Sylow's theorem proof Sylow's theorems state that the $p$-Sylow subgroups exist for a group G of order $p^km$, where $p$ is prime and does not divide $m$. My question is how to prove that there is at least 1 subgroup of order $p^n$ exists for every non-negative integer $n \le k$.

Actually, there exists a normal subgroup of order $p^n$ for any group of order $p^k$, where $n\leq k$. We know that the center of a nontrivial $p$-group is nontrivial, then you can use induction to prove the fact.
