Closures of permutation groups with restricted nonabelian composition factors
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908518619021312 |
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| author | Ponomarenko, Ilia Skresanov, Saveliy V. Vasil'ev, Andrey V. |
| author_facet | Ponomarenko, Ilia Skresanov, Saveliy V. Vasil'ev, Andrey V. |
| contents | Given a permutation group $G$ on a finite set $Ω$, let $G^{(k)}$ denote the $k$-closure of $G$, that is, the largest permutation group on $Ω$ having the same orbits in the induced action on $Ω^k$ as $G$. Recall that a group is $\mathrm{Alt}(d)$-free if it does not contain a section isomorphic to the alternating group of degree $d$. Motivated by some problems in computational group theory, we prove that the $k$-closure of an $\mathrm{Alt}(d)$-free group is again $\mathrm{Alt}(d)$-free for $k \geq 4$ and $d \geq 25$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03780 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Closures of permutation groups with restricted nonabelian composition factors Ponomarenko, Ilia Skresanov, Saveliy V. Vasil'ev, Andrey V. Group Theory 20B05 (Primary) 20B25, 20G40 (Secondary) Given a permutation group $G$ on a finite set $Ω$, let $G^{(k)}$ denote the $k$-closure of $G$, that is, the largest permutation group on $Ω$ having the same orbits in the induced action on $Ω^k$ as $G$. Recall that a group is $\mathrm{Alt}(d)$-free if it does not contain a section isomorphic to the alternating group of degree $d$. Motivated by some problems in computational group theory, we prove that the $k$-closure of an $\mathrm{Alt}(d)$-free group is again $\mathrm{Alt}(d)$-free for $k \geq 4$ and $d \geq 25$. |
| title | Closures of permutation groups with restricted nonabelian composition factors |
| topic | Group Theory 20B05 (Primary) 20B25, 20G40 (Secondary) |
| url | https://arxiv.org/abs/2406.03780 |