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

2025/07/03 by Azad, Morteza Baniasad, Arabtash, Mostafa · 1 citation
#20D60 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2507.02594

Abstract

Let G be a finite group. Let ρ(G) = ∏_{g ∈ G} o(g)=p₁^α₁ p₂^α₂ ⋯ p_k^α_k, where p₁, p₂, ⋯, p_k 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, ⋯,α_k\. In this paper, we give a new characterization for some groups by Exp_ρ(G). We prove that the groups \rm PSL(2, 5) × ℤ_p, \rm PSL(2, 7) and \rm PSL(2, 11) are uniquely determined by Exp_ρ(G). Furthermore, we prove that the groups \rm PSL(2, 5) and \rm PSL(2, 13) are uniquely determined by the parameters Exp_ρ(G) and |G|. Additionally, we prove that if Exp_ρ(G) = Exp_ρ(ℤ_2qr), then G ≅ \rm PSL(2, 5) or G ≅ ℤ_2qr, where q and r are distinct odd prime numbers.

Citations

• On the structure of finite groups determined by the arithmetic and geometric means of element orders
• Upper bounds for the product of element orders of finite groups
• Inequalities detecting structural properties of a finite group

Cited by

• The influence of the prime omega function on the product of element orders in finite groups

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