Editing Subgroup series (section)

==Examples==
{{category see also|Subgroup series}}

===Maximal series===
* A '''[[composition series]]''' is a maximal ''subnormal'' series.
:Equivalently, a subnormal series for which each of the ''A''<sub>''i''</sub> is a [[maximal subgroup|maximal]] normal subgroup of ''A''<sub>''i''&thinsp;+1</sub>. Equivalently, a composition series is a subnormal series for which each of the factor groups are [[simple group|simple]].
* A '''[[chief series]]''' is a maximal ''normal'' series.

===Solvable and nilpotent===
* A '''[[solvable group]]''', or soluble group, is one with a subnormal series whose factor groups are all [[Abelian group|abelian]].
* A '''[[nilpotent series]]''' is a subnormal series such that successive quotients are [[nilpotent group|nilpotent]].
:A nilpotent series exists if and only if the group is [[solvable group|solvable]].
* A '''[[central series]]''' is a subnormal series such that successive quotients are [[center (group)|central]], i.e. given the above series, <math>A_{i+1}/A_i \subseteq Z(G/A_i)</math> for <math>i=0, 1, \ldots, n-2</math>.
:A central series exists if and only if the group is [[nilpotent group|nilpotent]].

===Functional series===
Some subgroup series are defined [[:Category:Functional subgroups|functionally]], in terms of subgroups such as the center and operations such as the commutator. These include:
* [[Lower central series]]
* [[Upper central series]]
* [[Derived series]]
* [[Lower Fitting series]]
* [[Upper Fitting series]]

===''p''-series===
There are series coming from subgroups of prime power order or prime power index, related to ideas such as [[Sylow subgroup]]s.
* [[Lower p-series|Lower ''p''-series]]
* [[Upper p-series|Upper ''p''-series]]