Random generation of finite and profinite groups and group enumeration

• 成果类型: Article
• 署名作者: Jaikin-Zapirain, Andrei; Pyber, Laszlo
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.173.2.4
• 发表日期: 2011
• 页码: 769-814
• 关键词: maximal-subgroups; permutation-groups; random elements; probability; GROWTH; index

摘要

We obtain a surprisingly explicit formula for the number of random elements needed to generate a finite d-generator group with high probability. As a corollary we prove that if G is a d-generated linear group of dimension n then cd + log n random generators suffice. Changing perspective we investigate profinite groups F which can be generated by a bounded number of elements with positive probability. In responses to a question of Shalev we characterize such groups in terms of certain finite quotients with a transparent structure. As a consequence we settle several problems of Lucchini, Lubotzky, Mann and Segal. As a byproduct of our techniques we obtain that the number of r-relator groups of order n is at most n(cr) as conjectured by Mann.