---
tag: 0J53NE674D
kind: statement
role: theorem
name: The symmetric group on five or more letters is not solvable
msc: [20F16, 20B30]
uses: [0ARPRJQWZ2, 05Y0F9M0TS, 0R3H35G96G]
sources:
  - url: https://en.wikipedia.org/wiki/Solvable_group
    locator: Non-example
---

## Statement

Let $n \geq 5$.
Then the [[symmetric group|0515A6D2Y1]] $S_n$ is not [[solvable|0FXJ7YFMMX]].

## Remarks

The group $A_n$ is [[simple|0ARPRJQWZ2]] and not abelian, so its commutator subgroup is $A_n$, and the derived series of $A_n$ is constant.
So $A_n$ is not solvable, and then $S_n$ is not solvable, because [[subgroups of solvable groups are solvable|05Y0F9M0TS]].
The groups $S_1, S_2, S_3$ and $S_4$ are solvable.
By Galois theory, this is the reason why the general polynomial equation of degree $n \geq 5$ is not solvable by radicals.

## Explanations

### L1 human

From five letters on, the even permutations form a simple group that is not abelian.
Such a group cannot be cut into abelian layers.
So the full symmetric group cannot be cut into abelian layers either.