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.