---
tag: 0R3H35G96G
kind: definition
name: Derived series
msc: [20F14]
defines: 0Q5G5DJ4P2
uses: [08SJ5K2CBG]
sources:
  - url: https://en.wikipedia.org/wiki/Commutator_subgroup
    locator: Derived series
  - url: https://en.wikipedia.org/wiki/Solvable_group
    locator: Definition
---

## Definition

Let $G$ be a [[group|G5S1M9]].
The derived series of $G$ is the sequence of subgroups $G^{(0)} = G$ and $G^{(i+1)} = [G^{(i)}, G^{(i)}]$ for $i \geq 0$.
Here $[H, H]$ is the [[commutator subgroup|0ASGSTXSBW]] of $H$.

## Remarks

Each $G^{(i)}$ is a [[characteristic subgroup|0KABPMRHM2]] of $G$, and $G^{(i)} / G^{(i+1)}$ is abelian.
The group $G$ is [[solvable|0FXJ7YFMMX]] if and only if $G^{(n)} = \{e\}$ for some $n$, and the least such $n$ is the derived length.
For $S_4$, the derived series is $S_4 \supseteq A_4 \supseteq V \supseteq \{e\}$.
For $n \geq 5$, the derived series of $S_n$ is $S_n \supseteq A_n \supseteq A_n \supseteq \cdots$, so it never reaches $\{e\}$.

## Explanations

### L1 human

Each step of the derived series takes the commutator subgroup of the last step.
So each step removes an abelian layer from the group.
If you reach the identity, the group is solvable.