A New Characterization of PSL($2, q$) for Some $q$

Abstract

Let $G$ be a finite group and let $π_e (G)$ be the set of orders of elements from $G$. Let $k ∈ π_e (G)$ and let $m_k$ be the number of elements of order $k$ in $G$. We set nse $(G) := \{m_k | k ∈ π_e (G)\}$. It is proved that PSL($2, q$) are uniquely determined by nse (PSL($2, q$)), where $q ∈ \{5, 7, 8, 9, 11, 13\}$. As the main result of the paper, we prove that if $G$ is a group such that nse $(G) = nse (PSL(2, q))$, where $q ∈ {16, 17, 19, 23}$, then $G ≅ PSL(2, q)$.