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 in G we consider rho^{n}_g(G)={(x,y)| x,yin G, [x^{n},y]=g} . Then the probability that the commutator equation [x^{n},y]=g has a solution in a finite group G, written , P^{n}_g(G) is equal to frac{|rho^{n}_{g}(G)|}{|G|^۲} . 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(G), for some finite groups G .

Keywords:

Finite groups , Nilpotent groups , commutativity degree , GAP