Uniformly semi-rational simple groups

TL;DR

This paper classifies non-abelian simple groups, especially alternating groups, that have a uniform semi-rational property related to the conjugacy classes of their cyclic subgroups' generators.

Contribution

It provides a classification of uniformly semi-rational non-abelian simple groups, focusing on the structure of their conjugacy classes, particularly in alternating groups.

Findings

Identifies conditions under which simple groups are uniformly semi-rational.

Classifies all such groups among non-abelian simple groups.

Highlights the special case of alternating groups.

Abstract

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.