886 13. MID-SIZE GROUPS OVER F2(3) The proper overgroups of 'f' in L'f' = Auta(L) are L1'f' and L2T-except
when L ~ A5, when only L1'f' occurs. In particular, all proper overgroups of T in
LT are {2, 3}-groups.
(4) Statements analogous to (1)-(3) hold for any K E C1(G, T) and VK E
Irr+(KT,R 2 (KT),T) in the roles of ''L, V".
PROOF. Part (1) follows from an inspection of the modules listed in 13.3.2.3.
Then (2) follows since V E Irr +(LT, Rz(LT)). Part (3) follows from the well-
known fact that the overgroups of T in an untwisted group of Lie type over Fz
are parabolics, and as Out(L) is a 2-group. Finally (4) follows since 13.3.2.3 also
applies to each K and V K. DAs usual in the FSU, by 3.3.2.4, we may apply the results of section B.6 to
members HE H*(T, M). Recall that for v EV#, Gv = Ca(v) in Notation 12.2.5.3.
LEMMA 13.3.5. (1) If L ~ L3(2) or U3(3) then Gv 1,. M for each v EV#.
(2) If L ~ A5 then H*(T, M) ~ Co(Z), so Gz 1:. M for z generating Zn V =(3) If L ~ A5, then Gv 1:. M for some v E V1 - Cv(L).
PROOF. As Hypothesis 13.3.1 excludes the groups in conclusions (2)-(4) of
Theorem 12.2.13, conclusion (1) of that result holds: namely Gv 1:. M for some
v EV#. Next Vis described in 13.3.2.3. In particular Cv(L) = 1 unless Vis a 5-
dimensional module for L ~ A5, and Lis transitive on (V/Cv(L))# unless L ~ A5·
Therefore (1) holds, and if L is A5, then Gv 1:. M for some v E V1. Further if
Cv(L) # 1, then Cv(L) ::=; Z(LT), so as M = !M(LT), (3) holds. Finally when
L ~ A5, H* (T, M) ~ Co(Z) by 13.2. 7. Then as Zn V = \Ii by 13.3.4.1, Gz 1:. M
for z generating Zn V, so (2) holds. DBy 13.3.5:LEMMA 13.3.6. Either G1 1:. M, or Cv(L) # 1 so that V is a 5-dimensional
module for L ~ A5.As usual we let B(X) denote the subgroup generated by all elements of order 3
in a group X.
LEMMA 13.3.7. Assume L ~ A 6. Then either(1) Ca(V) is a 3' -group, or
(2) L/02(L) ~ A5, m3(Co(V)) = 1, and Lo= B(Co(V)).
PROOF. Let D := B(Co(V)) and PE Syl3(Co(V)). Recall that we may apply
12.2.8; then B(M) = L so that D ::=; L, and hence either D = 1, or L/0 2 (L) ~ A 6
with D = B(CL(V)) =Lo. In the first case, conclusion (1) holds. In the second, as
01 (P) ::=; D = Lo, 01 (P) is of order 3, so conclusion (2) holds. Thus the lemma is
established. DLEMMA 13.3.8. Assume KE C1(G, T), let MK := Na(K), and assume HE
. H(T, MK) and Y = 02 (Y) :::;I H with Y :S:: MK. Then
(1) K 1,. YCM,,(K/02(K)).
(2) Y is a {2, 3}-group.