On $A$-Groups with the Same Index Set as a Nilpotent Group
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915350030843904 |
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| author | Zhou, Wei Gorshkov, Ilya |
| author_facet | Zhou, Wei Gorshkov, Ilya |
| contents | Let $G$ be a finite group and $N(G)$ be the set of conjugacy class sizes of $G$. For a prime $p$, let $|G||_p$ be the highest $p$-power dividing some element of $N(G)$. and define $|G|| = Π_{p\in π(G)}|G||_p$. $G$ is said to be an $A$-group if all its Sylow subgroups are abelian. We prove that if $G$ is an $A$-group such that $N(G)$ contains $|G||_p$ for every $p\in π(G)$ as well as $|G||$, then $G$ must be abelian. This result gives a positive answer to a question posed by Camina and Camina in 2006. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_15250 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $A$-Groups with the Same Index Set as a Nilpotent Group Zhou, Wei Gorshkov, Ilya Group Theory 20D15, 20D60 Let $G$ be a finite group and $N(G)$ be the set of conjugacy class sizes of $G$. For a prime $p$, let $|G||_p$ be the highest $p$-power dividing some element of $N(G)$. and define $|G|| = Π_{p\in π(G)}|G||_p$. $G$ is said to be an $A$-group if all its Sylow subgroups are abelian. We prove that if $G$ is an $A$-group such that $N(G)$ contains $|G||_p$ for every $p\in π(G)$ as well as $|G||$, then $G$ must be abelian. This result gives a positive answer to a question posed by Camina and Camina in 2006. |
| title | On $A$-Groups with the Same Index Set as a Nilpotent Group |
| topic | Group Theory 20D15, 20D60 |
| url | https://arxiv.org/abs/2506.15250 |