Let p be a prime number. Prove that every power of a p-cycle is either a p-cycle or (1).
Let $\displaystyle \sigma$ be a $\displaystyle p$-cycle. Then you can form the subgroup (of the symmetric group) $\displaystyle \left< \sigma\right> $. Now $\displaystyle \sigma^n \in \left< \sigma \right>$. Since $\displaystyle \left< \sigma \right>$ is a cyclic group of prime order it means the order of $\displaystyle \sigma^n$ is either $\displaystyle 1$ or $\displaystyle p$. If it is $\displaystyle 1$ then $\displaystyle \sigma^n = (1)$. Otherwise we know we can factor $\displaystyle \sigma^n = \tau_1\cdot ... \tau_k$ where $\displaystyle \tau_i$'s are disjoint cycles, if $\displaystyle m_i$ is length of $\displaystyle \tau_i$ then $\displaystyle p=\text{ord}(\sigma^n) = \text{lcm}(m_1,...,m_k)$. But since $\displaystyle m_i\not | p$ if $\displaystyle k>1$ (because it is prime) it means it is not possible for $\displaystyle \text{lcm}(m_1,...,m_k) = p$. Therefore $\displaystyle k=1$ and so $\displaystyle \sigma^n$ must be a $\displaystyle p$-cycle.