A criterion for a normal subgroup to be hypercentral based on class sizes
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917581220216832 |
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| author | Beltrán, Antonio |
| author_facet | Beltrán, Antonio |
| contents | Let $G$ be a finite group and $N$ a normal subgroup of $G$. We prove that the knowledge of the sizes of the conjugacy classes of $G$ that are contained in $N$ and of their multiplicities provides information of $N$ in relation to the structure of $G$. Among other results, we obtain a criterion to determine whether a Sylow $p$-subgroup of $N$ lies in the hypercentre of $G$ for a fixed prime $p$, and therefore, whether the whole subgroup $N$ is hypercentral in $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_01358 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A criterion for a normal subgroup to be hypercentral based on class sizes Beltrán, Antonio Group Theory 20D15, 20E45 Let $G$ be a finite group and $N$ a normal subgroup of $G$. We prove that the knowledge of the sizes of the conjugacy classes of $G$ that are contained in $N$ and of their multiplicities provides information of $N$ in relation to the structure of $G$. Among other results, we obtain a criterion to determine whether a Sylow $p$-subgroup of $N$ lies in the hypercentre of $G$ for a fixed prime $p$, and therefore, whether the whole subgroup $N$ is hypercentral in $G$. |
| title | A criterion for a normal subgroup to be hypercentral based on class sizes |
| topic | Group Theory 20D15, 20E45 |
| url | https://arxiv.org/abs/2402.01358 |