Generalized torsion elements in infinite groups
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911305567305728 |
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| author | Bastos, Raimundo Mendonça, Luis |
| author_facet | Bastos, Raimundo Mendonça, Luis |
| contents | A group element is called generalized torsion if a finite product of its conjugates is equal to the identity. We show that in a finitely generated abelian-by-finite group, an element is generalized torsion if and only if its image in the abelianization is torsion. We also prove a quantitative version with sharp bounds to the generalized exponent of these groups. In particular, we provide many examples of finitely presentable torsion-free groups in which all elements are generalized torsion. We also discuss positive generalized identities in abelian-by-finite groups and related classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17918 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized torsion elements in infinite groups Bastos, Raimundo Mendonça, Luis Group Theory A group element is called generalized torsion if a finite product of its conjugates is equal to the identity. We show that in a finitely generated abelian-by-finite group, an element is generalized torsion if and only if its image in the abelianization is torsion. We also prove a quantitative version with sharp bounds to the generalized exponent of these groups. In particular, we provide many examples of finitely presentable torsion-free groups in which all elements are generalized torsion. We also discuss positive generalized identities in abelian-by-finite groups and related classes. |
| title | Generalized torsion elements in infinite groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2411.17918 |