Prove that two elements of $S_{n}$ are conjugate if and only if they have the same cycle type.

Describe (without proof) a necessary and sufficient condition for a permutation $\sigma \in A_{n}$ to have the same conjugacy class in $A_{n}$ as it has in $S_{n}$.

For which $\sigma \in S_{n}$ is $\sigma$ conjugate (in $S_{n}$ ) to $\sigma^{2} ?$

For every $\sigma \in A_{5}$, show that $\sigma$ is conjugate to $\sigma^{-1}$ (in $\left.A_{5}\right)$. Exhibit a positive integer $n$ and a $\sigma \in A_{n}$ such that $\sigma$ is not conjugate to $\sigma^{-1}$ (in $A_{n}$ ).