Some Considerations on the \(n\)-th Commutativity Degrees of Finite Groups

Abstract

Let \(G\) be a finite group and \(n\) a positive integer. The \(n\)-th commutativity degree \(P_n(G)\) of \(G\) is the probability that the \(n\)-th power of a random element of \(G\) commutes with another random element of \(G\). In 1968, P. Erdős and P. Turán investigated the case \(n = 1\), involving only methods of combinatorics. Later, several authors improved their studies and there is a growing literature on the topic in the last 10 years. We introduce the relative \(n\)-th commutativity degree \(P_n(H,G)\) of a subgroup \(H\) of \(G\). This is the probability that an \(n\)-th power of a random element in \(H\) commutes with an element in \(G\). The influence of \(P_n(G)\) and \(P_n(H,G)\) on the structure of \(G\) is the purpose of the present work.