Fixed point conditions for non-coprime actions
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866918002750914560 |
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| author | Burkhart, Michael C. |
| author_facet | Burkhart, Michael C. |
| contents | In the setting of finite groups, suppose $J$ acts on $N$ via automorphisms so that the induced semidirect product $N\rtimes J$ acts on some non-empty set $Ω$, with $N$ acting transitively. Glauberman proved that if the orders of $J$ and $N$ are coprime, then $J$ fixes a point in $Ω$. We consider the non-coprime case and show that if $N$ is abelian and a Sylow $p$-subgroup of $J$ fixes a point in $Ω$ for each prime $p$, then $J$ fixes a point in $Ω$. We also show that if $N$ is nilpotent, $N\rtimes J$ is supersoluble, and a Sylow $p$-subgroup of $J$ fixes a point in $Ω$ for each prime $p$, then $J$ fixes a point in $Ω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_12286 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fixed point conditions for non-coprime actions Burkhart, Michael C. Group Theory 05E18, 20E45 (Primary) 20F16, 20J06 (Secondary) In the setting of finite groups, suppose $J$ acts on $N$ via automorphisms so that the induced semidirect product $N\rtimes J$ acts on some non-empty set $Ω$, with $N$ acting transitively. Glauberman proved that if the orders of $J$ and $N$ are coprime, then $J$ fixes a point in $Ω$. We consider the non-coprime case and show that if $N$ is abelian and a Sylow $p$-subgroup of $J$ fixes a point in $Ω$ for each prime $p$, then $J$ fixes a point in $Ω$. We also show that if $N$ is nilpotent, $N\rtimes J$ is supersoluble, and a Sylow $p$-subgroup of $J$ fixes a point in $Ω$ for each prime $p$, then $J$ fixes a point in $Ω$. |
| title | Fixed point conditions for non-coprime actions |
| topic | Group Theory 05E18, 20E45 (Primary) 20F16, 20J06 (Secondary) |
| url | https://arxiv.org/abs/2308.12286 |