Derived series

Definition

Let $G$ be a group. 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 of $H$.

Remarks

Each $G^{(i)}$ is a characteristic subgroup of $G$, and $G^{(i)} / G^{(i+1)}$ is abelian. The group $G$ is solvable 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.