Thin systems of generators of groups

Abstract

A subset T of a group G with the identity e is called k-thin (k∈N) if |A∩gA| ≤ k, |A∩Ag| ≤ k for every g∈G, g≠e. We show that every infinite group G can be generated by some 2-thin subset. Moreover, if G is either Abelian or a torsion group without elements of order 2, then there exists a 1-thin system of generators of G. For every infinite group G, there exist a 2-thin subset X such that G=XX⁻¹ ∪ X⁻¹X, and a 4-thin subset Y such that G=YY⁻¹.