---
tag: 0GKQKF0FXJ
kind: example
name: The alternating group A5 is simple
msc: [20B30, 20D06]
instance_of: [0Z5905Q73D]
uses: [0ARPRJQWZ2, 0D1VWQGZZ7, 3C53TD]
sources:
  - url: https://en.wikipedia.org/wiki/Alternating_group
    locator: Basic properties
---

## Example

The group $A_5$ has order $60$.
Its [[conjugacy classes|0DXD4QSZ8A]] have sizes $1$, $15$, $20$, $12$ and $12$.
The class of size $1$ is $\{e\}$, and the class of size $20$ contains the $3$-cycles.
The class of size $15$ contains the products of two disjoint transpositions.
The $24$ five-cycles split into two classes of size $12$.
A normal subgroup $N$ of $A_5$ is a union of conjugacy classes, and it contains the class $\{e\}$.
By [[Lagrange's theorem|3C53TD]], $|N|$ divides $60$.
No sum of $1$ and some of the numbers $15, 20, 12, 12$ divides $60$, except $1$ and $60$.
Thus $N = \{e\}$ or $N = A_5$, and $A_5$ is simple.
It is the smallest simple group that is not abelian.