Excluded power graphs of groups
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918400620494848 |
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| author | Curtin, Brian |
| author_facet | Curtin, Brian |
| contents | Let $\mathcal{X}$ be a set of integers greater than one. The $\mathcal{X}$-excluded power graph of a group $G$ has vertex set $G$ and an edge from $g$ to each power of $g$ other than itself provided that the power is not divisible by any element of $\mathcal{X}$. When $G=H\times K$ for groups $H$ and $K$ with coprime orders, excluding the prime factors of $|H|$ yields a power graph with a quotient consisting of multiple copies of a quotient of the power graph (no exclusions) of $K$. Partial results for the semidirect product under the same conditions are given. We describe groups whose $\mathcal{X}$-excluded power graphs consist of disjoint directed cliques. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_09519 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Excluded power graphs of groups Curtin, Brian Combinatorics Group Theory 05C25, 05C20 Let $\mathcal{X}$ be a set of integers greater than one. The $\mathcal{X}$-excluded power graph of a group $G$ has vertex set $G$ and an edge from $g$ to each power of $g$ other than itself provided that the power is not divisible by any element of $\mathcal{X}$. When $G=H\times K$ for groups $H$ and $K$ with coprime orders, excluding the prime factors of $|H|$ yields a power graph with a quotient consisting of multiple copies of a quotient of the power graph (no exclusions) of $K$. Partial results for the semidirect product under the same conditions are given. We describe groups whose $\mathcal{X}$-excluded power graphs consist of disjoint directed cliques. |
| title | Excluded power graphs of groups |
| topic | Combinatorics Group Theory 05C25, 05C20 |
| url | https://arxiv.org/abs/2502.09519 |