On sampling generating sets of finite groups and product replacement algorithm

Abstract

The problem of determining the structure of Nk(G) is of interest in several contexts. The counting problem goes back to Philip Hall, who expressed Nk(G) as a Mobius type summation of Nk(H) over all maximal subgroups H ⊂ G (see [23]). Recently the counting problem has been studied for large simple groups where remarkable progress has been made (see [25, 27]). In this paper we analyze Nk for solvable groups and products of simple groups. The sampling problem, while often used in theory as a tool for approximate counting, recently began a life of its own. In this paper we will present an algorithm for exact sampling in case when G is nilpotent. When little about structure of G is known, one can only hope for approximate sampling. In [11] Celler et al. proposed a product replacement Markov chain on Nk(G) which is conjectured to be rapidly mixing to a uniform stationary distribution. The subject was further investigated in [6, 12, 17, 16], while the conjecture is fully established only when G ' Zp, p is a prime. We prove rapid mixing for all abelian groups G. Also, we disprove the folklore conjecture that the group elements in generating k-tuples are (nearly) uniformly distributed. Finally, we would like to remark that the generating k-tuples occur in connection with the so-called random random walks, which are the ordinary random walks on G with random support. The analysis of these “average case” random walks was inspired by Aldous and Diaconis in [1] and was continued in a number of papers (see e.g. [19, 33, 36, 39]). We explain how the sampling problem can be used to test convergence of random random walks.