Group equals Center iff Abelian

Theorem
Let $$G$$ be a group.

Then $$G$$ is abelian iff $$Z \left({G}\right) = G$$, that is, if $$G$$ equals its center.

Proof

 * Let $$G$$ be abelian.

Then $$\forall a \in G: \forall x \in G: a x = x a$$.

Thus $$\forall a \in G: a \in Z \left({G}\right) = G$$.


 * Let $$Z \left({G}\right) = G$$.

Then by the definition of center, $$\forall a \in G: \forall x \in G: a x = x a$$ and thus $$G$$ is abelian.