A composition series for a group $G$ is a chain of subgroups $\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$ where each quotient $G_{i+1}/G_i$ is a simple group (has no proper nontrivial normal subgroups). The quotients are called composition factors and $n$ is the length.

A group $G$ is solvable if it has a composition series with all factors abelian. Equivalently, the derived series $G \geq G' \geq G'' \geq \cdots$ terminates at $\{e\}$, where $G' = [G,G]$ is the commutator subgroup.

A group $G$ is nilpotent if its lower central series $G = G_1 \geq G_2 \geq \cdots$ terminates at $\{e\}$, where $G_{i+1} = [G_i, G]$. Equivalently, $G$ is a direct product of its Sylow subgroups. Every nilpotent group is solvable.