# Definition:Abelian Group/Definition 2

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## Definition

An **abelian group** is a group $G$ if and only if:

- $G = \map Z G$

where $\map Z G$ is the center of $G$.

## 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.

## 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

- 1967: John D. Dixon:
*Problems in Group Theory*... (previous) ... (next): Introduction: Notation