Lemma 3.2 (Kantor-Luks)(CFSG) For G = G/K, where K [??] G [less than or equal to] Sym([OMEGA]), given H,N [less than or equal to] G such that H normalizes N, one can find [C.sub.H](N) (including, in particular, Z(H)) in polynomial time.
Nevertheless, in [25, [section]4], with the help of Kantor's Sylow machinery and thus CFSG (Theorem 3.1(xi)), the problem was asserted to be in polynomial time; since the proof was omitted in  due to space limitations, we include it here.
Lemma 3.16 (CFSG) Problem 16 is solvable in time polynomial in [absolute value of [OMEGA]], n, log q and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].
Remark (i) If we hypothesize that G [member of] [[GAMMA].sub.d] in Problem 16, then the problem can be solved without invoking CFSG by using the method of Kantor and Taylor from , in place of Theorem 3.1(xi), to find Sylow subgroups.
For this, we appeal to the following fact from CFSG: For every nonabelian simple group T, we have [absolute value of Out(T)] = O(log [absolute value of T]) (see, e.g., ).
Corollary 3.19 (CFSG) Problem 24 is in polynomial time.
Lemma 3.20 (CFSG) Problem 25 is in polynomial time.
Lemma 3.21 (CFSG) Problem 26 is in polynomial time.
Corollary 3.24 (CFSG) Problem 29 is in polynomial time.
Corollary 3.25 (CFSG) Problem 30 is in polynomial time.
Lemma 3.26 (CFSG) Problem 31 is in polynomial time.
Lemma 3.27 (CFSG) Problem 32 is in polynomial time.