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 H_i is normal in G. 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 H_i be normal in G. 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, A_4, is solvable but not supersolvable.

Basic Properties

Some facts about supersolvable groups: