I have difficulties proving the following theorem:
Let $σ ∈ S_n$ satisfy $σ \neq (1)$, and $σ^p = (1)$, where p is a prime number such that $\frac{n}{2} < p ≤ n$. Prove that σ is a p-cycle.
There are many similar questions here, however they are all slightly different from this theorem. Obviously, if $\sigma$ is a p-cycle, then $σ^p = (1)$, but I'm not sure how to use $\frac{n}{2} < p ≤ n$ to prove that no other length of the cycle is possible.
Thanks in advance!