The **derived series** of a finite group $G$ is the chain of subgroups
\[ G =G^{(0)}\rhd G^{(1)} \rhd G^{(2)} \rhd \cdots \rhd G^{(k)} \]
where $G^{(i+1)}$ is the commutator subgroup of $G^{(i)}$ for all $i$, each inclusion is proper, and $G^{(k+1)}=G^{(k)}$.

A group is solvable if and only if $G^{(k)}=\langle e\rangle$. In this case, $k$ is the **solvable length** of $G$.

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**Knowl status:**

- Review status: beta
- Last edited by John Jones on 2019-05-24 00:00:02

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