Document Type : Original Article


Department of pure Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran



In this paper, we study the number of solutions of commutator equation [x^{n},y]=g

in 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 $frac{|rho^{n}_{g}(G)|}{|G|^2}$ . By using the numerical solutions of the equation $xy - zu equiv t(bmod~n)$ we derive formulas for calculating the probability of $P^{n}_g(G)$, for some finite groups $G$ .