subgroup series

A subgroup series is used in the subgroup method.

In mathematics, specifically group theory, a subgroup series is a chain of subgroups:

Subgroup series are a special example of the use of filtrations in abstract algebra.

Other totally ordered sets arise rarely, if ever, as indexing sets of subgroup series.

Some subgroup series are defined functionally, in terms of subgroups such as the center and operations such as the commutator.

For instance, one can define but rarely sees naturally occurring bi-infinite subgroup series (series indexed by the integers):

Infinite subgroup series can also be defined and arise naturally, in which case the specific (totally ordered) indexing set becomes important, and there is a distinction between ascending and descending series.

Subgroup series can simplify the study of a group to the study of simpler subgroups and their relations, and several subgroup series can be invariantly defined and are important invariants of groups.