I had to find two 5-cycles in $A_5$ that are not conjugate. I believe I found two, namely $a=(12345)$ and $b=(21345)$.
It wasn't part of the exercise, but I'd like to prove that they are not conjugate. But I'm not sure where to start. I think we have to start with that they are conjugate in $S_5$, so there exists a $g \in S_5$ such that $gag^{-1}=b$. And then prove all the $g's$ for which that is true are $\notin A_5$?
Someone stated this as a duplicate, but I don't think it is, because they don't really ask for/ give the algebraic proof that I'm looking for.