Conjugacy Class of An

Proposition

Let $x\in A_n$ with $n>1$.

• If $x$ commutes with some odd permutation in $S_n$, then $x^{S_n}=x^{A_n}$.
• If $x$ does not commute with any odd permutation in $S_n$ then $x^{S_n}$ splits into two conjugacy classes in $A_n$ of equal size, with representatives $x$ and $(12{)}^{-1}x\left(\begin{array}{l}1\end{array}2\right)$.

\begin{proof} See |here. \end{proof}