$p$-nilpotency criteria for some verbal subgroups
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866911241430106112 |
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| author | Rojas, Yerko Contreras Grazian, Valentina Monetta, Carmine |
| author_facet | Rojas, Yerko Contreras Grazian, Valentina Monetta, Carmine |
| contents | Let $G$ be a finite group, let $p$ be a prime and let $w$ be a group-word. We say that $G$ satisfies $P(w,p)$ if the prime $p$ divides the order of $xy$ for every $w$-value $x$ in $G$ of $p'$-order and for every non-trivial $w$-value $y$ in $G$ of order divisible by $p$. If $k \geq 2$, we prove that the $k$th term of the lower central series of $G$ is $p$-nilpotent if and only if $G$ satisfies $P(γ_k,p)$. In addition, if $G$ is soluble, we show that the $k$th term of the derived series of $G$ is $p$-nilpotent if and only if $G$ satisfies $P(δ_k,p)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_14474 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | $p$-nilpotency criteria for some verbal subgroups Rojas, Yerko Contreras Grazian, Valentina Monetta, Carmine Group Theory 20D12, 20F15 Let $G$ be a finite group, let $p$ be a prime and let $w$ be a group-word. We say that $G$ satisfies $P(w,p)$ if the prime $p$ divides the order of $xy$ for every $w$-value $x$ in $G$ of $p'$-order and for every non-trivial $w$-value $y$ in $G$ of order divisible by $p$. If $k \geq 2$, we prove that the $k$th term of the lower central series of $G$ is $p$-nilpotent if and only if $G$ satisfies $P(γ_k,p)$. In addition, if $G$ is soluble, we show that the $k$th term of the derived series of $G$ is $p$-nilpotent if and only if $G$ satisfies $P(δ_k,p)$. |
| title | $p$-nilpotency criteria for some verbal subgroups |
| topic | Group Theory 20D12, 20F15 |
| url | https://arxiv.org/abs/2105.14474 |