Symmetric groups and conjugacy 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.

7 pages