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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.02594 |
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| _version_ | 1866918082603122688 |
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| author | Azad, Morteza Baniasad Arabtash, Mostafa |
| author_facet | Azad, Morteza Baniasad Arabtash, Mostafa |
| contents | Let $G$ be a finite group. Let $ρ(G) = \prod_{g \in G} o(g)={p_1}^{α_1} {p_2}^{α_2} \cdots {p_k}^{α_k}$, 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}}_ρ(G)$, i.e., $ {\operatorname{Exp}}_ρ(G)=\{α_1,α_2, \cdots,α_k\}$. In this paper, we give a new characterization for some groups by $ {\operatorname{Exp}}_ρ(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}}_ρ(G)$. Furthermore, we prove that the groups ${\rm PSL}(2, 5)$ and ${\rm PSL}(2, 13)$ are uniquely determined by the parameters $ {\operatorname{Exp}}_ρ(G)$ and $|G|$. Additionally, we prove that if ${\operatorname{Exp}}_ρ(G) = {\operatorname{Exp}}_ρ(\mathbb{Z}_{2qr})$, then $G \cong {\rm PSL}(2, 5)$ or $G \cong \mathbb{Z}_{2qr}$, where $q$ and $r$ are distinct odd prime numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02594 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Recognition by the set of exponents in the prime factorization of the product of element orders Azad, Morteza Baniasad Arabtash, Mostafa Group Theory 20D60 Let $G$ be a finite group. Let $ρ(G) = \prod_{g \in G} o(g)={p_1}^{α_1} {p_2}^{α_2} \cdots {p_k}^{α_k}$, 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}}_ρ(G)$, i.e., $ {\operatorname{Exp}}_ρ(G)=\{α_1,α_2, \cdots,α_k\}$. In this paper, we give a new characterization for some groups by $ {\operatorname{Exp}}_ρ(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}}_ρ(G)$. Furthermore, we prove that the groups ${\rm PSL}(2, 5)$ and ${\rm PSL}(2, 13)$ are uniquely determined by the parameters $ {\operatorname{Exp}}_ρ(G)$ and $|G|$. Additionally, we prove that if ${\operatorname{Exp}}_ρ(G) = {\operatorname{Exp}}_ρ(\mathbb{Z}_{2qr})$, then $G \cong {\rm PSL}(2, 5)$ or $G \cong \mathbb{Z}_{2qr}$, where $q$ and $r$ are distinct odd prime numbers. |
| title | Recognition by the set of exponents in the prime factorization of the product of element orders |
| topic | Group Theory 20D60 |
| url | https://arxiv.org/abs/2507.02594 |