Probability that the commutator of two group elements is equal to a given element

In this paper we study the probability that the commutator of two randomly chosen elements in a finite group is equal to a given element of that group. Explicit computations are obtained for groups GG which |G′||G′| is prime and G′≤Z(G)G′≤Z(G) as well as for groups GG which |G′||G′| is prime and G′∩Z(G)=1G′∩Z(G)=1. This paper extends results of Rusin [see D.J. Rusin, What is the probability that two elements of a finite group commute? Pacific J. Math. 82 (1) (1979) 237–247].