Non-abelian tensor product of residually finite groups
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866908504023891968 |
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| author | Bastos, Raimundo Rocco, Noraí R. |
| author_facet | Bastos, Raimundo Rocco, Noraí R. |
| contents | Let $G$ and $H$ be groups that act compatibly on each other. We denote by $η(G,H)$ a certain extension of the non-abelian tensor product $G \otimes H$ by $G \times H$. Suppose that $G$ is residually finite and the subgroup $[G,H] = \langle g^{-1}g^h \ \mid g \in G, h\in H\rangle$ satisfies some non-trivial identity $f \equiv~1$. We prove that if $p$ is a prime and every tensor has $p$-power order, then the non-abelian tensor product $G \otimes H$ is locally finite. Further, we show that if $n$ is a positive integer and every tensor is left $n$-Engel in $η(G,H)$, then the non-abelian tensor product $G \otimes H$ is locally nilpotent. The content of this paper extend some results concerning the non-abelian tensor square $G \otimes G$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1709_03132 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Non-abelian tensor product of residually finite groups Bastos, Raimundo Rocco, Noraí R. Group Theory 20E26, 20F50, 20J06 Let $G$ and $H$ be groups that act compatibly on each other. We denote by $η(G,H)$ a certain extension of the non-abelian tensor product $G \otimes H$ by $G \times H$. Suppose that $G$ is residually finite and the subgroup $[G,H] = \langle g^{-1}g^h \ \mid g \in G, h\in H\rangle$ satisfies some non-trivial identity $f \equiv~1$. We prove that if $p$ is a prime and every tensor has $p$-power order, then the non-abelian tensor product $G \otimes H$ is locally finite. Further, we show that if $n$ is a positive integer and every tensor is left $n$-Engel in $η(G,H)$, then the non-abelian tensor product $G \otimes H$ is locally nilpotent. The content of this paper extend some results concerning the non-abelian tensor square $G \otimes G$. |
| title | Non-abelian tensor product of residually finite groups |
| topic | Group Theory 20E26, 20F50, 20J06 |
| url | https://arxiv.org/abs/1709.03132 |