The Relative n-th Commutativity Degree of $2$-Generator $p$-Groups of Nilpotency Class Two

Abstract

Let $H$ be a subgroup of a finite group $G$. The relative $n$-th commutativity degree, denoted as $P_n(H,G)$, is the probability of commuting the $n$-th power of a random element of $H$ with an element of $G$. Obviously, if $H=G$ then the relative $n$-th commutativity degree coincides with the $n$-th commutativity degree, $P_n(G)$. The purpose of this article is to compute the explicit formula for $P_n(G)$, where $G$ is a 2-generator $p$-group of nilpotency class two. Furthermore, we observe that if we have two pairs of relative isoclinic groups, then they have equal relative $n$-th commutativity degree.