Derived series of a group (reviewed)

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 $G$.