12.2. GROUPS OVER F2, AND THE CASE V A TI-SET IN G 803with U = [W,J] = [W, h]. Further [W, V] = [U, V] = V n U = V n Wis of rank
n-1.
We next claim that we may choose P invariant under v; and when Pis non-
abelian, that <I>(P) ~ NHnM(V). If P ~ Zp, then v acts on P and we are done.
Suppose P ~ EP2. Then (P,v) ~ N := NH((h)). Set N := N/(h). As v* inverts
P*, v t/. 02(N), so we may apply the Baer-Suzuki Theorem again in N, to conclude
that v inverts y of order pin CH(h) - (h). Thus (h, y) E Sylp(H) is v-invariant,and choosing P := (h, y), we are done in this case also. Finally, suppose P ~ p^1 +2.
Then <I>(P) = Z(P) centralizes h and so acts on [W, h] = U. Further 02 ,w(H) =02(H)(P) and V centralizes 02,w(H)/02(H), so that [(P), VJ ~ 02 (H). Thus
<I>(P) acts on 1 =f. Cu(02(H)V) ~ Un V, and so since V is a TI-set in M by
12.2.6, <I>(P) acts on V, establishing the final assertion of the claim. Next <I>(P)centralizes V/(U n V) of rank 1, and hence centralizes some v 0 E V - U, which
we may take to be v. Set P 1 := <I>(P)(h), H1 := NH(P 1 ), and H 1 := Hif P 1. We
apply the Baer-Suzuki Theorem one more time to H1: As v* inverts P* /<I>(P*),
v tj. 02(H 1 ), so v inverts an element k of order pin H1, and tl;ien the preimage of(k) is av-invariant Sylow p-group P of H. This completes the proof of the claim.
So in any event we may assume v acts on P. Thus V PW = ( v) PW is a
subgroup of H. Further [02(H), v] ~ Vn02(H) = VnU ~ W, so that v centralizes
02(H)/W. Then as P = [P, v], [0 2 (H), P] ~ W, and hence PW :::::! 02 (H)P.
Thus as K = 02 (H) ~ P0 2 (H), K ~ PW. We saw earlier that W = [W, P], so
W ~ 02 (PW) ~ 02 (H) = K, and hence K =PW. Summarizing:LEMMA 12.2.22. P ~ Zp, Ep2, or p1+^2 , and we may choose P so that P is
invariant under v EV - U, W = (UP) = [W, P] is elementary abelian, [W, V] =
V n Wis of rank n -1, K =PW, and <I>(P) ~ NHnM(V).LEMMA 12.2.23. P ~ Zp·PROOF. Assume Pis not Zp, and let Hp:= KV, <I>:= <I>(P), Wp := Cw(<I>),
and Hp:= Hp/CHp(Wp).
Suppose Wp =f. 1. By 12.2.22, W = [W, P], so as T* is irreducible on P* /<I>(P*),
<I> = C p (W p) and P ~ Ep2. Then by Generation by Centralizers of HyperplanesA.1.17, Wp is generated by nontrivial subgroups Wi := Cwp (Pi), where Pi runs
over a nonempty collection of subgroups of index pin P generating P. As v inverts
P/<P, v acts on each subgroup Pi and hence on each Wi; further as W = [W, P]
so that Cw(P) = 1, also Wi = [Wi, P], so that v is nontrivial on Wi. Thus
1 =f. [Wi, v] ~ WinV. Therefore as Vis a TI-set in G by 12.2.16, Pi~ Ca(WinV) ~
Na(V) ~ M, so H =KT= PT~ M, contrary to H f:o M.
Therefore Wp = 1, so P ~ p1+^2 and W = [W, ]. By 12.2.22, ~ NHnM(V),
so W n V = [W n V, <P] is of rank n - 1by12.2.22, and then m([V, ])= n - l. In
particular,. is faithful on V. Now ~ Mv = L = L/02(L) = GL(V) ~ Ln(2)
by 12.2.20. Therefore as T = T, p = 3 and 'f is a rank one parabolic of L.
However for any X of order 3 in a rank one parabolic, [V, X] is of rank 2; so as
m([V, ])= n - 1 we conclude n = 3.
As n = 3, Un Vis of rank 2. Now KV= WPV = (V, V"', VY) for x, y chosen