A result on certain sums of element orders in finite groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909561435193344 |
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| author | Tărnăuceanu, Marius |
| author_facet | Tărnăuceanu, Marius |
| contents | Given a finite group $G$ of order $p^nm$, where $p$ is a prime and $p\nmid m$, we denote by $ψ_p(G)$ the sum of orders of $p$-parts of elements in $G$. In the current note, we prove that $ψ_p(G)\leqψ_p(C_{p^nm})$, where $C_{p^nm}$ is the cyclic group of order $p^nm$, and the equality holds if and only if $G$ is $p$-nilpotent of a particular type. A generalization of this result is also presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_01682 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A result on certain sums of element orders in finite groups Tărnăuceanu, Marius Group Theory Given a finite group $G$ of order $p^nm$, where $p$ is a prime and $p\nmid m$, we denote by $ψ_p(G)$ the sum of orders of $p$-parts of elements in $G$. In the current note, we prove that $ψ_p(G)\leqψ_p(C_{p^nm})$, where $C_{p^nm}$ is the cyclic group of order $p^nm$, and the equality holds if and only if $G$ is $p$-nilpotent of a particular type. A generalization of this result is also presented. |
| title | A result on certain sums of element orders in finite groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2504.01682 |