The probability that the commutator equation [x,y]=g has solution in a finite group

Let G be a finite group. For g\in G, an ordered pair (x_۱,y_۱)\in G\times G is called a solution of the commutator equation [x,y]=g if [x_۱,y_۱]=g. We consider \rho_g(G)=\{(x,y)| x,y\in 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|^۲}. 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 ۲.

Keywords:

GAP , Alternating groups , Symmetric groups , Nilpotent groups