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.
Subgroup series are a special example of the use of filtrations in abstract algebra.
In a standard notation There is no requirement made that Ai be a normal subgroup of G, only a normal subgroup of Ai +1.
If the series has no repetition then the length is n. For a subnormal series, the length is the number of non-trivial factor groups.
Every nontrivial group has a normal series of length 1, namely
The ACC is equivalent to the maximal condition: every non-empty collection of subgroups has a maximal member, and the DCC is equivalent to the analogous minimal condition.
Analogous results hold for Artinian groups.
Noetherian groups are equivalently those such that every subgroup is finitely generated, which is stronger than the group itself being finitely generated: the free group on 2 or finitely more generators is finitely generated, but contains free groups of infinite rank.
[1] 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.
are indexed by the natural numbers may simply be called an infinite ascending series, and conversely for an infinite descending series.
Two subnormal series are said to be equivalent or isomorphic if there is a bijection between the sets of their factor groups such that the corresponding factor groups are isomorphic.
The existence of the supremum of two subnormal series is the Schreier refinement theorem.
These include: There are series coming from subgroups of prime power order or prime power index, related to ideas such as Sylow subgroups.