that
Mv := NM(V) and Mv := Mv/CM(V). So there is an Mv-invariant symplectic
form on V, and when n = 5, an invariant quadratic form. Thus we use terminology
(e.g., of isotropic or singular vectors) associated to those forms.
As in Notation 13.2.1, Vi is the T-invariant subspace of V of dimension i and
Gi := Na(V'.i).
LEMMA 13.5.4. WhenL/02(L) ~ A 6 or A5, seth := 02(G1)L2 or02(G1)L2,+,
respectively. Then:
(1) 12 = (02(G1)a^2 ) :::] G2.
(2) C1 2 (Vi) = 02(12) and 12/02(h) ~ 83.
(3) m3(Ca(Vi)) ~ 1.
(4) Ca(V3) ~ Mv. Hence [V, Ca(V3)] ~ V1.
(5) [02(G1), Vi] =I-1.
(6) 02 (12) = L2 or 02 (L2,+), respectively, 02 (1 2 ) = L 2 , and 02 (0^2 (1 2 )) is
nonabelian.
PROOF. The equalities in (6) follow from the definition of 12 and the fact that
L2 and L2,+ are T-invariant. Then as V = [V,L 2 ] and 02 (L 2 ) =f. 1, the remaining
statement in (6) follows. Hypothesis 13.3.13 is satisfied by 13.5.2.3, so (5) follows
from 13.3.14. Then, parts (1)-(4) of the lemma follow from 13.3.15. D
LEMMA 13.5.5. G 1 n G 3 ~ Mv.
PROOF. When n = 5, G3 ~ Mv by 13.2.3.2. When n = 6, AutL 1 r(V3) =
CaL(V 3 )(Vi), so as Ca(V3) ~ Mv by 13.5.4.4, and as LlT ~ Mv, G 1 n G 3 <
~. D
As Cv(L) = 1:
Vi= Zn Vis of order 2.
Let z be a generator for V 1. By 13.3.6,
G1 = Ca(z) i. M,
so G1 E Hz =I-0, where as usual
Hz:= {HE H(L1T): H ~ G1 and Hi. M}