Let $g ∈ S_n$ be a permutation. Describe a method for answering the following question: does there exist a permutation $f ∈ S_n$ such that $f ◦ f = g$?
I don't want to be spoonfed the answer, but can someone give me a direction? It is getting late and my brain is running on fumes, so I can't think of a point to start.
Start by taking a cycle decomposition of $g$. What needs to be true of the cycles?