On finitely generated profinite groups, II: products in quasisimple groups

• 成果类型: Article
• 署名作者: Nikolov, Nikolay; Segal, Dan
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2007.165.239
• 发表日期: 2007
• 页码: 239-273
• 关键词: powers

摘要

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 vertical bar S/ Z (S) vertical bar > C, beta j (j = 1,..., M) are any automorphisms of S, and q(j) (j = 1,..., M) are any divisors of q, then there exist inner automorphisms alpha(j) of S such that S = Pi F-1(M)[S, (a(j)beta(j))(qj)]. These results, which rely on the classification of finite simple groups, are needed to complete the proofs of the main theorems of Part 1.