# 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 | \(\ds \forall x, y \in G:\) | \(\ds x + y \in G \) | |||||

\((\text G 1)\) | $:$ | 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-**, 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

This page may be the result of a refactoring operation.As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.In particular: Check Definition:Additive Group of Integers in the belowIf you have access to any of these works, then you are invited to review this list, and make any necessary corrections.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{SourceReview}}` from the code. |

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