Derived series

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 $\{e\}$.