gi2 i3. MID-SIZE GROUPS OVER F2
PROOF. Assume [V, Vt] # 1. By hypothesis Vt S Rz and V S Rg, so Vt
and V normalize each other. Let Hg* := Hg /0 2 (Hg). By 13.4.14, case (i) or (ii)
of 13.4.7.1 holds. Now 1 =f. V S Rg. But in case (i), as Rg centralizes Lg,
Hg ~ 83 x 83 , V is of order 2, and [Vt, V] S [Vt, Kg] = [Z,K2]^9. Similarly
in case (ii), V S Rg ~ E 4 , so V = R^9 if IVI > 2; further by 13.4.7.1, Hg
contains no transvections on Vt. ·
Suppose first that IV* I = 2. Then Vt induces a nontrivial group of transvec-
tions on V with axis Cv(Vt), so as V is a 5-dimensional module for L ~ A5, it
follows that [V, Vt] = (v) with v of weight 2 in V. Conjugating in L, we may assume
v E Vi. Further 2 = Wt I = !Vt/ Cvci' (V) I· Hence V is a group of transvections
on Vt with center (v), so Hg~ 83 x 83 and v E [Vt, V] S [Z,K2]^9 by paragraph
on,e. But by 13.4.14.1, Zv S [Z, K 2 ], so (v) is conjugate in G to Zv of weight 6,
contradicting 13.2.2.5 since v has weight 2.
Therefore IVl > 2, so by paragraph one, H8 ~ L3(2) and V = R^9 is
of order 4. From the action of Ho on Vo, [V, Vt] = Cvt (V) and Vt /Cvt (V)
are of rank 3: this is clear if Vo is semisimple, and it follows from H.5.2 if Vo
is the core of the permut~tion module. Hence Vt = R2 and m(R2) = 3. As
m(R2) = 3, Zv S [V, Rz] = [V, Vt]= Cv(Vt). Then Zv is weakly closed in [V, Vt]
by 13.2.2.5. Also (L 2 T)9 acts on [Rg, Vt]= [V, Vt], and then also on the subgroup
Zv weakly closed in [V, Vt], so (L 2 T)9 SM. Then T9 is conjugate to Tin M, so
as Na(T) S M by Theorem 3.3.1, g E M. Now as R 2 = Vt S flg = 02(L^9 T^9 ),
LzT = (L2T)^9. As M =LT by 13.4.11, L 2 T is maximal in M, so g E LzT S Ho,
so Ho= Hg. Thus Vt= Vo ::1 T, so as Cvt(V) = [V, Vt] S V n Vt,
[02(LT), Vo] S Cv 0 (V) S V.
Therefore [0 2 (LT), L] = V and L is an A 6 -block. Set Ko := 02 (H 0 ); similarly
[02(Ho), VJ S Vo and then [02(Ho), Ko] =Vo.
If Vo = Ui EB U2 is the sum of non-isomorphic 3-dimensional modules for Ko, we
saw that Zv S U := Ui for i := 1 or 2 during the proof of 13.4.14.1. If instead Vo is
the core of the permutation module and Zv S 8oc(V 0 ), set U := 8oc(V 0 ). In either
of these two cases, since V* = R2 and Lz centralizes Zv, [U, VJ = Cu(L2) = Zv =
Cu(V), so U induces a 4-group oftransvections on V with center Zv, impossible as
CM(V/Zv) = CM(V) by 13.4.2.4. Therefore we are in case (i) of 13.4.14.2, where
Vo is the core of the permutation module and Zv i. 8oc(Vo); so by that result, z is
of weight 4 in V.
As L is an A 6 -block, Li has two noncentral 2-chief factors, so K is an L 3 (2)-
block or an A5-block using 13.4.13.1. Further as z is of weight 4 in V, (z) = [V n
02(Li), 02(Li)J, so that z E VH. Therefore since z E Z(H), VH is the 5-dimensional
module for the A 6 -block K. By 13.4.12.1, Z = Zv(z) is of order 4, and by symmetry
between L and K, Z S VH and Zn 02 (Li) i. Z(K) n VH = (z); so as z E Li,
Z S Li. Calculating in the A 6 -block K, [Z(K)[ S 4 and 02 (Li/Z(K)) ~ Q~, so
IZ(02(Li)/(z))I S 4. Therefore as [Vn0 2 (Li),02(Li)] = (z) and jVn02(Li)[ =
8IZv n 02(Li)l = 16, we have a contradiction. D
LEMMA 13.4.16. If g E G with Vt S Rz and Vo S Rg, then [Vo, Vt] = 1.
PROOF. Assume Vt is a counterexample, and let H 0 := Ho/CH 0 (Vo). Inter-
changing Vo and V^9 if necessary, we may assume that m(Vt*) 2:: m(Vo/Cv 0 (Vt)),
so Vo is an FF-module for H 0. The modules Vo in case (ii) of 13.4.7.1 are not