Recognition by the set of exponents in the prime factorization of the product of element orders

Abstract

Let $G$ be a finite group. Let $\rho (G)={\prod }_{g\in G}o(g)={p_1}^{{\alpha }_1}{p_2}^{{\alpha }_2}\cdots {p_k}^{{\alpha }_k}$, where $p_1,p_2,\cdots ,p_k$ are distinct prime numbers and $o(g)$ denotes the order of $g\in G$. The set of exponents in the prime factorization of the product of element orders is denoted by ${\mathrm{Exp}}_{\rho }(G)$, i.e., ${\mathrm{Exp}}_{\rho }(G)=\{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_k\}$. In this paper, we give a new characterization for some groups by ${\mathrm{Exp}}_{\rho }(G)$. We prove that the groups $\mathrm{PSL}(2,5)\times {\mathbb{Z}}_p$, $\mathrm{PSL}(2,7)$ and $\mathrm{PSL}(2,11)$ are uniquely determined by ${\mathrm{Exp}}_{\rho }(G)$. Furthermore, we prove that the groups $\mathrm{PSL}(2,5)$ and $\mathrm{PSL}(2,13)$ are uniquely determined by the parameters ${\mathrm{Exp}}_{\rho }(G)$ and $|G|$. Additionally, we prove that if ${\mathrm{Exp}}_{\rho }(G)={\mathrm{Exp}}_{\rho }({\mathbb{Z}}_{2qr})$, then $G\cong \mathrm{PSL}(2,5)$ or $G\cong {\mathbb{Z}}_{2qr}$, where $q$ and $r$ are distinct odd prime numbers.