Generators and commutators in finite groups; abstract quotients of compact groups

• 成果类型: Article
• 署名作者: Nikolov, Nikolay; Segal, Dan
• 署名单位: University of Oxford; Imperial College London
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-012-0383-6
• 发表日期: 2012
• 页码: 513-602
• 关键词: quasisimple groups; profinite groups; chevalley-groups; PRODUCTS; ELEMENTS; POWERS; bounds

摘要

The first part of the paper establishes results about products of commutators in a d-generator finite group G, for example: if HaS(2)G=aOE (c) g (1),aEuro broken vertical bar,g (r) > then every element of the subgroup [H,G] is a product of f(r) factors of the form with . Under certain conditions on H, a similar conclusion holds with the significantly weaker hypothesis that G=HaOE (c) g (1),aEuro broken vertical bar,g (r) >, where f(r) is replaced by f (1)(d,r). The results are applied in the second part of the paper to the study of normal subgroups in finitely generated profinite groups, and in more general compact groups. Results include the characterization of (topologically) finitely generated compact groups which have a countably infinite image, and of those which have a virtually dense normal subgroup of infinite index. As a corollary it is deduced that a compact group cannot have a finitely generated infinite abstract quotient.