# Definition:Abelian Group

## Definition

### 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 $G$.

## Additive Notation

When discussing abelian groups, it is customary to use additive notation, where:

- $x + y$ is used to indicate the result of the operation $+$ on $x$ and $y$.

- $e$ or $0$ is used for the identity element. Note that in this context, $0$ is
**not**a zero element.

- $-x$ is used for the inverse element.

- $n x$ is used to indicate the $n$th power of $x$.

## Abelian Group Axioms

Under this regime, the group axioms read:

\((\text G 0)\) | $:$ | Closure | \(\displaystyle \forall x, y \in G:\) | \(\displaystyle x + y \in G \) | ||||

\((\text G 1)\) | $:$ | Associativity | \(\displaystyle \forall x, y, z \in G:\) | \(\displaystyle x + \paren {y + z} = \paren {x + y} + z \) | ||||

\((\text G 2)\) | $:$ | Identity | \(\displaystyle \exists 0 \in G: \forall x \in G:\) | \(\displaystyle 0 + x = x = x + 0 \) | ||||

\((\text G 3)\) | $:$ | Inverse | \(\displaystyle \forall x \in G: \exists \paren {-x}\in G:\) | \(\displaystyle x + \paren {-x} = 0 = \paren {-x} + x \) | ||||

\((\text C)\) | $:$ | Commutativity | \(\displaystyle \forall x, y \in G:\) | \(\displaystyle x + y = y + x \) |

This notation gains in importance and usefulness when discussing rings.

## Also known as

The usual way of spelling **abelian group** is without a capital letter, but **Abelian** is frequently seen.

The term **commutative group** can occasionally be seen.

## Also see

- Results about
**abelian groups**can be found here.

## Source of Name

This entry was named for Niels Henrik Abel.

## Historical Note

The importance of **abelian groups** was discovered by Niels Henrik Abel during the course of his investigations into the theory of equations.

## Linguistic Note

The pronunciation of **abelian** in the term **abelian group** is usually either **a- bee-lee-an** or

**a-**, putting the emphasis on the second syllable.

*bell*-ee-anNote that the term **abelian** has thus phonetically lost the connection to its eponym Abel (correctly pronounced ** aah-bl**).

## Sources

- 1992: William A. Adkins and Steven H. Weintraub:
*Algebra: An Approach via Module Theory*... (previous) ... (next): $\S 1.1$: Example $1$