Actions of nilpotent groups on nilpotent groups
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909560618352640 |
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| author | Burkhart, Michael C. |
| author_facet | Burkhart, Michael C. |
| contents | For finite nilpotent groups $J$ and $N$, suppose $J$ acts on $N$ via automorphisms. We exhibit a decomposition of the first cohomology set in terms of the first cohomologies of the Sylow $p$-subgroups of $J$ that mirrors the primary decomposition of $H^1(J,N)$ for abelian $N$. We then show that if $N \rtimes J$ acts on some non-empty set $Ω$, where the action of $N$ is transitive and for each prime $p$ a Sylow $p$-subgroup of $J$ fixes an element of $Ω$, then $J$ fixes an element of $Ω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16941 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Actions of nilpotent groups on nilpotent groups Burkhart, Michael C. Group Theory 05E18, 20D15, 20J06 (Primary) 20D45 (Secondary) For finite nilpotent groups $J$ and $N$, suppose $J$ acts on $N$ via automorphisms. We exhibit a decomposition of the first cohomology set in terms of the first cohomologies of the Sylow $p$-subgroups of $J$ that mirrors the primary decomposition of $H^1(J,N)$ for abelian $N$. We then show that if $N \rtimes J$ acts on some non-empty set $Ω$, where the action of $N$ is transitive and for each prime $p$ a Sylow $p$-subgroup of $J$ fixes an element of $Ω$, then $J$ fixes an element of $Ω$. |
| title | Actions of nilpotent groups on nilpotent groups |
| topic | Group Theory 05E18, 20D15, 20J06 (Primary) 20D45 (Secondary) |
| url | https://arxiv.org/abs/2501.16941 |