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

Let G be a finite group. Let ρ(G) = ∏g ∈ G o(g)=p₁α₁ p₂α₂ ⋯ pₖαₖ, where p₁, p₂, ⋯, pₖ are distinct prime numbers and $o(g)$ denotes the order of g ∈ G. The set of exponents in the prime factorization of the product of element orders is denoted by Exp_ρ(G), i.e., Exp_ρ(G)=\α₁,α₂, ⋯,αₖ\. In this paper, we give a new characterization for some groups by Exp_ρ(G). We prove that the groups PSL(2, 5) × Zₚ, PSL(2, 7) and PSL(2, 11) are uniquely determined by Exp_ρ(G). Furthermore, we prove that the groups PSL(2, 5) and PSL(2, 13) are uniquely determined by the parameters Exp_ρ(G) and $|G|$. Additionally, we prove that if Exp_ρ(G) = Exp_ρ(Z2qr), then G PSL(2, 5) or G Z2qr, where q and r are distinct odd prime numbers.