Solvable group (reviewed)

A group $G$ is solvable if there exists a chain of subgroups \[ \langle e\rangle =H_0\leq H_1 \leq H_2 \leq \cdots \leq H_n=G\] such that for all $i<n$, $H_i$ is a normal subgroup of $H_{i+1}$ (i.e., it is a subnormal series) and each quotient $H_{i+1}/H_i$ is abelian.