A nilpotency criterion for some verbal subgroups
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866917055627788288 |
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| author | Monetta, Carmine Tortora, Antonio |
| author_facet | Monetta, Carmine Tortora, Antonio |
| contents | The word $w=[x_{i_1},x_{i_2},\dots,x_{i_k}]$ is a simple commutator word if $k\geq 2, i_1\neq i_2$ and $i_j\in \{1,\dots,m\}$, for some $m>1$. For a finite group $G$, we prove that if $i_{1} \neq i_j$ for every $j\neq 1$, then the verbal subgroup corresponding to $w$ is nilpotent if and only if $|ab|=|a||b|$ for any $w$-values $a,b\in G$ of coprime orders. We also extend the result to a residually finite group $G$, provided that the set of all $w$-values in $G$ is finite. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1812_02123 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | A nilpotency criterion for some verbal subgroups Monetta, Carmine Tortora, Antonio Group Theory 20F18, 20F45 The word $w=[x_{i_1},x_{i_2},\dots,x_{i_k}]$ is a simple commutator word if $k\geq 2, i_1\neq i_2$ and $i_j\in \{1,\dots,m\}$, for some $m>1$. For a finite group $G$, we prove that if $i_{1} \neq i_j$ for every $j\neq 1$, then the verbal subgroup corresponding to $w$ is nilpotent if and only if $|ab|=|a||b|$ for any $w$-values $a,b\in G$ of coprime orders. We also extend the result to a residually finite group $G$, provided that the set of all $w$-values in $G$ is finite. |
| title | A nilpotency criterion for some verbal subgroups |
| topic | Group Theory 20F18, 20F45 |
| url | https://arxiv.org/abs/1812.02123 |