The probability of generating some common families of finite groups

Abstract

Let G be a finite group. Define e(G) to be the expected number of elements of G which have to be drawn at random with replacement from G before a set of generators is found. Define λ_n(G) to be the probability that n elements drawn at random with replacement from G generate G. In this paper we discuss some general approaches to computing e(G) and λ_n(G). We apply these approaches to some common classes of finite groups, including the p-groups and the nilpotent groups.