On finitely generated profinite groups, II: products in quasisimple groups
نویسنده
چکیده
We prove two results. (1) There is an absolute constant D such that for any finite quasisimple group S, given 2D arbitrary automorphisms of S, every element of S is equal to a product of D ‘twisted commutators’ defined by the given automorphisms. (2) Given a natural number q, there exist C = C(q) and M = M(q) such that: if S is a finite quasisimple group with |S/Z(S)| > C, βj (j = 1, . . . ,M) are any automorphisms of S, and qj (j = 1, . . . ,M) are any divisors of q, then there exist inner automorphisms αj of S such that S = ∏M 1 [S, (αjβj) qj ]. These results, which rely on the Classification of finite simple groups, are needed to complete the proofs of the main theorems of Part I.
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