Power-conjugate equations in symmetric groups

Abstract

First we consider the solutions of the general “cubic” equation (with ) in the symmetric group . In certain cases this equation can be rewritten as or as , where depends on the αi ’s and the new unknown permutation is a product of x (or ) and one of the permutations . Using combinatorial arguments and some basic number theoretical facts, we obtain results about the solutions of the so-called power-conjugate equation in , where is an integer exponent. A divisibility condition involving the type of α provide solutions with and a further condition gives the complete list of solutions. Some other divisibility assumptions concerning the type of α ensure that the solutions of are exactly the solutions of in the centralizer of α. Slightly stronger assumptions provide a complete answer to the question, when our equation has only the trivial solution y = 1.