# If $sigma in A_n$ has a split conjugacy class, then $tau sigma tau^{-1}$ is not conjugate to $sigma$ for odd $tau$

Mathematics Asked by jskattt797 on November 27, 2020

Let $$sigma in A_n$$ such that the disjoint cycle decomposition of $$sigma$$ contains no even cycles or cycles of the same length. Although there is one conjugacy class of $$sigma$$ in $$S_n$$, there are two conjugacy classes of $$sigma$$ in $$A_n$$. We can choose any permutation in the original $$S_n$$ conjugacy class to represent one of the two $$A_n$$ classes. But how can we find another permutation to represent the other $$A_n$$ class?
These answers suggest the following to be true:

Let $$sigma in A_n$$. If the conjugacy class of $$sigma$$ splits in $$A_n$$, then $$tau sigma tau^{-1}$$ is not conjugate (in $$A_n$$) to $$sigma$$ for all $$tau in S_n setminus A_n$$.

What is a proof of this fact?

From here we know that $$tau sigma tau^{-1} neq sigma$$ for all odd $$tau$$. But proving $$tau sigma tau^{-1}$$ is not conjugate (in $$A_n$$) to $$sigma$$ seems to be a stronger claim.