Let be a group.
whenever is a limit ordinal.
Given a descending 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 descending series of is called a descendant subgroup of .
A descending series of in which all terms are normal in is called a descending normal series.
Let be a property of groups. A group is said to be hypo- if it has a descending normal series whose factors all have property . 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 .
|Date of creation||2013-03-22 16:14:52|
|Last modified on||2013-03-22 16:14:52|
|Last modified by||yark (2760)|
|Defines||descending normal series|