Random generation of finite and profinite groups and group enumeration

Abstract

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 response 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.

Locations

• Biblos-e Archivo (Universidad Autónoma de Madrid) - View - PDF
• Annals of Mathematics - View - PDF