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.