Commuting probability for the Sylow subgroups of a profinite group
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917778404933632 |
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| author | Detomi, Eloisa Morigi, Marta Shumyatsky, Pavel |
| author_facet | Detomi, Eloisa Morigi, Marta Shumyatsky, Pavel |
| contents | Given two subgroups $H,K$ of a compact group $G$, the probability that a random element of $H$ commutes with a random element of $K$ is denoted by $Pr(H,K)$.
We show that if $G$ is a profinite group containing a Sylow $2$-subgroup $P$, a Sylow $3$-subgroup $Q_3$ and a Sylow $5$-subgroup $Q_5$ such that $Pr(P,Q_3)$ and $Pr(P,Q_5)$ are both positive, then $G$ is virtually prosoluble (Theorem 1.1).
Furthermore, if $G$ is a prosoluble group in which for every subset $π\subseteqπ(G)$ there is a Hall $π$-subgroup $H_π$ and a Hall $π'$-subgroup $H_{π'}$ such that $Pr(H_π,H_{π'})>0$, then $G$ is virtually pronilpotent (Theorem 1.2). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11165 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Commuting probability for the Sylow subgroups of a profinite group Detomi, Eloisa Morigi, Marta Shumyatsky, Pavel Group Theory Given two subgroups $H,K$ of a compact group $G$, the probability that a random element of $H$ commutes with a random element of $K$ is denoted by $Pr(H,K)$. We show that if $G$ is a profinite group containing a Sylow $2$-subgroup $P$, a Sylow $3$-subgroup $Q_3$ and a Sylow $5$-subgroup $Q_5$ such that $Pr(P,Q_3)$ and $Pr(P,Q_5)$ are both positive, then $G$ is virtually prosoluble (Theorem 1.1). Furthermore, if $G$ is a prosoluble group in which for every subset $π\subseteqπ(G)$ there is a Hall $π$-subgroup $H_π$ and a Hall $π'$-subgroup $H_{π'}$ such that $Pr(H_π,H_{π'})>0$, then $G$ is virtually pronilpotent (Theorem 1.2). |
| title | Commuting probability for the Sylow subgroups of a profinite group |
| topic | Group Theory |
| url | https://arxiv.org/abs/2409.11165 |