Characterization of $\PSL(2,q)$ by the number of singular elements

Abstract

Given a finite group $G$, let $\pi (G)$ denote the set of all primes that divide the order of $G$. For a prime $r\in \pi (G)$, we define $r$-singular elements as those elements of $G$ whose order is divisible by $r$. Denote by $S_r(G)$ the number of $r$-singluar elements of $G$. We denote the proportion $S_r(G)/|G|$ of $r$-singular elements in $G$ by ${\mu }_r(G)$. Let $\mu (G):=\{{\mu }_r(G)|r\in \pi (G)\}$ be the set of all proportions of $r$-singular elements for each prime $r$ in $\pi (G)$. In this paper, we prove that if a finite group $G$ has the same set $\mu (G)$ as the simple group $\PSL(2,q)$, then $G$ is isomorphic to $\PSL(2,q)$.