Let be a group.
whenever is a limit ordinal.
Given an ascending series , the subgroups are called the terms of the series and the quotients (http://planetmath.org/QuotientGroup) are called the factors of the series.
A subgroup of that is a term of some ascending series of is called an ascendant subgroup of . The notation is sometimes used to indicate that is an ascendant subgroup of .
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 in which all terms are normal in is called an ascending normal series.
Let be a property of groups. A group is said to be hyper- if it has an ascending normal series whose factors all have property . So, for example, a hyperabelian group is a group that has an ascending normal series with abelian factors. Hyperabelian groups are sometimes called -groups.
|Date of creation||2013-03-22 16:14:55|
|Last modified on||2013-03-22 16:14:55|
|Last modified by||yark (2760)|
|Defines||ascending normal series|