Finite groups, commuting probability, and coprime automorphisms
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908641666269184 |
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| author | Detomi, Eloisa Guralnick, Robert M. Morigi, Marta Shumyatsky, Pavel |
| author_facet | Detomi, Eloisa Guralnick, Robert M. Morigi, Marta Shumyatsky, Pavel |
| contents | Given two subgroups $H,K$ of a finite group $G$, the probability that a pair of random elements from $H$ and $K$ commutes is denoted by $Pr(H,K)$. Suppose that a finite group $G$ admits a group of coprime automorphisms $A$ and let $ε>0$. We show that, if for any distinct primes $p,q\inπ(G)$ there is an $A$-invariant Sylow $p$-subgroup $P$ and an $A$-invariant Sylow $q$-subgroup $Q$ of $G$ for which $Pr([P,A],[Q,A])\geε$, then $F_2([G,A])$ has $ε$-bounded index in $[G,A]$ (Theorem 1.2). Here $F_2(K)$ stands for the second term of the upper Fitting seris of a group $K$. We also show that, if $G=[G,A]$ and for any prime $p$ dividing the order of $G$ there is an $A$-invariant Sylow $p$-subgroup $P$ such that $\Pr([P,A], [P,A]^x)\geqε$ for all $x\in G$, then $G$ is bounded-by-abelian-by-bounded (Theorem 1.4). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07597 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite groups, commuting probability, and coprime automorphisms Detomi, Eloisa Guralnick, Robert M. Morigi, Marta Shumyatsky, Pavel Group Theory 20D20, 20D45, 20P05 Given two subgroups $H,K$ of a finite group $G$, the probability that a pair of random elements from $H$ and $K$ commutes is denoted by $Pr(H,K)$. Suppose that a finite group $G$ admits a group of coprime automorphisms $A$ and let $ε>0$. We show that, if for any distinct primes $p,q\inπ(G)$ there is an $A$-invariant Sylow $p$-subgroup $P$ and an $A$-invariant Sylow $q$-subgroup $Q$ of $G$ for which $Pr([P,A],[Q,A])\geε$, then $F_2([G,A])$ has $ε$-bounded index in $[G,A]$ (Theorem 1.2). Here $F_2(K)$ stands for the second term of the upper Fitting seris of a group $K$. We also show that, if $G=[G,A]$ and for any prime $p$ dividing the order of $G$ there is an $A$-invariant Sylow $p$-subgroup $P$ such that $\Pr([P,A], [P,A]^x)\geqε$ for all $x\in G$, then $G$ is bounded-by-abelian-by-bounded (Theorem 1.4). |
| title | Finite groups, commuting probability, and coprime automorphisms |
| topic | Group Theory 20D20, 20D45, 20P05 |
| url | https://arxiv.org/abs/2511.07597 |