---
tag: 0T13HM377F
kind: definition
name: Alternating group
msc: [20B30]
defines: 0Z5905Q73D
uses: [0771AJP4BX, 0JEJJCB0EX, 0Z424HK8CQ]
sources:
  - url: https://en.wikipedia.org/wiki/Alternating_group
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Alternating_group
    locator: Basic properties
---

## Definition

Let $n$ be a positive integer.
The alternating group $A_n$ is the set of the [[even permutations|0SQTMQTSRZ]] in $S_n$, with composition as the operation.

## Remarks

For $n \geq 2$, $A_n$ is the [[kernel|0W88CEGQP1]] of the [[sign homomorphism|0JEJJCB0EX]], so it is a [[normal subgroup|0CB0DX5BVV]] of $S_n$.
For $n \geq 2$, $A_n$ is also the [[commutator subgroup|0ASGSTXSBW]] of $S_n$.
The group $A_3$ is cyclic of order $3$, and $A_4$ has order $12$.
The group $A_n$ is abelian if and only if $n \leq 3$.

## Explanations

### L1 human

The alternating group keeps only the even permutations.
The product of two even permutations is even, so these form a group.
For $n \geq 5$ it is one of the most important examples of a simple group.