ascending series

Let G be a group.

An ascending series of G is a family (Hα)αβ of subgroupsMathworldPlanetmathPlanetmath of G, where β is an ordinalMathworldPlanetmathPlanetmath, such that H0={1} and Hβ=G, and HαHα+1 for all α<β, and


whenever δβ is a limit ordinalMathworldPlanetmath.

Note that this is a generalizationPlanetmathPlanetmath of the concept of a subnormal series. Compare also the dual concept of a descending series.

Given an ascending series (Hα)αβ, the subgroups Hα are called the terms of the series and the quotients ( Hα+1/Hα are called the factors of the series.

A subgroup of G that is a term of some ascending series of G is called an ascendant subgroup of G. The notation HascG is sometimes used to indicate that H is an ascendant subgroup of G.

The groups in which every subgroup is ascendant are precisely the groups that satisfy the normalizer condition. Groups in which every cyclic subgroup is ascendant are called Gruenberg groups. It can be shown that in a Gruenberg group, every finitely generated subgroup is ascendant and nilpotent (and so, in particular, Gruenberg groups are locally nilpotent).

An ascending series of G in which all terms are normal in G is called an ascending normal series.

Let 𝔛 be a property of groups. A group is said to be hyper-X if it has an ascending normal series whose factors all have property 𝔛. So, for example, a hyperabelian group is a group that has an ascending normal series with abelianMathworldPlanetmath factors. Hyperabelian groups are sometimes called SI*-groups.

Title ascending series
Canonical name AscendingSeries
Date of creation 2013-03-22 16:14:55
Last modified on 2013-03-22 16:14:55
Owner yark (2760)
Last modified by yark (2760)
Numerical id 12
Author yark (2760)
Entry type Definition
Classification msc 20E15
Classification msc 20F22
Related topic DescendingSeries
Related topic SubnormalSeries
Related topic SubnormalSubgroup
Defines ascending normal series
Defines ascendant subgroup
Defines ascendant
Defines hyperabelian group
Defines hyperabelian
Defines Gruenberg group