# Math Help - prove permutation is a cycle

1. ## prove permutation is a cycle

Let p be a prime number. Prove that every power of a p-cycle is either a p-cycle or (1).

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