Uniformly semi-rational simple groups

A finite group \(G\) is called uniformly semi-rational if there exists an integer \(r\) such that the generators of every cyclic sugroup \(\langle x \rangle\) of \(G\) lie in at most two conjugacy classes, namely \(x^G\) or \((x^r)^G\). In this paper, we provide a classification of uniformly semi-rational non-abelian simple groups with particular focus on alternating groups.