descending series

Let G be a group.

A descending series of G is a family (Hα)αβ of subgroupsMathworldPlanetmathPlanetmath of G, where β is an ordinalMathworldPlanetmathPlanetmath, such that H0=G and Hβ={1}, and Hα+1Hα 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 an ascending series.

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

A subgroup of G that is a term of some descending series of G is called a descendant subgroup of G.

A descending series of G in which all terms are normal in G is called a descending normal series.

Let 𝔛 be a property of groups. A group is said to be hypo-X if it has a descending normal series whose factors all have property 𝔛. So, for example, a hypoabelian group is a group that has a descending normal series with abelianMathworldPlanetmath factors. Hypoabelian groups are sometimes called SD-groups; they are precisely the groups that have no non-trivial perfect subgroups, and they are also precisely the groups in which the transfinite derived series eventually reaches {1}.

Title descending series
Canonical name DescendingSeries
Date of creation 2013-03-22 16:14:52
Last modified on 2013-03-22 16:14:52
Owner yark (2760)
Last modified by yark (2760)
Numerical id 7
Author yark (2760)
Entry type Definition
Classification msc 20E15
Classification msc 20F22
Related topic AscendingSeries
Related topic SubnormalSeries
Related topic SubnormalSubgroup
Defines descending normal series
Defines descendant subgroup
Defines descendant
Defines hypoabelian group
Defines hypoabelian
Defines SD-group