Editing Subgroup series

{{More citations needed|date=April 2025}}
In [[mathematics]], specifically [[group theory]], a '''subgroup series''' of a [[group (mathematics)|group]] <math>G</math> is a [[Chain (order theory)|chain]] of [[subgroup]]s:
:<math>1 = A_0 \leq A_1 \leq \cdots \leq A_n = G</math>
where <math>1</math> is the [[trivial group|trivial subgroup]]. Subgroup series can simplify the study of a group to the study of simpler subgroups and their relations, and [[#Functional series|several subgroup series]] can be invariantly defined and are important invariants of groups.  A subgroup series is used in the [[subgroup method]].

Subgroup series are a special example of the use of [[filtration (mathematics)|filtration]]s in [[abstract algebra]].

==Definition==
===Normal series, subnormal series===
A '''subnormal series''' (also '''normal series''', '''normal tower''', '''subinvariant series''', or just '''series''') of a [[group (mathematics)|group]] ''G'' is a sequence of [[subgroup]]s, each a [[normal subgroup]] of the next one. In a standard notation

:<math>1 = A_0\triangleleft A_1\triangleleft \cdots \triangleleft A_n = G.</math>

There is no requirement made that ''A''<sub>''i''</sub> be a normal subgroup of ''G'', only a normal subgroup of  ''A''<sub>''i''&thinsp;+1</sub>. The [[quotient group]]s ''A''<sub>''i''&thinsp;+1</sub>/''A''<sub>''i''</sub> are called the '''factor groups''' of the series.

If in addition each ''A''<sub>''i''</sub> is normal in ''G'', then the series is called a '''normal series''', when this term is not used for the weaker sense, or an '''invariant series'''.

===Length===
A series with the additional property that ''A''<sub>''i''</sub> ≠ ''A''<sub>''i''&thinsp;+1</sub> for all ''i'' is called a series ''without repetition''; equivalently, each ''A''<sub>''i''</sub> is a proper subgroup of ''A''<sub>''i''&thinsp;+1</sub>. The ''length'' of a series is the number of strict inclusions ''A''<sub>''i''</sub> &lt; ''A''<sub>''i''&thinsp;+1</sub>. If the series has no repetition then the length is ''n''.

For a subnormal series, the length is the number of [[Trivial group|non-trivial]] factor groups. Every nontrivial group has a normal series of length 1, namely <math>1 \triangleleft G</math>, and any nontrivial proper normal subgroup gives a normal series of length 2. For [[simple group]]s, the trivial series of length 1 is the longest subnormal series possible.

===Ascending series, descending series===
Series can be notated in either ascending order:
:<math>1 = A_0\leq A_1\leq \cdots \leq A_n = G</math>
or descending order:
:<math>G = B_0\geq B_1\geq \cdots \geq B_n = 1.</math>

For a given finite series, there is no distinction between an "ascending series" or "descending series" beyond notation. For ''infinite'' series however, there is a distinction: the ascending series
:<math>1 = A_0\leq A_1\leq \cdots \leq G</math>
has a smallest term, a second smallest term, and so forth, but no largest proper term,  no second largest term, and so forth, while conversely the descending series
:<math>G = B_0\geq B_1\geq \cdots \geq 1</math>
has a largest term, but no smallest proper term.

Further, given a recursive formula for producing a series, the terms produced are either ascending or descending, and one calls the resulting series an ascending or descending series, respectively. For instance the [[derived series]] and [[lower central series]] are descending series, while the [[upper central series]] is an ascending series.

===Noetherian groups, Artinian groups===
A group that satisfies the [[ascending chain condition]] (ACC) on subgroups is called a '''Noetherian group''', and a group that satisfies the [[descending chain condition]] (DCC) is called an '''Artinian group''' (not to be confused with [[Artin group]]s), by analogy with [[Noetherian ring]]s and [[Artinian ring]]s. The ACC is equivalent to the '''maximal condition''': every [[Empty set|non-empty]] collection of subgroups has a maximal member, and the DCC is equivalent to the analogous '''minimal condition'''.

A group can be Noetherian but not Artinian, such as the [[infinite cyclic group]], and unlike for [[Ring (mathematics)|rings]], a group can be Artinian but not Noetherian, such as the [[Prüfer group]]. Every finite group is clearly Noetherian and Artinian.

[[Group homomorphism|Homomorphic]] [[Image (mathematics)|images]] and subgroups of Noetherian groups are Noetherian, and an [[group extension|extension]] of a Noetherian group by a Noetherian group is Noetherian. Analogous results hold for Artinian groups.

Noetherian groups are equivalently those such that every subgroup is [[finitely generated group|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.

Noetherian groups need not be finite extensions of [[polycyclic group]]s.<ref>{{cite journal | author = Ol'shanskii, A. Yu. | year = 1979 | title = Infinite Groups with Cyclic Subgroups | journal = Soviet Math. Dokl. | volume = 20 | pages = 343&ndash;346}} (English translation of ''Dokl. Akad. Nauk SSSR'', '''245''', 785&ndash;787)</ref>

===Infinite and transfinite series===
Infinite subgroup series can also be defined and arise naturally, in which case the specific ([[Total order|totally ordered]]) indexing set becomes important, and there is a distinction between ascending and descending series. An ascending series <math>1 = A_0\leq A_1\leq \cdots \leq G</math> where the <math>A_i</math> are indexed by the [[natural number]]s may simply be called an '''infinite ascending series''', and conversely for an '''infinite descending series'''. If the subgroups are more generally [[Ordinal number#Indexing classes of ordinals|indexed by ordinal numbers]], one obtains a '''transfinite series''',<ref>
{{cite arXiv |last=Sharipov | first=R.A. |eprint=0908.2257 |class=math.GR |title=Transfinite normal and composition series of groups  |year=2009 }}</ref> such as this ascending series:
:<math>1 = A_0\leq A_1\leq \cdots \leq A_\omega \leq A_{\omega+1} = G</math>

Given a recursive formula for producing a series, one can define a transfinite series by [[transfinite recursion]] by defining the series at [[limit ordinal]]s by 
<math>A_\lambda := \bigcup_{\alpha < \lambda} A_\alpha</math> (for ascending series) or <math>A_\lambda := \bigcap_{\alpha < \lambda} A_\alpha</math> (for descending series). Fundamental examples of this construction are the transfinite [[lower central series]] and [[upper central series]].

Other totally ordered sets arise rarely, if ever, as indexing sets of subgroup series.{{Citation needed|date=January 2008}} For instance, one can define but rarely sees naturally occurring bi-infinite subgroup series (series indexed by the [[integer]]s):
:<math>1 \leq \cdots \leq A_{-1} \leq A_0\leq A_1 \leq \cdots \leq G</math>

==Comparison of series==
A ''refinement'' of a series is another series containing each of the terms of the original 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 [[group isomorphism|isomorphic]]. Refinement gives a [[partial order]] on series, up to equivalence, and they form a [[Lattice (order)|lattice]], while subnormal series and normal series form sublattices. The existence of the supremum of two subnormal series is the [[Schreier refinement theorem]]. Of particular interest are ''maximal'' series without repetition.

==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]]

== Subgroup method ==

{{unreferenced section|date=April 2025}}
The '''subgroup method''' is an algorithm used in the [[mathematical]] field of [[group theory]].  It is used to find the [[word (group theory)|word]] of an element.  It doesn't always return the minimal word, but it can return optimal words based on the [[subgroup series|series of subgroups]] that is used.  The code looks like this:

 '''function''' operate(element, generator)
     <returns generator operated on element>
 
 '''function''' subgroup(g)
     sequence := (set of subgroups that will be used, depending on the method.)
     word := []
     '''for''' subgroup in sequence
         coset_representatives := []
         <fill coset_representatives with coset representatives of (next subgroup)/subgroup>
         '''for''' operation in coset_representatives
             '''if''' '''operate'''(g, operation) is in the next subgroup '''then'''
                 append operation onto word
                 g = '''operate'''(g, operation)
                 '''break'''
     '''return''' word

==References==
{{Reflist}}

{{DEFAULTSORT:Subgroup Series}}
[[Category:Subgroup series| ]]