Finite simple groups have many classes of $p$-elements
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909624339267584 |
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| author | Giudici, Michael Morgan, Luke Praeger, Cheryl E. |
| author_facet | Giudici, Michael Morgan, Luke Praeger, Cheryl E. |
| contents | For an element $x$ of a finite group $T$, the $\mathrm{Aut}(T)$-class of $x$ is the set $\{ x^σ\mid σ\in \mathrm{Aut}(T)\}$. We prove that the order $|T|$ of a finite nonabelian simple group $T$ is bounded above by a function of the parameter $m(T)$, where $m(T)$ is the maximum, over all primes $p$, of the number of $\mathrm{Aut}(T)$-classes of elements of $T$ of $p$-power order. This bound is a substantial generalisation of results of Pyber, and of Héthelyi and Külshammer, and it has implications for relative Brauer groups of finite extensions of global fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18863 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite simple groups have many classes of $p$-elements Giudici, Michael Morgan, Luke Praeger, Cheryl E. Group Theory For an element $x$ of a finite group $T$, the $\mathrm{Aut}(T)$-class of $x$ is the set $\{ x^σ\mid σ\in \mathrm{Aut}(T)\}$. We prove that the order $|T|$ of a finite nonabelian simple group $T$ is bounded above by a function of the parameter $m(T)$, where $m(T)$ is the maximum, over all primes $p$, of the number of $\mathrm{Aut}(T)$-classes of elements of $T$ of $p$-power order. This bound is a substantial generalisation of results of Pyber, and of Héthelyi and Külshammer, and it has implications for relative Brauer groups of finite extensions of global fields. |
| title | Finite simple groups have many classes of $p$-elements |
| topic | Group Theory |
| url | https://arxiv.org/abs/2411.18863 |