# 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 $G$.

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 $\ds \forall x, y \in G:$ $\ds x + y \in G$ $(\text G 1)$ $:$ Associativity $\ds \forall x, y, z \in G:$ $\ds x + \paren {y + z} = \paren {x + y} + z$ $(\text G 2)$ $:$ Identity $\ds \exists 0 \in G: \forall x \in G:$ $\ds 0 + x = x = x + 0$ $(\text G 3)$ $:$ Inverse $\ds \forall x \in G: \exists \paren {-x}\in G:$ $\ds x + \paren {-x} = 0 = \paren {-x} + x$ $(\text C)$ $:$ Commutativity $\ds \forall x, y \in G:$ $\ds x + y = y + x$

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-bell-ee-an, putting the emphasis on the second syllable.

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