A class of finite $p$-groups and the normalized unit groups of group algebras
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916078405287936 |
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| author | Wang, Yulei Liu, Heguo |
| author_facet | Wang, Yulei Liu, Heguo |
| contents | Let $p$ be a prime and $\mathbb{F}_p$ be a finite field of $p$ elements. Let $\mathbb{F}_pG$ denote the group algebra of the finite $p$-group $G$ over the field $\mathbb{F}_p$ and $V(\mathbb{F}_pG)$ denote the group of normalized units in $\mathbb{F}_pG$. Suppose that $G$ is a finite $p$-group given by a central extension of the form $$1\longrightarrow \mathbb{Z}_{p^n}\times \mathbb{Z}_{p^m} \longrightarrow G \longrightarrow \mathbb{Z}_p\times \cdots\times \mathbb{Z}_p \longrightarrow 1$$ and $G'\cong \mathbb{Z}_p$, $n, m\geq 1$ and $p$ is odd. In this paper, the structure of $G$ is determined. And the relations of $V(\mathbb{F}_pG)^{p^l}$ and $G^{p^l}$, $Ω_l(V(\mathbb{F}_pG))$ and $Ω_l(G)$ are given. Furthermore, there is a direct proof for $V(\mathbb{F}_pG)^p\bigcap G=G^p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00638 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A class of finite $p$-groups and the normalized unit groups of group algebras Wang, Yulei Liu, Heguo Group Theory Let $p$ be a prime and $\mathbb{F}_p$ be a finite field of $p$ elements. Let $\mathbb{F}_pG$ denote the group algebra of the finite $p$-group $G$ over the field $\mathbb{F}_p$ and $V(\mathbb{F}_pG)$ denote the group of normalized units in $\mathbb{F}_pG$. Suppose that $G$ is a finite $p$-group given by a central extension of the form $$1\longrightarrow \mathbb{Z}_{p^n}\times \mathbb{Z}_{p^m} \longrightarrow G \longrightarrow \mathbb{Z}_p\times \cdots\times \mathbb{Z}_p \longrightarrow 1$$ and $G'\cong \mathbb{Z}_p$, $n, m\geq 1$ and $p$ is odd. In this paper, the structure of $G$ is determined. And the relations of $V(\mathbb{F}_pG)^{p^l}$ and $G^{p^l}$, $Ω_l(V(\mathbb{F}_pG))$ and $Ω_l(G)$ are given. Furthermore, there is a direct proof for $V(\mathbb{F}_pG)^p\bigcap G=G^p$. |
| title | A class of finite $p$-groups and the normalized unit groups of group algebras |
| topic | Group Theory |
| url | https://arxiv.org/abs/2401.00638 |