Equivalence of Definitions of Abelian Group

Definition 1 implies Definition 2
Let $G$ be an abelian group by definition 1.

Then by definition:
 * $\forall a \in G: \forall x \in G: a x = x a$

Thus:
 * $\forall a \in G: a \in \map Z G$

By definition of subset:
 * $G \subseteq \map Z G$

By definition of center:
 * $\map Z G \subseteq G$

So by definition of set equality:
 * $G = \map Z G$

Thus $G$ is an abelian group by definition 2.

Definition 2 implies Definition 1
Let $G$ be an abelian group by definition 2.

Then by definition:
 * $G = \map Z G$

So by the definition of center:
 * $\forall a \in G: \forall x \in G: a x = x a$

Thus $G$ is an abelian group by definition 1.