Exercise 4.3.22 (If $n$ is odd then the set of all $n$-cycles consists of two conjugacy classes of equal size in $A_n$)

Proof. Assume that $n$ is odd. The set of all $n$-cycles is the conjugacy class $𝒦$ of the $n$-cycle $\sigma =(12\dots n)$ in $S_n$, where $\sigma \in A_n$ since $n$ is odd. The cycle type of $\sigma$ is given by $(n)$, so the cycle type of $\sigma$ consists of distinct odd integers. By Exercise 21, $𝒦$ consists of two conjugacy classes in $A_n$ (of equal size by Exercise 19).

If $n$ is odd then the set of all $n$-cycles consists of two conjugacy classes of equal size in $A_n$. □