# On finitely generated profinite groups, II: products in quasisimple groups | Annals of Mathematics

Abstract 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=∏M1[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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### 数学年刊（Annals of Mathematics）

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