On semi-rational finite simple groups

An element of a group G is called semi-rational if all generators of ⟨x⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle x\rangle $$\end{document} lie in the union of two conjugacy classes of G. If all elements of G are semi-rational, then G is called a semi-rational group. In this paper, we determine all semi-rational simple groups. Our study in this article generalises Feit and Seitz’s result (Ill J Math 33(1):103–131, 1989) to semi-rational groups.