Supersolvable group

In mathematics, a group is supersolvable (or supersoluble) if it has an invariant normal series where all the factors are cyclic groups. Supersolvablility is stronger than the notion of solvability.

Definition

Let "G" be a group. "G" is supersolvable is there exists a normal series

:$\{1\}\; =\; H\_0\; riangleleft\; H\_1\; riangleleft\; cdots\; riangleleft\; H\_\{s-1\}\; riangleleft\; H\_s\; =\; G$

such that each quotient group $H\_\{i+1\}/H\_i\; ;$ 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 normal series with each quotient cyclic, but there is no requirement that each $H\_i$ be normal in $G$. As every finite soluble 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:

* 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 supersoluble 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 supersoluble.
* Every irreducible complex representation of a finite supersoluble group is monomial, that is, induced from a linear character of a subgroup. In other words, every supersoluble group is a monomial group.
*Every maximal subgroup in a supersoluble group has prime index.
*A finite group is supersoluble if and only if every maximal subgroup has prime index.
*A finite group is supersoluble 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 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.