Equivalence of Definitions of Abelian Group

Theorem

The following definitions of the concept of Abelian Group are equivalent:

Definition 1

An abelian group is a group $G$ where:

$\forall a, b \in G: a b = b a$

That is, every element in $G$ commutes with every other element in $G$.

Definition 2

An abelian group is a group $G$ if and only if:

$G = \map Z G$

where $\map Z G$ is the center of $G$.

Proof

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.

$\Box$

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.

$\blacksquare$