788 12. LARGER GROUPS OVER F2 IN .Cj(G, T)an F 2 L-module. Let T 1 := Nr(V 1 ), and for v EV, let Mv := CM(v), Gv := Ca(v),
and Lv := 02 (CL(v)).
By 3.2.5.3, V :SJ M; hence
M=Na(V)
asMEM.Recall that the module V is described in section H.9 of Volume I. We adopt
the notation of that section, including the description of the orbits Oi (1 :::; i :::; 3)
of Mon V#.
We now proceed to analyze our group G. Eventually we obtain a contradiction,
and hence show no such group exists.LEMMA 12.1.1. Let v E 03. Then Gv = Ca(v):::; M.
PROOF. First from lemma H.9.1.4, Q = 02(LvTv), where Tv := Cr(v) E
Syl2(Mv)· Now C(G, Q) :::; M by 1.4.1.1. Thus as Lv :SJ Mv, we conclude fromA.4.2.7 that Q E B 2 (Gv) and Q is Sylow in (QMv). Therefore Hypothesis C.2.3 is
satisfied by Gv, Mv, Qin the roles of "H, MH, R".
Let W := [V,Lv]· BylemmaH.9.1.4, Lv/02(Lv) ~ Ln-1(2) and W = W1E9W2
with W1 the natural module for Lv/02(Lv) and W2 its dual. Let z generate Cw(Tv),
and observe z is 2-central in G by H.9.1.2.
For each value of n we define a subgroup Kv E C(Gv) with Lv :::; Kv :SJ Gv: If
n = 4, then Wis not an FF-module for Lv by Theorem B.5.1.1, so J(Tv) = J(Q)by B.2.7. Then as C(G, Q):::; M, Na(Tv) :::; Mv and hence Tv E Syh(Gv)· So by
1.2.4, Lv :::; Kv E C(Gv), and as Tv acts on Lv, Kv :SJ Gv by 1.2.1.3. On the other
hand, if n = 5, then by 1.2.1.1, Lv projects nontrivially on some Kv E C(Gv), so
Kv has a section isomorphic to Lv/02(Lv) ~ L4(2). Therefore Kv = 031 (Gv) byA.3.18, so that again Lv :::; Kv :SJ Gv.
Suppose that there is a component K of Gv. Then Lv = 02 (Lv) acts on K
by 1.2.1.3. By A.1.6, 02(M) :::; Q :::; Gv, so Mv E He by 1.1.4.4; thus K 1:_ Mv.
Similarly G,, E He by 1.1.4.6, so Gv,z := Gv n G,, E He by 1.1.3.2; thus Ki Gv,z,
so K 1:_ G,,. Butz E W = [W,Lv]:::; Lv, so [K,Lv] =/= 1. Therefore [K,Kv] =/= l, and
hence K = Kv by 1.2.1.2. Then Cw(K) :::; Cw(Lv) = 1. Set G~ := Gv/Cav (K).
Then W* ~Was an L~-module, and L~ :SJ M;. But no group with such a 2-local
M; appears on the list of Theorem C (A.2.3).
This contradiction shows that E( Gv) = 1. Next W = [W, LvJ, so W centralizes
O(Gv) by A.1.26. Therefore O(Gv) :::; Gv,z, and hence O(Gv) = 1 as Gv,z E He.
Thus we have shown 02 (F*(Gv)) = 1, so that Gv E He.
We now assume that Gv i M, and derive a contradiction.
Suppose that Lv :SJ Gv and set Y := CaJLv/02(Lv)). Then as Aut(Lv/02(Lv))
is induced in LvTv, Gv = LvTvY, so Y 1: Mas Gv 1: M. Next embed Tv:::; XE
Syh(Gv); then Nx(Tv) normalizes Crv (Lv/02(Lv)) = Q, and so lies in Mv-hence
Tv = X E Syl2(Gv)· Thus Q = Tv n Y is Sylow in Y, so we conclude from the
C(G, T)-Theorem C.1.29 that there is a xo-block B of Y with B 1:_ M. If B
is an L2(2n)-block, then a Cartan subgroup D of B lying in B n M centralizes
Lv/02(Lv); hence D centralizes V, as M = NaL(V)(L) and CM(Lv) = 1. Thus
V :::; Crv (B n M) = Crv (B), and hence B :::; Ca(V) :::; M, contrary to the choice
of B. If B is an A 5 -block, then 02 (B n M) :::; 031 (Mv) = Lv by A.3.18, whereas
Z(Lv/02(Lv)) = 1. Thus Bis an A3-block. Notice since B centralizes Lv/02(Lv)