Supersolvable group

Definition

Basic Properties

• 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, 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.