---
tag: 0R0X73XHAT
kind: statement
role: proposition
name: Abelian groups are solvable
msc: [20F16]
uses: [0P3KZK0FTV, 071DDFVG4J]
sources:
  - url: https://en.wikipedia.org/wiki/Solvable_group
    locator: Abelian groups
---

## Statement

Let $G$ be an [[abelian group|082GHNPJ53]].
Then $G$ is [[solvable|0FXJ7YFMMX]].

## Remarks

The series $\{e\} \trianglelefteq G$ has one factor, $G$ itself, and this factor is abelian.
Equivalently, the commutator subgroup $G^{(1)}$ is already $\{e\}$.
The converse is false: $S_3$ is solvable, by the series $\{e\} \trianglelefteq A_3 \trianglelefteq S_3$, but it is not abelian.

## Explanations

### L1 human

An abelian group is already one abelian layer.
So it is solvable with a series of length one.
Solvable groups generalise abelian groups.