Center of Group is Abelian Subgroup

Theorem

Let $G$ be a group.

Let $\map Z G$ be the center of $G$.

Then $\map Z G$ is an abelian subgroup of $G$.

Proof

By Center of Group is Subgroup, $\map Z G$ is a subgroup of $G$.

The definition of the center $\map Z G$ grants that all elements of $\map Z G)$ commute with all elements of $G$.

In particular, all elements of $\map Z G$ commute with all elements of $\map Z G$ as $\map Z G \subseteq G$.

Therefore $\map Z G$ is abelian.

$\blacksquare$

Also see

• Group equals Center iff Abelian

Sources

• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.9$: Exercise $5.14$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Examples of groups $\text{(vii)}$