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.