Characterization of $\PSL(2,q)$ by the number of singular elements
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| Format: | Preprint |
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2025
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| _version_ | 1866910899884785664 |
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| author | Shen, Rulin Yan, Deyu |
| author_facet | Shen, Rulin Yan, Deyu |
| contents | Given a finite group $G$, let $π(G)$ denote the set of all primes that divide the order of $G$. For a prime $r \in π(G)$, we define $r$-singular elements as those elements of $G$ whose order is divisible by $r$. Denote by $S_r(G)$ the number of $r$-singluar elements of $G$. We denote the proportion $S_r(G)/|G|$ of $r$-singular elements in $G$ by ${μ_r}(G)$. Let $μ(G) := {\{μ_r}(G) | r\in π(G)\}$ be the set of all proportions of $r$-singular elements for each prime $r$ in $π(G)$. In this paper, we prove that if a finite group $G$ has the same set $μ(G)$ as the simple group $\PSL(2,q)$, then $G$ is isomorphic to $\PSL(2,q)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_24212 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterization of $\PSL(2,q)$ by the number of singular elements Shen, Rulin Yan, Deyu Group Theory Given a finite group $G$, let $π(G)$ denote the set of all primes that divide the order of $G$. For a prime $r \in π(G)$, we define $r$-singular elements as those elements of $G$ whose order is divisible by $r$. Denote by $S_r(G)$ the number of $r$-singluar elements of $G$. We denote the proportion $S_r(G)/|G|$ of $r$-singular elements in $G$ by ${μ_r}(G)$. Let $μ(G) := {\{μ_r}(G) | r\in π(G)\}$ be the set of all proportions of $r$-singular elements for each prime $r$ in $π(G)$. In this paper, we prove that if a finite group $G$ has the same set $μ(G)$ as the simple group $\PSL(2,q)$, then $G$ is isomorphic to $\PSL(2,q)$. |
| title | Characterization of $\PSL(2,q)$ by the number of singular elements |
| topic | Group Theory |
| url | https://arxiv.org/abs/2503.24212 |