A "power" conjugate equation in the symmetric group

Abstract: First we consider the solutions of the general "cubic" equation a_{1}x^{r1}a_{2}x^{r2}a_{3}x^{r3}=1 (with r1,r2,r3 in {1,-1}) in the symmetric group S_{n}. In certain cases this equation can be rewritten as aya^{-1}=y^{2} or as aya^{-1}=y^{-2}, where a in S_{n} depends on the a_{i}'s and the new unknown permutation y in S_{n} is a product of x (or x^{-1}) and one of the permutations a_{i}^{1} and a_{i}^{-1}. Using combinatorial arguments and some basic number theoretical facts, we obtain results about the solutions of the so-called power conjugate equation aya^{-1}=y^{e} in S_{n}, where e is an integer exponent. Under certain conditions, the solutions are exactly the solutions of y^{e-1}=1 in the centralizer of a.