descending series
Let G be a group.
A descending series of G
is a family (Hα)α≤β of subgroups of G,
where β is an ordinal
,
such that H0=G and Hβ={1},
and Hα+1⊴Hα for all α<β,
and
⋂α<δHα=Hδ |
whenever δ≤β is a limit ordinal.
Note that this is a generalization of the concept of a subnormal series.
Compare also the dual concept of an ascending series.
Given a descending series (Hα)α≤β, the subgroups Hα are called the terms of the series and the quotients (http://planetmath.org/QuotientGroup) 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 abelian 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 |