Group equals Center iff Abelian

Theorem
Let $G$ be a group.

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

Necessary Condition
Let $G$ be abelian.

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

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

Sufficient Condition
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.

Also see

 * Center of Group is Abelian Subgroup