Finite and profinite groups with small Engel sinks of p-elements
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911046323666944 |
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| author | Berto, Lucas Dal Caldeira, Jhone Shumyatsky, Pavel |
| author_facet | Berto, Lucas Dal Caldeira, Jhone Shumyatsky, Pavel |
| contents | A (left) Engel sink of an element g of a group G is a subset containing all sufficiently long commutators [...[[x,g],g],...,g], where x ranges over G. We prove that if p is a prime and G a finite group in which, for some positive integer m, every p-element has an Engel sink of cardinality at most m, then G has a normal subgroup N such that G/N is a p'-group and the index [N:O_p(G)] is bounded in terms of m only.
Furthermore, if G is a profinite group in which every p-element possesses a finite Engel sink, then G has a normal subgroup N such that N is virtually pro-p while G/N is a pro-p' group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_01409 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite and profinite groups with small Engel sinks of p-elements Berto, Lucas Dal Caldeira, Jhone Shumyatsky, Pavel Group Theory A (left) Engel sink of an element g of a group G is a subset containing all sufficiently long commutators [...[[x,g],g],...,g], where x ranges over G. We prove that if p is a prime and G a finite group in which, for some positive integer m, every p-element has an Engel sink of cardinality at most m, then G has a normal subgroup N such that G/N is a p'-group and the index [N:O_p(G)] is bounded in terms of m only. Furthermore, if G is a profinite group in which every p-element possesses a finite Engel sink, then G has a normal subgroup N such that N is virtually pro-p while G/N is a pro-p' group. |
| title | Finite and profinite groups with small Engel sinks of p-elements |
| topic | Group Theory |
| url | https://arxiv.org/abs/2505.01409 |