A Note on Finite Nilpotent Groups
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929418098704384 |
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| author | Werner, Nicholas J. |
| author_facet | Werner, Nicholas J. |
| contents | It is well known that if $G$ is a group and $H$ is a normal subgroup of $G$ of finite index $k$, then $x^k \in H$ for every $x \in G$. We examine finite groups $G$ with the property that $x^k \in H$ for every subgroup $H$ of $G$, where $k$ is the index of $H$ in $G$. We prove that a finite group $G$ satisfies this property if and only if $G$ is nilpotent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05498 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Note on Finite Nilpotent Groups Werner, Nicholas J. Group Theory It is well known that if $G$ is a group and $H$ is a normal subgroup of $G$ of finite index $k$, then $x^k \in H$ for every $x \in G$. We examine finite groups $G$ with the property that $x^k \in H$ for every subgroup $H$ of $G$, where $k$ is the index of $H$ in $G$. We prove that a finite group $G$ satisfies this property if and only if $G$ is nilpotent. |
| title | A Note on Finite Nilpotent Groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2407.05498 |