Compact groups in which commutators have finite right Engel sinks
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909342311120896 |
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| author | Khukhro, Evgeny Shumyatsky, Pavel |
| author_facet | Khukhro, Evgeny Shumyatsky, Pavel |
| contents | A right Engel sink of an element $g$ of a group $G$ is a subset containing all sufficiently long commutators $[...[[g,x],x],\dots ,x]$. We prove that if $G$ is a compact group in which, for some $k$, every commutator $[...[g_1,g_2],\dots ,g_k]$ has a finite right Engel sink, then $G$ has a locally nilpotent open subgroup. If in addition, for some positive integer $m$, every commutator $[...[g_1,g_2],\dots ,g_k]$ has a right Engel sink of cardinality at most $m$, then $G$ has a locally nilpotent subgroup of finite index bounded in terms of $m$ only. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_05840 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Compact groups in which commutators have finite right Engel sinks Khukhro, Evgeny Shumyatsky, Pavel Group Theory 20D25, 20E18, 20F45, 22C05 A right Engel sink of an element $g$ of a group $G$ is a subset containing all sufficiently long commutators $[...[[g,x],x],\dots ,x]$. We prove that if $G$ is a compact group in which, for some $k$, every commutator $[...[g_1,g_2],\dots ,g_k]$ has a finite right Engel sink, then $G$ has a locally nilpotent open subgroup. If in addition, for some positive integer $m$, every commutator $[...[g_1,g_2],\dots ,g_k]$ has a right Engel sink of cardinality at most $m$, then $G$ has a locally nilpotent subgroup of finite index bounded in terms of $m$ only. |
| title | Compact groups in which commutators have finite right Engel sinks |
| topic | Group Theory 20D25, 20E18, 20F45, 22C05 |
| url | https://arxiv.org/abs/2410.05840 |