---
type: "concept"
title: "Group"
slug: "group"
language: "en"
translationGroupId: "cmqw3fxg10001s001zo8wpbnm"
domain: "Algebra"
status: "usable"
aliases: []
lastEditedBy: "ancient-tree"
---

A group is a [[Set|set]] $G$ equipped with a [[Binary operation|binary operation]] $*:G\times G \rightarrow G$ satisfying the following properties :
1) Associativity : for all $a,b,c\in G$, $(a*b)*c=a*(b*c)$.
2) Identity element : there exists an element $e\in G$ such that $e*a=a*e=a$ for every $a\in G$.
3) Inverses : for every $a\in G$, there exists an element $a^{-1}\in G$ such that $a*a^{-1}=a^{-1}*a=e$.

##### Remarks
- When the context is clear, the operation $*$ is often ommited, writing $ab$ for $a*b$.