802 12. LARGER GROUPS OVER F2 IN .Cj(G, T)the noncentral chief factors for K on Vz are A 5 -modules.^1 Therefore case (ii)
of 12.2.7.3 occurs, so Y ~ A 7 , and each J E Irr +(Y, Vy, T) is T-invariant and of
rank 4 or 6. Again using the fact that the noncentral chief factors for K on Vz
are L 2 (4)-modules, we conclude that J is of rank 4 and the natural module fork.
Therefore [J, T n K] = [J, w] ~ E4 for each w EV - 02(K). Now [J, w] ~ V, andfrom the action of A 7 on J,
Y = (Cy(v): v EV n J#) ~ M,
whereas K ::::; Y with K 1:. M. This contradiction completes the proof of 12.2.19.
D
By 12.2.19, His solvable, so we may apply B.6.8.2 to conclude that
H=PT,where P is a p-group for some odd prime p, P = F(H*), and (P) = P n M.
Thus [P*, V*] f=. 1 by 12.2.18, and hence P* = [P*, V*], since V ::::! T and T* is
irreducible on P* /<I>(P*) by B.6.8.2. By Coprime Action, we may pick h E P-<I>(P)
so that Cv· (h*) is a hyperplane of V*. Let I:= (V, Vh) and U := 02(I). By I.6.2.2,
U = (V n U) x (Vh n U) is a sum of natural modules of I /U £=! D 2 p. Thus Vh n U
is of rank m(V) -1 ~ 2, and induces the full group of transvections in GL(V) with
axis V n U. Therefore we may apply the dual of G.3.1 to the action of LT on V,
to restrict the cases in 12.2.2.3 to:
LEMMA 12.2.20. L = GL(V) ~ Ln(2) for n = 3, 4, or 5, V is the natural mod-
ule for L, Un V is a hyperplane of V, and U induces the full group of transvections
with axis Un V on V. In particular as T::::; Na(V), T::::; LCr(V).Observe that we are beginnning to show that G has a 2-local structure similar
to that of one of the groups in conclusions (2)-(4) of Theorem 12.2.13.
LEMMA 12.2.21. U::::; 02 (H), so V is of order 2 and inverts P /(P*).
PROOF. We saw U = [U, 02 (!)], so if U £. 02 (H), then U = [U, 02 (!*)] f=. 1,
impossible as H is 2-nilpotent. We also saw that P = [P, V], so as V n U is a
hyperplane of U by 12.2.20, V* is of order 2, and V* inverts P* /<I>(P*). D
Next we obtain some restrictions on the structure of Hand its action on ((Un
V)H).We observed earlier for each h E P - (P) = P - M that I* ~ D 2 p, so that
h has order p. Then by A.1.24, P ~ Zp, Ep2, or p1+^2. Let v E V - U. By the
Baer-Suzuki Theorem, v inverts an element h' of order p in K; replacing P by
a Sylow group containing h', we may take h' E P. Then as V* is of order 2 by
12.2.21, we may take h' = h in the definition of I. Let W := ((Un V)H). Then
W ::::; 02 (H) by 12.2.21, and U = (V n U)(Vh n U) ::::; ((Un V)P) ::::; W. Indeed
as Tacts on Un v = 02(H) n v, (UnV)H = (UnV)TP =(Un V)P = uP, so
W =(UP). In particular as U::::; [W,0^2 (!)]::::; [W,P], W =(UP)::::; [W,P], and
hence W = [W, P]. Now 1 f=. UnV = 02(H)nV ::::! 02 (H), so UnV commutes with
its H-conjugates by I.6.2.2, and hence W is elementary abelian. From the action
of I on U in I.6.2.2a, [U, v] =[Un Vh, v] =Un V. Also [W, v]::::; W n V =Un V,
so W = (Un Vh) EB Cw(v). Similady W =(Un V) EB Cw(vh), so W = U EB Cw(I)(^1) This is where Hypoth~sis 12.2.3 eliminates M22 and M23 as possible conclusions in Theorem
12.2.13.