Rational conjugacy classes and rational characters for some finite simple groups

Abstract

If $G$ is a finite group, an irreducible complex-valued character $\ddot{I}‡$ is called rational if $\ddot{I}‡(g)$ is rational for all $g\in G$. Also, a conjugacy class $x^G$ is called rational, if for all irreducible complex-valued character $\ddot{I}‡$, the value $\ddot{I}‡(x^G)$ is rational. We prove that for $q$, a power of prime, the group ${\mathrm{PSL}}_2(q)$ has same number of rational characters and rational conjugacy classes. Furthermore, we verify that this equality holds for all finite simple groups whose character tables appear in the $\text{ATLAS of Finite Groups}$, except for the Tits group.

Preliminary version; 14 pages; 1 table; comments welcome