Symmetric groups and conjugacy classes

TL;DR

This paper investigates the structure of conjugacy class products in symmetric groups, proving they are never a single conjugacy class and characterizing when they form unions of two or more classes.

Contribution

It provides new results on the behavior of conjugacy class products in symmetric groups, including conditions for when these products form unions of multiple classes.

Findings

Product of two nontrivial conjugacy classes is never a single class.

If n is not divisible by 2 or 3, the product is at least three classes.

Characterization of elements when the product is exactly two classes.

Abstract

Let S_n be the symmetric group on n-letters. Fix n>5. Given any nontrivial $\alpha,\beta\in S_n$, we prove that the product $\alpha^{S_n}\beta^{S_n}$ of the conjugacy classes $\alpha^{S_n}$ and $\beta^{S_n}$ is never a conjugacy class. Furthermore, if n is not even and $n$ is not a multiple of three, then $\alpha^{S_n}\beta^{S_n}$ is the union of at least three distinct conjugacy classes. We also describe the elements $\alpha,\beta\in S_n$ in the case when $\alpha^{S_n}\beta^{S_n}$ is the union of exactly two distinct conjugacy classes.