8.3. ELIMINATING La(2) I 2 ON 9LEMMA 8.3.13. If V9 n v n z^0 -:/:-0, then V9 ::::; Ca(V).
PROOF. This is a consequence of 8.3.12 and 3.2.10.6.
LEMMA 8.3.14. W2(8, V)::::; Ca(V).727D
PROOF. If not, there is V9 with m(V9 /A) = 2 and A := V9 n M satisfies
1 -:/:-A ::::; S. As A 1: Ca(V) we may assume without loss that A 1: Co(V1)-so
A has non-trivial projection A 1 on 81. If A 1 = 81 , then for any hyperplane B
of A1, A is non-trivial on the proper subspace Cv1 (B) of V1. On the other hand,
if A 1 is of rank 1, the same is true for the hyperplane B := 82 n A of A with
Cv 1 (B) = V1. Since vt ~ z^0 , without loss we may assume z E [Cv 1 (B),A]. By
construction, m(V^9 / B) = 3, so as r > 3 by 8.3.11, Cv 1 (B) ::::; Na(V^9 ). Therefore
z E [Cv 1 (B), A] ::::; V n V9. But now 8.3.13 says V9 ::::; C(V), contrary to our choice
of V9. This establishes the lemma. D
Now we are in a position to complete the proof of Theorem 8.3.1. Recall
8 = 02 (XT), so from the embedding of X in Kz in 8.3.10, 8 is Sylow in Kz8
and n(Kz) = 2. From 8.3.14 and E.3.16.3, Co(C2(8, V)) ::::; M; and from 8.3.5,
Na(W 0 (8, V)) ::::; M. Therefore ass= 3 by 8.3.2, E.3.19 says Kz ::::; M, contradict-
ing 8.3.10.