If $G$ is a finite group, an irreducible complex-valued character $\chi$ is called rational if $\chi(g)$ is rational for all $g\in G$. Also, a conjugacy class $x^G$ is called rational, if for all irreducible complex-valued character $\chi$, the value $\chi(x^G)$ 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 $\textit{ ATLAS of Finite Groups}$, except for the Tits group.
Categories: math.GR math.RT