Group equals Center iff Abelian

Theorem

Let $G$ be a group.

Then $G$ is abelian if and only if $\map Z G = G$, that is, if and only if $G$ equals its center.

Proof

Necessary Condition

Let $G$ be abelian.

Then:

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

Thus:

$\forall a \in G: a \in \map Z G = G$

$\Box$

Sufficient Condition

Let $\map Z G = G$.

Then by the definition of center:

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

and thus $G$ is abelian by definition.

$\blacksquare$

Also see

• Center of Group is Abelian Subgroup

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Examples of groups $\text{(vii)}$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: conjugacy class