s. a. gorjian

Let G be a finite group. For gin G, an ordered pair $(x_1,y_1)in Gtimes G$ is called a solution of the commutator equation $[x,y]=g$ if $[x_1,y_1]=g$. We consider rho_g(G)={(x,y)| x,yin G, [x,y]=g}, then the probability that the commutator equation $[x,y]=g$ has solution in a finite group $G$, written P_g(G), is equal to frac{|rho_{g}(G)|}{|G|^2}. In this paper, we present two methods for the computing P_g(G). First by $GAP, we give certain explicit formulas for P_g(A_n) and P_g(S_n). Also we note that this method can be applied to any group of small order. Then by using the numerical solutions of the equation xy-zu equiv t (mod~n), we derive formulas for calculating the probability of $rho_g(G)$ where $G$ is a two generated group of nilpotency class 2.the formula is not displayed correctly!