Non-abelian tensor square and related constructions of $p$-groups
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2019
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| _version_ | 1866915463821262848 |
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| author | Bastos, Raimundo de Melo, Emerson Gonçalves, Nathália Nunes, Ricardo |
| author_facet | Bastos, Raimundo de Melo, Emerson Gonçalves, Nathália Nunes, Ricardo |
| contents | Let $G$ be a group. We denote by $ν(G)$ a certain extension of the non-abelian tensor square $[G,G^φ]$ by $G \times G$. We prove that if $G$ is a finite potent $p$-group, then $[G,G^φ]$ and the $k$-th term of the lower central series $γ_k(ν(G))$ are potently embedded in $ν(G)$ (Theorem A). Moreover, we show that if $G$ is a potent $p$-group, then the exponent $\exp(ν(G))$ divides $p \cdot \exp(G)$ (Theorem B). We also study the weak commutativity construction of powerful $p$-groups (Theorem C). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1906_07830 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Non-abelian tensor square and related constructions of $p$-groups Bastos, Raimundo de Melo, Emerson Gonçalves, Nathália Nunes, Ricardo Group Theory 20D15, 20E06 Let $G$ be a group. We denote by $ν(G)$ a certain extension of the non-abelian tensor square $[G,G^φ]$ by $G \times G$. We prove that if $G$ is a finite potent $p$-group, then $[G,G^φ]$ and the $k$-th term of the lower central series $γ_k(ν(G))$ are potently embedded in $ν(G)$ (Theorem A). Moreover, we show that if $G$ is a potent $p$-group, then the exponent $\exp(ν(G))$ divides $p \cdot \exp(G)$ (Theorem B). We also study the weak commutativity construction of powerful $p$-groups (Theorem C). |
| title | Non-abelian tensor square and related constructions of $p$-groups |
| topic | Group Theory 20D15, 20E06 |
| url | https://arxiv.org/abs/1906.07830 |