On the distribution of the elements of a finite group generated by covers

Abstract

Let G be a finite group and let A be a finite sequence of subsets of G. We consider covers for the group G of the type A ^k, where A ^k is the concatenation of k copies of A. We show that the distribution of the elements of G generated by A ^k approaches the uniform distribution as k → ∞ (in the ℓ _∞-norm). If G = PSL(2, p) and ${{\bf A}=( \langle \alpha \rangle, \langle \beta \rangle )}$ , where α and β are two non-commuting generators of order p, we provide the exact distribution of the elements generated by A ^k.