---
tag: T4PCQ68C5K
kind: example
name: The alternating group A3 is normal in S3
msc: [20B30]
instance_of: [0CB0DX5BVV]
illustrates: [0V547YQRMV]
uses: [0Z5905Q73D, 0515A6D2Y1, 04QEBF8S49]
sources:
  - url: https://en.wikipedia.org/wiki/Normal_subgroup
    locator: Examples
---

## Example

In the [[symmetric group|0515A6D2Y1]] $S_3$, let $A_3 = \{e, (1\,2\,3), (1\,3\,2)\}$. This is the [[alternating group|0Z5905Q73D]] of degree $3$.
$A_3$ has $3$ elements and $S_3$ has $6$, so $A_3$ has [[index|04QEBF8S49]] $2$.
The two left cosets are $A_3$ and the set of the three transpositions. The two right cosets are the same two sets.
Thus $gA_3 = A_3g$ for each $g \in S_3$, and $A_3$ is normal in $S_3$.

## Remarks

This shows the result that [[subgroups of index two are normal|0V547YQRMV]].