A generalization of the n^th- commutativity degree in finite groups

In this paper, we study the number of solutions of commutator equation [x^{n},y]=gin two classes of finite groups. For g∈G we consider p^{n}_G={(x,y)| x,∈G, [x^{n},y] . Then the probability that the commutator equation [x^{n},y] has a solution in a finite group G, written , P^{n}_G is equal to p^{n}_G|G|^2} . By using the numerical solutions of the equation xy - zu equiv t(bmod~n) we derive formulas for calculating the probability of P^{n}_G, for some finite groups G .