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.