Powers in the wreath product of $G$ with $S_n$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866910167424040960 |
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| author | Kundu, Rijubrata Mondal, Sudipa |
| author_facet | Kundu, Rijubrata Mondal, Sudipa |
| contents | In this paper we compute powers in the wreath product $G\wr S_n$, for any finite group $G$. For $r\geq 2$, a prime, consider $ω_r: G\wr S_n\to G\wr S_n$ defined by $g \mapsto g^r$. Let $P_{r}(G\wr S_n)=\frac{|ω_r(G\wr S_n)|}{|G|^n n!}$, be the probability that a randomly chosen element in $G\wr S_n$ is a $r^{th}$ power. We prove, $P_r(G\wr S_{n+1})=P_r(G\wr S_n)$ for all $n\not \equiv -1(\text{mod } r)$ if, order of $G$ is coprime to $r$. We also give a formula for the number of conjugacy classes that are $r^{th}$ powers in $G\wr S_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_04954 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Powers in the wreath product of $G$ with $S_n$ Kundu, Rijubrata Mondal, Sudipa Group Theory Combinatorics 20B05, 05A15 (Primary) 20P05, 20B30 (Secondary) In this paper we compute powers in the wreath product $G\wr S_n$, for any finite group $G$. For $r\geq 2$, a prime, consider $ω_r: G\wr S_n\to G\wr S_n$ defined by $g \mapsto g^r$. Let $P_{r}(G\wr S_n)=\frac{|ω_r(G\wr S_n)|}{|G|^n n!}$, be the probability that a randomly chosen element in $G\wr S_n$ is a $r^{th}$ power. We prove, $P_r(G\wr S_{n+1})=P_r(G\wr S_n)$ for all $n\not \equiv -1(\text{mod } r)$ if, order of $G$ is coprime to $r$. We also give a formula for the number of conjugacy classes that are $r^{th}$ powers in $G\wr S_n$. |
| title | Powers in the wreath product of $G$ with $S_n$ |
| topic | Group Theory Combinatorics 20B05, 05A15 (Primary) 20P05, 20B30 (Secondary) |
| url | https://arxiv.org/abs/2010.04954 |