13.3. STARTING MID-SIZED GROUPS OVER F 2 , AND ELIMINATING U 3 (3) 89S.iii := (r)Ao and Ao are the only such subgroups in case Mv ~ G 2 (2). Further
G.Mv (Vi) is Ao or Ai in the respective case.
By F.9.2, UH s ni(Z(QH)), 02(H*) = 1, <P(UH) s V1, and QH = GH(UH)·
Thus if V centralizes UH, then V S QH S kerH(NH(V)), and then for h E H,
[V, Vh] S V n Vh, so that [V, Vh] = 1 by 13.3.17.3. But then VH is abelian,
contrary to our assumption. Therefore [UH, VJ "/= 1, so that V* "/= 1, and as
<P(UH) S V1, 1 "/= UH is a normal elementary abelian subgroup of M1, so by
the previous paragraph, UH = Ao or Ai. In particular UH centralizes Vi, so
Vi S Z(UH) and thus UH= (Vl) is elementary abelian.
Let ZH := Z(QH). By 13.3.17.4, [QH, V2] i= l, so as L 1 is irreducible on Vs,
Vin ZH =Vi. Furthermore as V2 SUH, Qc := GqH(UH) is properly contained in
QH.For x E Qc, define cp(xQc) : UH/ZH---; Vi by cp(xQc) : uZH f--+ [x,u] for
u E UH. By F.9.7, cp is an H-equivariant isomorphism between QH/Qc and the
F2-dual space of UH/ZH.
As [V,Ao] = [V,A1] =Vi, [V*,UH] = V3. Then as VinZH = V1, V* isnontrivial on UH/ Z H and hence also on Q H / Q c as cp is an equivariant isomorphism.
But QH s TS Nc(V), so [QH/Qc, V] S Qc(V n QH)/Qc, and hence V n QH i
Qc.
Next Vs = Vl is a hyperplane of V, with [v, Ri]Vi/Vi = Vs/Vi for each
v E V - Vs. Thus as L1 is irreducible on Vs/Vi and V n QH i Qc 2: V3, we
conclude that Vs= V n QH and Vi= V n Qc = V n UH. Also by 13.3.15.4,
Vs S GqH(V2) S 02(G2),
so V = (V/^2 ) S 02(G2).
Now V is of order 2 and 02 ( H) = 1, so by the Baer-Suzuki Theorem we
can pick h EH so that for I:= (V, Vh), I ~ D 2 m with m > 1 odd. Then V is
conjugate to V*h in I, so we may assume h EI.
Suppose [Vi, Vt] "/= 1. Then as G.Mv (Vs) - S G.Mv (Vi), [Vs, Vs] h contains a
hyperplane of Vi containing V 1. As all such hyperplanes are fused under L1, we may
take V2 S [Vs, Vsh]. Now Vs= V n QH is normal in QH, and hence Vsh = Vh n QH
is normal in QH, so that V2 S Vs n Vsh S V n Vh S Z(I). Thus IS G2, impossible
as I is not a 2-group, while VS 02(G2) and h EI.
This contradiction shows that v;.,h centralizes Vi, and hence by symmetry, Vs Vsh
centralizes Vi V:f. In particular Vsh s Ai. Therefore [V, v;.,h] S Vi, and by symmetry
I centralizes Vs v;.,h /Vi V 3 h. Hence as h E I, Vs V 3 h = Vs (Vi V 3 h) = v;.,h (Vi V'3h) = Vsh Vi.
Therefore as Vi S V'3\ Vt = V 3 h is of rank m(V 3 h) S m(V'3h /Vi) S 2. Therefore as
r(G, V) > 3 by 13.3.17.2, Vs Ga(Gv;(V)) s Na(Vh), once again contradicting I
not a 2-group. This completes the proof of 13.3.18. D
LEMMA 13.3.19. (1) If g E G with V n V9 "/= 1, then [V, V9] = 1.
(2) W2(T, V) centralizes V.
(3) HS M for each H E H(T) with n(H) S 2.PROOF. Suppose 1 i= V n V9. As Lis transitive on V#, we may take z E V9
and g E G 1 by A.1.7.1. But then [V, V9] = 1by13.3.18.2. Thus (1) is established.
Let A := V9 n M s T with m(V^9 /A) =: k S 2, and suppose A i= 1. Let
U := Nv(V9). By (1), V n V9 = 1, so as [U,A] s V n V9, U s Gv(A). In
particular U < V as A "/= 1. On the other hand, if B S A with m(A/ B) < 4 - k,