On the residual nilpotence of generalized free products of groups
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866910456148393984 |
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| author | Sokolov, E. V. |
| author_facet | Sokolov, E. V. |
| contents | Let $G$ be the generalized free product of two groups with an amalgamated subgroup. We propose an approach that allows one to use results on the residual $p$-finiteness of $G$ for proving that this generalized free product is residually a finite nilpotent group or residually a finite metanilpotent group. This approach can be applied under most of the conditions on the amalgamated subgroup that allow the study of residual $p$-finiteness. Namely, we consider the cases where the amalgamated subgroup is a) periodic, b) locally cyclic, c) central in one of the free factors, d) normal in both free factors, or e) is a retract of one of the free factors. In each of these cases, we give certain necessary and sufficient conditions for $G$ to be residually a) a finite nilpotent group, b) a finite metanilpotent group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_00285 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the residual nilpotence of generalized free products of groups Sokolov, E. V. Group Theory 20E26, 20E06 (Primary) 20F14 (Secondary) Let $G$ be the generalized free product of two groups with an amalgamated subgroup. We propose an approach that allows one to use results on the residual $p$-finiteness of $G$ for proving that this generalized free product is residually a finite nilpotent group or residually a finite metanilpotent group. This approach can be applied under most of the conditions on the amalgamated subgroup that allow the study of residual $p$-finiteness. Namely, we consider the cases where the amalgamated subgroup is a) periodic, b) locally cyclic, c) central in one of the free factors, d) normal in both free factors, or e) is a retract of one of the free factors. In each of these cases, we give certain necessary and sufficient conditions for $G$ to be residually a) a finite nilpotent group, b) a finite metanilpotent group. |
| title | On the residual nilpotence of generalized free products of groups |
| topic | Group Theory 20E26, 20E06 (Primary) 20F14 (Secondary) |
| url | https://arxiv.org/abs/2312.00285 |