I'm looking to show that for any integer $n > 1$, every element of order $n$ in $S_n$ is an $n$-cycle if and only if $n = p^r$ for some prime $p$ and some integer $r \geq 1$.
I am very new to group theory and I believe I need only surface level knowledge to prove this but I'm very stuck with how to start.
Hint:
Suppose $n=p^r$. Let $\sigma$ be an element of order $n$. We can write $$\sigma=\alpha_1\dots\alpha_k$$ where $\alpha_i$'s are disjoint cycles. Then note that $$p^r=|\sigma|=\text{lcm}(|\alpha_1|,\dots,|\alpha_k|).$$ Also note that $|\alpha_i|$ divides $p^r$. Use this information to show that one of the $\alpha_i$ must be $n$-cycle while the rest must be identity.
Conversely, suppose to the contrary that $n=p^rq$ where $\gcd(p,q)=1$ and $q>1$. Find an element of order $n$ which is not an $n$-cycle. The suitable choice will be the product of a $p^r$-cycle and a $q$-cycle.