Q:

Assume $G$ is abelian. Prove that $p$ divides $|Z(G)|$.

Let $G$ be a group and let $p$ be a positive prime number. Suppose $|G| = p^
n$. for some positive integer $n$.
I know to be abelian you have to be commutative, where $ab=ba$ and $a,b \in$ a group. And we have that the center:  $Z(G) = \{z ∈ G : ∀g \in G, zg = gz\}$. Its looks so easy to connect but I am not sure how to connect.

A:

Since $G$ is abelian, $G=Z(G)$, so it's obvious.
But I want to tell you that even if $G$ is not abelian the theorem also holds.
It follows from the Class Equation.

