Thin systems of generators of groups

Abstract

A subset \(T\) of a group \(G\) with the identity \(e\) is called \(k\)-thin (\(k\in\mathbb{N}\)) if \(|A\cap gA|\leqslant k\), \(|A\cap Ag|\leqslant k\) for every  \(g\in G\), \(g\ne e\). 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 \(G=XX^{-1}\cup X^{-1}X\), and a 4-thin subset \(Y\) such that \(G=YY^{-1}\).