TITLE: Understanding what the Sylow theorems say about $p$-groups
QUESTION [5 upvotes]: I have a simple question.
If we consider a group $G$ with order $p^k$ for a prime $p$. For example $125=5^3$.
What we can obtain from sylows theorem? (I already understood it for the other cases, where we have something like $p^km$).
Thanks.

REPLY [6 votes]: I think Morgan has answered your question completely. So even though Sylow Theorems don't give new information about $p$-groups, there is a lemma in the proof of Sylow Theorems that you might find interesting:
One way to prove Sylow Theorems (at least the first Sylow theorem) is to use the fact that $[G:H]\equiv [N_{G}(H):H] \textrm{ (mod } p)$ whenever $H$ is a $p$-subgroup of $G$. To prove this lemma, consider the canonical action of $H$ on the set of left-cosets $G/H$, and look at the fixed points.
When $G=p^{m}$ itself a $p$-group, you can apply this lemma to conclude that $G$ has a normal subgroup of order $p^{m-1}$. The congruence $[G:H]\equiv [N_{G}(H):H] \textrm{ (mod } p)$ in general proves that normalizer of a proper subgroup $H$ of $p$-group is strictly larger than $H$. Hence, $p$-groups are nilpotent.