Power-conjugate equations in symmetric groups

TL;DR

This paper studies solutions to conjugate equations in symmetric groups, revealing how the existence of solutions depends on the exponent and the fixed element involved.

Contribution

It provides a detailed analysis of conjugate equations in symmetric groups, highlighting conditions for the existence of solutions based on group element types and exponents.

Findings

Existence of solutions depends on the exponent e and the type of fixed element a.

Certain types of a lead to non-trivial solutions for specific exponents.

The results characterize when conjugate equations have solutions in symmetric groups.

Abstract

We investigate the solutions of the conjugate equation aya^(-1)=y^e in the symmetric group S_{n}. Here a is a fixed (constant), e is an integer exponent and y is a single unknown permutation (in S_{n}). It turns out that the existence of a non-trivial solution y heavily depends on e and the type of a.