---
tag: 071DDFVG4J
kind: definition
name: Abelian group
msc: [20Kxx]
defines: 082GHNPJ53
uses: [G5S1M9, 04E4V9G5FR]
sources:
  - url: https://en.wikipedia.org/wiki/Abelian_group
    locator: Definition
---

## Definition

An abelian group is a [[group|G5S1M9]] $(G, \ast)$ whose operation is [[commutative|099E72KTF8]].
Thus $a \ast b = b \ast a$ for all $a, b \in G$.

## Remarks

The integers, the rationals and the real numbers with addition are abelian groups.
The operation of an abelian group is often written $+$, with identity $0$ and inverse $-a$.
The group of invertible $2 \times 2$ real matrices under multiplication is not abelian.

## Explanations

### L1 human

In an abelian group, the order of two elements in a product does not matter.
Addition of numbers is the model: $a + b = b + a$.
The name honours Niels Henrik Abel.