In mathematics, a group is supersolvable (or supersoluble) if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is stronger than the notion of solvability. Definition Let G be a group. G is supersolvable if there exists a normal series such that each quotient group is cyclic and each is normal in . By contrast, for a solvable group the definition requires each quotient to be abelian. In another direction, a polycyclic group must have a subnormal series with each quotient cyclic, but there is no requirement that each be normal in . As every finite solvable group is polycyclic, this can be seen as one of the key differences between the definitions. For a concrete example, the alternating group on four points, , is solvable but not supersolvable. Basic Properties Some facts about supersolvable groups: Supersolvable groups are always polycyclic, and hence solvable. Every finitely generated nilpotent group is supersolvable. Every metacyclic group is supersolvable. The commutator subgroup of a supersolvable group is nilpotent. Subgroups and quotient groups of supersolvable groups are supersolvable. A finite supersolvable group has an invariant normal series with each factor cyclic of prime order. In fact, the primes can be chosen in a nice order: For every prime p, and for π the set of primes greater than p, a finite supersolvable group has a unique Hall π-subgroup. Such groups are sometimes called ordered Sylow tower groups. Every group of square-free order, and every group with cyclic Sylow subgroups (a Z-group), is supersolvable. Every irreducible complex representation of a finite supersolvable group is monomial, that is, induced from a linear character of a subgroup. In other words, every finite supersolvable group is a monomial group. Every maximal subgroup in a supersolvable group has prime index. A finite group is supersolvable if and only if every maximal subgroup has prime index. A finite group is supersolvable if and only if every maximal chain of subgroups has the same length. This is important to those interested in the lattice of subgroups of a group, and is sometimes called the Jordan–Dedekind chain condition. By Baum's theorem, every supersolvable finite group has a DFT algorithm running in time O(n log n). References Schenkman, Eugene. Group Theory. Krieger, 1975. Schmidt, Roland. Subgroup Lattices of Groups. de Gruyter, 1994. Keith Conrad, SUBGROUP SERIES II, Section 4 , http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/subgpseries2.pdf Solvable groups