On the subgroup separability of the free product of groups
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912210246172672 |
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| author | Sokolov, E. V. |
| author_facet | Sokolov, E. V. |
| contents | Suppose that $\mathcal{C}$ is a root class of groups (i.e., a class of groups that contains non-trivial groups and is closed under taking subgroups and unrestricted wreath products), $G$ is the free product of residually $\mathcal{C}$-groups $A_{i}$ ($i \in \mathcal{I}$), and $H$ is a subgroup of $G$ satisfying a non-trivial identity. We prove a criterion for the $\mathcal{C}$-separability of $H$ in $G$. It follows from this criterion that, if $\{\mathcal{V}_{j} \mid j \in \mathcal{J}\}$ is a family of group varieties, each $\mathcal{V}_{j}$ ($j \in \mathcal{J}$) is distinct from the variety of all groups, and $\mathcal{V} = \bigcup_{j \in \mathcal{J}} \mathcal{V}_{j}$, then one can give a description of $\mathcal{C}$-separable $\mathcal{V}$-subgroups of $G$ provided such a description is known for every group $A_{i}$ ($i \in \mathcal{I}$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17342 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the subgroup separability of the free product of groups Sokolov, E. V. Group Theory 20E26, 20E06 Suppose that $\mathcal{C}$ is a root class of groups (i.e., a class of groups that contains non-trivial groups and is closed under taking subgroups and unrestricted wreath products), $G$ is the free product of residually $\mathcal{C}$-groups $A_{i}$ ($i \in \mathcal{I}$), and $H$ is a subgroup of $G$ satisfying a non-trivial identity. We prove a criterion for the $\mathcal{C}$-separability of $H$ in $G$. It follows from this criterion that, if $\{\mathcal{V}_{j} \mid j \in \mathcal{J}\}$ is a family of group varieties, each $\mathcal{V}_{j}$ ($j \in \mathcal{J}$) is distinct from the variety of all groups, and $\mathcal{V} = \bigcup_{j \in \mathcal{J}} \mathcal{V}_{j}$, then one can give a description of $\mathcal{C}$-separable $\mathcal{V}$-subgroups of $G$ provided such a description is known for every group $A_{i}$ ($i \in \mathcal{I}$). |
| title | On the subgroup separability of the free product of groups |
| topic | Group Theory 20E26, 20E06 |
| url | https://arxiv.org/abs/2501.17342 |