# Supersolvable group

supersoluble group

A group $G$ with a finite series of normal subgroups (cf. Subgroup series)

$$G=G_1\supseteq G_2\supseteq\dots\supseteq G_{n+1}=E,$$

in which each quotient group $G_{i-1}/G_i$ is cyclic. Every supersolvable group is a polycyclic group. Subgroups and quotient groups of a supersolvable group are also supersolvable, and the commutator subgroup of a supersolvable group is nilpotent. A finite group is supersolvable if and only if all of its maximal subgroups have prime index (Huppert's theorem).