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By a), Op(m) ~ C"(ß(Op( m)), where (fj = (fj/op' (m). d. 6 Theorem. Suppose that W is a positive characteristic plunctor. Q of mwhich 55 § 7. o). Then W controls transfer in (ß. Proof Suppose that this is false and that (ß is a counterexample of minimal order. Let 6 E Sp((ß), 91 = N»(W(6)). 1, 6 n (ß' #6 n 91', since W does not control transfer in (ß. We obtain a contradiction in a number of steps. (1) W controls transfer in every proper section of (ß. For every proper section of (ß satisfies the hypothesis of the theorem and is of smaller order than (ß.

40 X. Local Finite Group Theory Thus 6 n (fj' n Since ~ ::; 3 ::; [~, (fj] 6'. 3, it follows that ~ n (fj' ::; ~ n [~, (fj] 6' = [~, (fj] (~ n 6'), whence the assertion. d. Next we prove the important "focal subgroup" theorem. 2 Theorem (D. G. HIGMAN[l]). 1/6 E Sp(fj), 6 n (fj' = <[lxgIX E 6, g E (fj, x g E 6). Proof Let X = {x-1xglx E 6, gE (fj, x g E 6}. Since x-1x g = [x, g], X s; (fj'; hence

Thus 6 n N(fj(~)' N(fj(W(N;;(~)))'. d. 1 (viii). 4 Lemma. Suppose that ~, Sl are subgroups of the finite group (fj such that (fj = ~Sl and ~ n Sl ~ 6 E Sp(fj). a) 1f A s;;; 6, gE (fj and Ag s;;; 6, there exist u E ~, V E Sl such that g = uv and AU s;;; 6. b) 1f '" is an equivalence relation on 9"(6) and '" does not contain fusion in (fj, then either '" does not contain fusion in ~ or '" does not contain fusion in R Proof a) Since (fj = ~Sl, there exist x E~, Y E Sl such that g = xy. It folIo ws from Ag s;;; 6 that AX s;;; 6 y - 1 • Thus A X s;;; ~x n Sly-l = ~ n R Since A S;;; 6,

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