896 13. MID-SIZE GROUPS OVER Fzthen as r(G, V) > 3 by 13.3.17.2, Ca(B) S Nc(V^9 ), so that Cv(B) = U. Thus
A E ~-k(T, V), so by B.4.6.9, k = 2 and A E .Af. Then without loss, A= A1, so
from the action of A 1 on V,Yr;.= (Cv(a) : a E .Af) s U,
and hence Yr; = U as U < V. As k = 2, W 1 (T, V) centralizes V. Therefore
as m(V/U) = 1 and U = Nv(V9), U centralizes V^9. Then since r(G, V) > 3,
V9 S Ca(U) S Na(V), so that V9 =A, contrary to k = 2. This proves (2). Finally
by (2) and 13.3.17.2, min{w(G, V),r(G, V)} :'.": 3, so E.3.35.1 implies (3). D
LEMMA 13.3.20. n(H) S 2 for each HE 1-t*(T, M) n G1.
PROOF. By 13.3.17.1, G 1 n G 3 s M, so hypothesis (c) of 12.2.11 is satisfied.
Therefore as HS G 1 , we may apply 12.2.11 with V 1 in the role of "U" to concludethat n(H) S 2. D
We can now complete the proof of Theorem 13.3.16: Recall T S G1, and
G1 f:_ M by 13.3.6, so there exists H E 7-t* (T, M) n G1. Then n(H) S 2 by 13.3.20,
so that HS M by 13.3.19.3, for our final contradiction.13.4. The treatment of the 5-dimensional module for A 6
In section 13.4 we prove:
THEOREM 13.4.1. Assume Hypothesis 13.3.1 with Cv(L) =I- 1. Then G ~
8p5(2).Set Zv := Cv(L). By hypothesis, Zv =I-1, so by 13.3.2.3,
Vis a 5-dimensional module for L/CL(V) ~ A5.
Recall this means that V is the core of the permutation module for A 6 acting on
D := {1, ... , 6}. Accordingly we adopt the notational conventions of section B.3.
We also adopt the conventions of Notations 12.2.5 and 13.2.1.
Of course the parabolic of the target group 8p 6 (2) stabilizing a point in the
natural module has this structure. Eventually we identify G with 8p 6 (2) duringthe proof of Proposition 13.4.9. We begin that process by setting up some notation
to discuss 8p5(2).
Let G = 8p5(2), T E 8yl2(G), and Pi, 1 S i S 3, the maximal parabolics
of G over T stabilizing an i-dimensional subspace of the natural module for G.
The pair (G, {A, P2, F3}) is a C 3 -system in the sense of section I.5. Notice L ~
A5 ~ P{/02(P1). We will produce a corresponding C 3 -system for G, and then useTheorem I.5.1 to conclude that G ~ Sp 6 (2). To do so, we will need to study the
centralizer Gz of a suitable involution z E V 1 - Zv, and show Gz/0 2 (Gz) ~ 83 x
83 ~ P2/02(P2). We must also construct a third 2-local Ho and show H 0 /0 2 (H 0 ) ~
£3(2) ~ F3/02(F3). Then it is not difficult to construct our C3-system.13.4.1. Preliminary results on the structure of certain 2-local sub-
groups. As usual Z = D 1 (Z(T)) from Notation 12.2.5. Notice Zv S ZL := Cz(L).
Recall that Zv is of order 2 and is of index 2 in V 1 =Zn V by 13.3.4.1.