# Definition:Abelian Group/Definition 1

## Definition

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 $G$.

## 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.

## 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).