Probability of Mutually Commuting Two Finite Subsets of a Finite Group

Abstract

For a finite group $G$, let $P(m,n,G)$ denote the probability that a $m$-subset and an $n$-subset of $G$ commute elementwise, and let $P(n,G)=P(1,n,G)$ be the probability that an element commutes with an $n$-subset of $G$. Some lower and upper bounds are given for $P(m,n,G)$, and it is shown that $\{P(m,n,G){\}}_{m,n}$ is decreasing with respect to $m$ and $n$. Also, $P(m,n,G)$ is computed for some classes of finite groups, including groups with a central factor of order $p^2$ and $P(n,G)$ is computed for groups with a central factor of order $p^3$ and wreath products of finite abelian groups.