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 $ { \operatorname{ Exp}}_{ \rho}(G)$, i.e., $ { \operatorname{ Exp}}_{ \rho}(G)=\{ \alpha_1,\alpha_2, \cdots,\alpha_k\}$. In this paper, we give a new characterization for some groups by $ { \operatorname{ Exp}}_{ \rho}(G)$. We prove that the groups ${ \rm PSL}(2, 5) \times \mathbb{ Z}_p$, ${ \rm PSL}(2, 7)$ and ${ \rm PSL}(2, 11)$ are uniquely determined by $ { \operatorname{ Exp}}_{ \rho}(G)$. Furthermore, we prove that the groups ${ \rm PSL}(2, 5)$ and ${ \rm PSL}(2, 13)$ are uniquely determined by the parameters $ { \operatorname{ Exp}}_{ \rho}(G)$ and $|G|$. Additionally, we prove that if ${ \operatorname{ Exp}}_{ \rho}(G) = { \operatorname{ Exp}}_{ \rho}(\mathbb{ Z}_{ 2qr})$, then $G \cong { \rm PSL}(2, 5)$ or $G \cong \mathbb{ Z}_{ 2qr}$, where $q$ and $r$ are distinct odd prime numbers.
Categories: math.GR