Asymptotics of the powers in finite reductive groups
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866917629037379584 |
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| author | Kulshrestha, Amit Kundu, Rijubrata Singh, Anupam |
| author_facet | Kulshrestha, Amit Kundu, Rijubrata Singh, Anupam |
| contents | Let $G$ be a connected reductive group defined over $\mathbb F_q$. Fix an integer $M\geq 2$, and consider the power map $x\mapsto x^M$ on $G$. We denote the image of $G(\mathbb F_q)$ under this map by $G(\mathbb F_q)^M$ and estimate what proportion of regular semisimple, semisimple and regular elements of $G(\mathbb F_q)$ it contains. We prove that as $q\to\infty$, all of these proportions are equal and provide a formula for the same. We also calculate this more explicitly for the groups $\text{GL}(n,q)$ and $\text{U}(n,q)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_12616 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Asymptotics of the powers in finite reductive groups Kulshrestha, Amit Kundu, Rijubrata Singh, Anupam Group Theory 20G40 Let $G$ be a connected reductive group defined over $\mathbb F_q$. Fix an integer $M\geq 2$, and consider the power map $x\mapsto x^M$ on $G$. We denote the image of $G(\mathbb F_q)$ under this map by $G(\mathbb F_q)^M$ and estimate what proportion of regular semisimple, semisimple and regular elements of $G(\mathbb F_q)$ it contains. We prove that as $q\to\infty$, all of these proportions are equal and provide a formula for the same. We also calculate this more explicitly for the groups $\text{GL}(n,q)$ and $\text{U}(n,q)$. |
| title | Asymptotics of the powers in finite reductive groups |
| topic | Group Theory 20G40 |
| url | https://arxiv.org/abs/2004.12616 |