# Supersolvable group

__: Group with series of normal subgroups where all factors are cyclic__

**Short description**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

- [math]\displaystyle{ \{1\} = H_0 \triangleleft H_1 \triangleleft \cdots \triangleleft H_{s-1} \triangleleft H_s = G }[/math]

such that each quotient group [math]\displaystyle{ H_{i+1}/H_i \; }[/math] is cyclic and each [math]\displaystyle{ H_i }[/math] is normal in [math]\displaystyle{ G }[/math].

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 [math]\displaystyle{ H_i }[/math] be normal in [math]\displaystyle{ G }[/math]. 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, [math]\displaystyle{ A_4 }[/math], 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.
- Moreover, a finite group is supersolvable if and only if its lattice of subgroups is a supersolvable lattice, a significant strengthening of the Jordan-Dedekind chain condition.
- By Baum's theorem, every supersolvable finite group has a DFT algorithm running in time
*O*(*n*log*n*).^{[clarification needed]}

## 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

Original source: https://en.wikipedia.org/wiki/Supersolvable group.
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