# Equivalence of Definitions of Abelian Group

## Contents

## 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.

$\blacksquare$

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 37.3$ Some important general examples of subgroups