THE PROBABILITY THAT THE COMMUTATOR EQUATION [x, y] = g HAS SOLUTION IN A FINITE GROUP

Let G be a finite group. For g ∈ G, an ordered pair (x1, y1) ∈ G × G is called a solution of the commutator equation [x, y] = g if [x1, y1] = g. We consider ρg(G) = {(x, y)|x, y ∈ G, [x, y] = g}, then the probability that the commutator equation [x, y] = g has solution in a finite group G, written Pg(G), is equal to |ρg(G)| |G| 2 . In this paper, we present two methods for the computing Pg(G). First by GAP, we calculate Pg(G) for G = An, Sn and g ∈ G. 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 ≡ t(mod n), we derive formulas for calculating the probability of ρg(G) where G = Hm, Gm, Km and g ∈ G.