Center of Abelian Group is Whole Group

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.

$\blacksquare$

Sources

• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 37.3$