$σ$-properties of finite groups in polynomial time
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909220421500928 |
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| author | Murashka, Viachaslau I. |
| author_facet | Murashka, Viachaslau I. |
| contents | Let $H, K$ be subgroups of the permutation group $G$ of degree $n$ with $K\trianglelefteq G$ and $σ$ be a partition of the set of all different prime divisors of $|G/K|$. We prove that in polynomial time (in $n$) one can check $G/K$ for $σ$-nilpotency and $σ$-solubility; $H/K$ for $σ$-subnormality and $σ$-$p$-permutability in $G/K$. Moreover one can find the least partition $σ$ of $π(G/K)$ for which $G/K$ is $σ$-nilpotent. Also one can find the least partition $σ$ of $π(G/K)$ for which $H/K$ is $σ$-$p$-permutable in $G/K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06466 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $σ$-properties of finite groups in polynomial time Murashka, Viachaslau I. Group Theory 20D20, 20B40 Let $H, K$ be subgroups of the permutation group $G$ of degree $n$ with $K\trianglelefteq G$ and $σ$ be a partition of the set of all different prime divisors of $|G/K|$. We prove that in polynomial time (in $n$) one can check $G/K$ for $σ$-nilpotency and $σ$-solubility; $H/K$ for $σ$-subnormality and $σ$-$p$-permutability in $G/K$. Moreover one can find the least partition $σ$ of $π(G/K)$ for which $G/K$ is $σ$-nilpotent. Also one can find the least partition $σ$ of $π(G/K)$ for which $H/K$ is $σ$-$p$-permutable in $G/K$. |
| title | $σ$-properties of finite groups in polynomial time |
| topic | Group Theory 20D20, 20B40 |
| url | https://arxiv.org/abs/2406.06466 |