Uniformly semi-rational simple groups

Abstract A finite group G is called uniformly semi-rational if there exists an integer r such that the generators of every cyclic subgroup $$\langle x \rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>x</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> of G lie in at most two conjugacy classes, namely $$x^G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>G</mml:mi> </mml:msup> </mml:math> or $$(x^r)^G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>r</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>G</mml:mi> </mml:msup> </mml:math> . In this paper, we provide a classification of uniformly semi-rational non-abelian simple groups with particular focus on alternating groups.