Group equals Center iff Abelian

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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


Sources