932 i3. MID-SIZE GROUPS OVER F2LEMMA 13.6.21. Ki. M.
PROOF. By 13.6.20, K* does not act on L~, so as Lv :::l Mv, 13.6.21 holds. DPROOF. Assume otherwise. Let Ko be minimal subject to Ko E £(KTv, Tv)
and K 0 = K*. Then K 0 /0 2 (K 0 ) is quasisimple by minimality of Ko, and Ko E Yz
as K 0 = K*, so replacing K by K 0 , we may assume K/02(K) is quasisimple.
If K+ is not Ar, then using 13.6.6 and 13.6.16.2, (KTv, Tv) is an MS-pair in
the sense of Definition C.1.31. So by C.1.32, either K is a block of type .,4.5, A5, or
G 2 (2), or K+ is L 3 (2) and by C.1.32.5, K is described in C.1.34. Similarly if K+ isAr, C.1.24 says K is an Ar-block or exceptional Ar-block. Set U := [Z(02(K)),K].
When K is a block, UK= U E Irr +(K, WK). If K is not a block, then K/02(K) ~
L3(2) and K is described in C.1.34, so U =UK is a sum of at most two isomorphic
natural modules for L3(2).
As Lv ~ K by 13.6.20, Lt is a T,l-invariant {2, 3}-subgroup of K+ with Sylow
3-group of order 3. Set P := Lv(Tv n K). When K+ is L3(2), Sp4.(2)', or G2(2)',p+ is a minimal parabolic.
Suppose first that K is an Ar-block. Then by B.3.2.4 and B.2.9.1, J(Tv)+
is the subgroup of T;j generated by.its three transpositions, and s+ = J(Tv)+
by B.2.20. Further NK+(S+) = NK(S)+ by a Frattini Argument, and NK(S) ~
Mv by 13.6.5. From the structure of Sr, NKTJS) is maximal in KTv subject to
containing a normal subgroup {2, 3}-subgroup which is not a 2-group, so it follows
that NKTJS) = (TvnK)Lv and Lv = 02 (NK(S)). Now LvTv ~ KiTv ~ KTv with
Ki/02(Ki) ~ A5, and as [z, Lv] =/:-1, [z, Ki] =/:-1. Further Kt = [Kt, J(Tv)+], soKi E Yz; thus replacing K by Ki, we may assume K is not an ordinary Ar-block.
Similarly if K is an .A. 6 -block, then U has the structure of a 3-dimensional
F 4K-module and J(Tv)+ is the 4-subgroup of T,j n K+ centralizing an F 4-line U 2
of u, sos+= J(Tv)+, NK(s)+ = NK(U2)+, and hence NK(U2) ~ Mv by 13.6.5.Let l := [V,Lv]· Then l is an LvTv-invariant line in [z,Lv] ~ [z,K] ~ U with
l = [Z, Lv]. It follows that if K/02(K) ~ L3(2), then m(U) =I-4, since in that
case no minimal parabolic of K+ acts on such a line (cf. B.4.8.2). If K is an
A5-block, then from the previous paragraph, NK(U 2 ) ~ Mv, so NK(U 2 ) acts on l,
a contradiction as NK(U 2 ) acts on no E 4 -subgroup of U.
Let 0 := U/Cu(K). In the remaining cases, if K is irreducible on 0, then there
is a unique Tv-invariant line in U, soi is that line. Then if K is not an exceptional
Ar-block, p+ is the parabolic stabilizing f, while if K is an exceptional Ar-block,
then Lt is one of the three Tv-invariant subgroups Lo = 02 (L 0 ) of NK(i) with
Lo/02(Lo) ~ Z3 and f = [Z, Lo]. If K is not irreducible on 0, then U is the sum
of two isomorphic modules for K+ ~ L 3 (2), i is a Tv-invariant line in one of those
irreducibles, and P = NK(l). For our purposes the important fact is that in each
case Cu(Lv) = 0, so Co(Lv) = Co(K).
Let ZK := Z(02(K)) and Zo the preimage in K of Z(02(K)). If K is a block,
then by definition U = [ZK, K] = [0 2 (K), K], so [Z 0 , K] = U. If K is not a block,
then from the description of K in C.1.34, again [Z 0 ,K] = U. So in any event
[Zo,K] =U.