A "power" conjugate equation in the symmetric group

TL;DR

This paper investigates solutions to a class of conjugate equations in the symmetric group, providing conditions under which solutions correspond to elements satisfying specific power relations within centralizers.

Contribution

It introduces a framework for solving power conjugate equations in symmetric groups, linking solutions to centralizer properties and extending previous understanding of such equations.

Findings

Solutions are characterized by elements satisfying y^{e-1}=1 in the centralizer of a.

Reformulation of equations as conjugation relations involving powers of permutations.

Conditions identified under which solutions can be explicitly described.

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.