A completeness criterion for the common divisor graph on $p$-regular class sizes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915724485722112 |
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| author | Sotomayor, Víctor |
| author_facet | Sotomayor, Víctor |
| contents | Let $G$ be a finite group. For some fixed prime $p$, let $Γ_p(G)$ be the common divisor graph built on the set of sizes of $p$-regular conjugacy classes of $G$: this is the simple undirected graph whose vertices are the class sizes of those non-central elements of $G$ such that $p$ does not divide their order, and two distinct vertices are adjacent if and only if they are not coprime. In this note we prove that if $Γ_p(G)$ is a $k$-regular graph with $k\geq 1$, then it is a complete graph with $k+1$ vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09083 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A completeness criterion for the common divisor graph on $p$-regular class sizes Sotomayor, Víctor Group Theory 20E45 Let $G$ be a finite group. For some fixed prime $p$, let $Γ_p(G)$ be the common divisor graph built on the set of sizes of $p$-regular conjugacy classes of $G$: this is the simple undirected graph whose vertices are the class sizes of those non-central elements of $G$ such that $p$ does not divide their order, and two distinct vertices are adjacent if and only if they are not coprime. In this note we prove that if $Γ_p(G)$ is a $k$-regular graph with $k\geq 1$, then it is a complete graph with $k+1$ vertices. |
| title | A completeness criterion for the common divisor graph on $p$-regular class sizes |
| topic | Group Theory 20E45 |
| url | https://arxiv.org/abs/2412.09083 |