Inner and Outer Derivations of $\mathbb{F}V_{8n}$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912058593771520 |
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| author | Manju, Praveen Sharma, Rajendra Kumar |
| author_facet | Manju, Praveen Sharma, Rajendra Kumar |
| contents | Let $\mathbb{F}$ be a field of characteristic $0$ or an odd rational prime $p$. In this article, we give an explicit classification of all the inner and outer derivations of the group algebra $\mathbb{F}V_{8n}$, where $V_{8n}$ is a group of order $8n$ ($n$ a positive integer) with presentation $\langle a, b \mid a^{2n} = b^{4} = 1, ba = a^{-1}b^{-1}, b^{-1}a = a^{-1}b \rangle$. First, we explicitly classify all the $\mathbb{F}$-derivations of $\mathbb{F}V_{8n}$ by giving the dimension and a basis of the derivation algebra consisting of all $\mathbb{F}$-derivations of $\mathbb{F}V_{8n}$. Consequently, we classify all inner and outer derivations of $\mathbb{F}V_{8n}$ when $\mathbb{F}$ is an algebraic extension of a prime field. Thus, we establish that all the derivations of $\mathbb{F}V_{8n}$ are inner when the characteristic of $\mathbb{F}$ is $0$ or $p$ with $p$ relatively prime to $n$, and that non-zero outer derivations exist only in the case when the characteristic of $\mathbb{F}$ is $p$ with $p$ dividing $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_03467 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inner and Outer Derivations of $\mathbb{F}V_{8n}$ Manju, Praveen Sharma, Rajendra Kumar Rings and Algebras Group Theory Representation Theory 16S34, 16W25, 20C05 Let $\mathbb{F}$ be a field of characteristic $0$ or an odd rational prime $p$. In this article, we give an explicit classification of all the inner and outer derivations of the group algebra $\mathbb{F}V_{8n}$, where $V_{8n}$ is a group of order $8n$ ($n$ a positive integer) with presentation $\langle a, b \mid a^{2n} = b^{4} = 1, ba = a^{-1}b^{-1}, b^{-1}a = a^{-1}b \rangle$. First, we explicitly classify all the $\mathbb{F}$-derivations of $\mathbb{F}V_{8n}$ by giving the dimension and a basis of the derivation algebra consisting of all $\mathbb{F}$-derivations of $\mathbb{F}V_{8n}$. Consequently, we classify all inner and outer derivations of $\mathbb{F}V_{8n}$ when $\mathbb{F}$ is an algebraic extension of a prime field. Thus, we establish that all the derivations of $\mathbb{F}V_{8n}$ are inner when the characteristic of $\mathbb{F}$ is $0$ or $p$ with $p$ relatively prime to $n$, and that non-zero outer derivations exist only in the case when the characteristic of $\mathbb{F}$ is $p$ with $p$ dividing $n$. |
| title | Inner and Outer Derivations of $\mathbb{F}V_{8n}$ |
| topic | Rings and Algebras Group Theory Representation Theory 16S34, 16W25, 20C05 |
| url | https://arxiv.org/abs/2410.03467 |