Finite groups with high commuting probability for Sylow subgroups
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arXiv
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| Format: | Preprint |
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2026
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| author | Detomi, Eloisa Senise, Débora Shumyatsky, Pavel |
| author_facet | Detomi, Eloisa Senise, Débora Shumyatsky, Pavel |
| contents | Given two subsets $X,Y$ of a finite group $G$, we write $\Pr(X,Y)$ for the probability that random elements $x \in X$ and $y \in Y$ commute. If $X,Y$ are subgroups, we denote by $\Pr^*(X,Y)$ the maximum real number $ε$ with the property that for every pair of distinct primes $p\inπ(X)$ and $q\inπ(Y)$ there is a Sylow $p$-subgroup $P$ of $X$ and a Sylow $q$-subgroup $Q$ of $Y$ such that $\Pr(P,Q) \geq ε$.
In this paper we handle, among other things, finite groups $G$ with high probabilities $\Pr^*(T,G)$, where $T$ is either a term of the lower central series of $G$ or the generalized Fitting subgroup $F_i^*(G)$. Our main results show that the structure of such groups is similar, in some precise sense, to that of nilpotent groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_22955 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Finite groups with high commuting probability for Sylow subgroups Detomi, Eloisa Senise, Débora Shumyatsky, Pavel Group Theory 20D20, 20D25, 20P05 Given two subsets $X,Y$ of a finite group $G$, we write $\Pr(X,Y)$ for the probability that random elements $x \in X$ and $y \in Y$ commute. If $X,Y$ are subgroups, we denote by $\Pr^*(X,Y)$ the maximum real number $ε$ with the property that for every pair of distinct primes $p\inπ(X)$ and $q\inπ(Y)$ there is a Sylow $p$-subgroup $P$ of $X$ and a Sylow $q$-subgroup $Q$ of $Y$ such that $\Pr(P,Q) \geq ε$. In this paper we handle, among other things, finite groups $G$ with high probabilities $\Pr^*(T,G)$, where $T$ is either a term of the lower central series of $G$ or the generalized Fitting subgroup $F_i^*(G)$. Our main results show that the structure of such groups is similar, in some precise sense, to that of nilpotent groups. |
| title | Finite groups with high commuting probability for Sylow subgroups |
| topic | Group Theory 20D20, 20D25, 20P05 |
| url | https://arxiv.org/abs/2605.22955 |