Editing Nilpotent group

{{Short description|Mathematical concept}}
{{Group theory sidebar |Basics}}

In [[mathematics]], specifically [[group theory]], a '''nilpotent group''' ''G'' is a [[Group (mathematics)|group]] that has an [[upper central series]] that terminates with ''G''. Equivalently, it has a [[central series]] of finite length or its [[lower central series]] terminates with {1}.

Intuitively, a nilpotent group is a group that is "almost [[Abelian group|abelian]]". This idea is motivated by the fact that nilpotent groups are [[Solvable group|solvable]], and for [[finite group|finite]] nilpotent groups, two elements having [[relatively prime]] [[order (group theory)|orders]] must [[commutative property|commute]]. It is also true that finite nilpotent groups are [[supersolvable group|supersolvable]]. The concept is credited to work in the 1930s by Russian mathematician [[Sergei Chernikov]].<ref name="Dixon">{{cite journal|last1=Dixon|first1=M. R.|last2=Kirichenko|first2=V. V.|last3=Kurdachenko|first3=L. A.|last4=Otal|first4=J.|last5=Semko|first5=N. N.|last6=Shemetkov|first6=L. A.|last7=Subbotin|first7=I. Ya.|title=S. N. Chernikov and the development of infinite group theory|journal=Algebra and Discrete Mathematics|date=2012|volume=13|issue=2|pages=169–208}}</ref>

Nilpotent groups arise in [[Galois theory]], as well as in the classification of groups. They also appear prominently in the classification of [[Lie group]]s.

Analogous terms are used for [[Lie algebra]]s (using the [[Lie bracket of vector fields|Lie bracket]]) including '''[[nilpotent Lie algebra|nilpotent]]''', '''lower central series''', and '''upper central series'''.

==Definition==
The definition uses the idea of a [[central series]] for a group. The following are equivalent definitions for a nilpotent group {{mvar|G}}:{{unordered list
| {{mvar|G}} has a [[central series]] of finite length. That is, a series of [[normal subgroup]]s
: <math>\{1\} = G_0 \triangleleft G_1 \triangleleft \dots \triangleleft G_n = G</math>

where <math>G_{i+1}/G_i \leq Z(G/G_i)</math>, or equivalently <math>[G,G_{i+1}] \leq G_i</math>.
| {{mvar|G}} has a [[lower central series]] terminating in the [[trivial group|trivial]] [[subgroup]] after finitely many steps. That is, a series of normal subgroups 
: <math>G = G_0 \triangleright G_1 \triangleright \dots \triangleright G_n = \{1\}</math>
where <math>G_{i+1} = [G_i, G]</math>.
| {{mvar|G}} has an [[upper central series]] terminating in the whole group after finitely many steps. That is, a series of normal subgroups 
: <math>\{1\} = Z_0 \triangleleft Z_1 \triangleleft \dots \triangleleft Z_n = G</math>
where <math>Z_1 = Z(G)</math> and <math>Z_{i+1}</math> is the subgroup such that <math>Z_{i+1}/Z_i = Z(G/Z_i)</math>.
}}

For a nilpotent group, the smallest {{mvar|n}} such that {{mvar|G}} has a central series of length {{mvar|n}} is called the '''nilpotency class''' of {{mvar|G}}; and {{mvar|G}} is said to be '''nilpotent of class {{mvar|n}}'''. (By definition, the length is {{mvar|n}} if there are <math>n + 1</math> different subgroups in the series, including the trivial subgroup and the whole group.)

Equivalently, the nilpotency class of {{mvar|G}} equals the length of the lower central series or upper central series.
If a group has nilpotency class at most {{mvar|n}}, then it is sometimes called a '''nil-{{mvar|n}} group'''.

It follows immediately from any of the above forms of the definition of nilpotency, that the trivial group is the unique group of nilpotency class&nbsp;{{math|0}}, and groups of nilpotency class&nbsp;{{math|1}} are exactly the non-trivial abelian groups.<ref name="Suprunenko-76"/><ref>{{cite book|author=Tabachnikova & Smith |title= Topics in Group Theory (Springer Undergraduate Mathematics Series)|year=2000|url={{Google books|plainurl=y|id=DD0TW28WjfQC|page=169|text=The trivial group has nilpotency class 0}}|page=169}}</ref>

==Examples==
[[File:HeisenbergCayleyGraph.png|thumb|right|A portion of the [[Cayley graph]] of the discrete [[Heisenberg group]], a well-known nilpotent group.]]

* As noted above, every abelian group is nilpotent.<ref name="Suprunenko-76">{{cite book|author=Suprunenko |title=Matrix Groups|year=1976|url={{Google books|plainurl=y|id=cTtuPOj5h10C|page=205|text=abelian group is nilpotent}}|page=205}}</ref><ref>{{cite book|author=Hungerford |title=Algebra|year=1974|url={{Google books|plainurl=y|id=t6N_tOQhafoC|page=100|text=every abelian group G is nilpotent}}|page=100}}</ref>
* For a small non-abelian example, consider the [[quaternion group]] ''Q''<sub>8</sub>, which is a smallest non-abelian ''p''-group. It has [[center (group theory)|center]] {1, −1} of [[order of a group|order]] 2, and its upper central series is {1}, {1, −1}, ''Q''<sub>8</sub>; so it is nilpotent of class 2.
* The [[direct product]] of two nilpotent groups is nilpotent.<ref name="Zassenhaus">{{cite book|author=Zassenhaus |title=The theory of groups|year=1999|url={{Google books|plainurl=y|id=eCBK6tj7_vAC|page=143|text=The direct product of a finite number of nilpotent groups is nilpotent}}|page=143}}</ref>
* All finite [[p-group|''p''-group]]s are in fact nilpotent ([[p-group#Non-trivial center|proof]]). For ''n'' > 1, the maximal nilpotency class of a group of order ''p''<sup>''n''</sup> is ''n'' - 1 (for example, a group of order ''p''<sup>''2''</sup> is abelian). The 2-groups of maximal class are the generalised [[quaternion group]]s, the [[dihedral group]]s, and the [[semidihedral group]]s.
* Furthermore, every finite nilpotent group is the direct product of ''p''-groups.<ref name="Zassenhaus"/>
* The multiplicative group of upper [[Triangular matrix#Unitriangular matrix|unitriangular]] ''n'' × ''n'' matrices over any field ''F'' is a [[Unipotent algebraic group|nilpotent group]] of nilpotency class ''n'' − 1. In particular, taking ''n'' = 3 yields the [[Heisenberg group]] ''H'', an example of a non-abelian<ref>{{cite book|author=Haeseler |title=Automatic Sequences (De Gruyter Expositions in Mathematics, 36)|year=2002|url={{Google books|plainurl=y|id=wmh7tc6uGosC|page=15|text=The Heisenberg group is a non-abelian}}|page=15}}</ref> infinite nilpotent group.<ref>{{cite book|author=Palmer |title= Banach algebras and the general theory of *-algebras|year=2001|url={{Google books|plainurl=y|id=zn-iZNNTb-AC|page=1283|text=Heisenberg group this group has nilpotent length 2 but is not abelian}}|page=1283}}</ref> It has nilpotency class 2 with central series 1, ''Z''(''H''), ''H''.
* The multiplicative group of [[Borel subgroup|invertible upper triangular]] ''n'' × ''n'' matrices over a field ''F'' is not in general nilpotent, but is [[solvable group|solvable]].
* Any nonabelian group ''G'' such that ''G''/''Z''(''G'') is abelian has nilpotency class 2, with central series {1}, ''Z''(''G''), ''G''.

The [[natural number]]s ''k'' for which any group of order ''k'' is nilpotent have been characterized {{OEIS|A056867}}.

==Explanation of term==
Nilpotent groups are called so because the "adjoint action" of any element is [[nilpotent]], meaning that for a nilpotent group <math>G</math> of nilpotence degree <math>n</math> and an element <math>g</math>, the function <math>\operatorname{ad}_g \colon G \to G</math> defined by <math>\operatorname{ad}_g(x) := [g,x]</math> (where <math>[g,x]=g^{-1} x^{-1} g x</math> is the [[commutator]] of <math>g</math> and <math>x</math>) is nilpotent in the sense that the <math>n</math>th iteration of the function is trivial: <math>\left(\operatorname{ad}_g\right)^n(x)=e</math> for all <math>x</math> in <math>G</math>.

This is not a defining characteristic of nilpotent groups: groups for which <math>\operatorname{ad}_g</math> is nilpotent of degree <math>n</math> (in the sense above) are called <math>n</math>-[[Engel group]]s,<ref>For the term, compare [[Engel's theorem]], also on nilpotency.</ref> and need not be nilpotent in general. They are proven to be nilpotent if they have finite [[order (group theory)|order]]<!-- Zorn's lemma, 1936-->, and are [[conjecture]]d to be nilpotent as long as they are [[finitely generated group|finitely generated]]<!-- by Havas, Vaughan-Lee, Kappe, Nickel, etc. -->.

An abelian group is precisely one for which the adjoint action is not just nilpotent but trivial (a 1-Engel group).

==Properties==

Since each successive [[factor group]] ''Z''<sub>''i''+1</sub>/''Z''<sub>''i''</sub> in the [[central series|upper central series]] is abelian, and the series is finite, every nilpotent group is a [[solvable group]] with a relatively simple structure.

Every subgroup of a nilpotent group of class ''n'' is nilpotent of class at most ''n'';<ref name="theo7.1.3">Bechtell (1971), p. 51, Theorem 5.1.3</ref> in addition, if ''f'' is a [[group homomorphism|homomorphism]] of a nilpotent group of class ''n'', then the image of ''f'' is nilpotent<ref name="theo7.1.3" /> of class at most ''n''.

The following statements are equivalent for finite groups,<ref>Isaacs (2008), Thm. 1.26</ref> revealing some useful properties of nilpotency:{{ordered list
 | list-style-type=lower-alpha
 | ''G'' is a nilpotent group.
 | If ''H'' is a proper subgroup of ''G'', then ''H'' is a proper [[normal subgroup]] of ''N''<sub>''G''</sub>(''H'') (the [[normalizer]]  of ''H'' in ''G'').  This is called the '''normalizer property''' and can be phrased simply as "normalizers grow".
 | Every [[Sylow subgroup]] of ''G'' is normal.
 | ''G'' is the [[direct product of groups|direct product]] of its Sylow subgroups.
 | If ''d'' divides the [[Order of a group|order]] of ''G'', then ''G'' has a [[normal subgroup]] of order ''d''.
}}

Proof:
; (a)→(b): By induction on |''G''|. If ''G'' is abelian, then for any ''H'', ''N''<sub>''G''</sub>(''H'') = ''G''. If not, if ''Z''(''G'') is not contained in ''H'', then ''h''<sub>''Z''</sub>''H''<sub>''Z''</sub><sup>−1</sup>''h<sup>−1</sup>'' = ''h''''H''''h<sup>−1</sup>'' = ''H'', so ''H''·''Z''(''G'') normalizers ''H''. If ''Z''(''G'') is contained in ''H'', then ''H''/''Z''(''G'') is contained in ''G''/''Z''(''G''). Note, ''G''/''Z''(''G'') is a nilpotent group. Thus, there exists a subgroup of ''G''/''Z''(''G'') which normalizes ''H''/''Z''(''G'')  and ''H''/''Z''(''G'') is a proper subgroup of it. Therefore, pullback this subgroup to the subgroup in ''G'' and it normalizes ''H''. (This proof is the same argument as for ''p''-groups{{snd}}the only fact we needed was if ''G'' is nilpotent then so is ''G''/''Z''(''G''){{snd}}so the details are omitted.)
; (b)→(c): Let ''p''<sub>1</sub>,''p''<sub>2</sub>,...,''p''<sub>''s''</sub> be the distinct primes dividing its order and let ''P''<sub>''i''</sub> in ''Syl''<sub>''p''<sub>''i''</sub></sub>(''G''), 1 ≤ ''i'' ≤ ''s''. Let ''P'' = ''P''<sub>''i''</sub> for some ''i'' and let ''N'' = ''N''<sub>''G''</sub>(''P''). Since ''P'' is a normal Sylow subgroup of ''N'', ''P'' is [[characteristic subgroup|characteristic]] in ''N''. Since ''P'' char ''N'' and ''N'' is a normal subgroup of ''N''<sub>''G''</sub>(''N''), we get that ''P'' is a normal subgroup of ''N''<sub>''G''</sub>(''N''). This means ''N''<sub>''G''</sub>(''N'') is a subgroup of ''N'' and hence ''N''<sub>''G''</sub>(''N'') = ''N''. By (b) we must therefore have ''N'' = ''G'', which gives (c).
; (c)→(d): Let ''p''<sub>1</sub>,''p''<sub>2</sub>,...,''p''<sub>''s''</sub> be the distinct primes dividing its order and let ''P''<sub>''i''</sub> in ''Syl''<sub>''p''<sub>''i''</sub></sub>(''G''), 1 ≤ ''i'' ≤ ''s''. For any ''t'', 1 ≤ ''t'' ≤ ''s'' we show inductively that ''P''<sub>1</sub>''P''<sub>2</sub>···''P''<sub>''t''</sub> is isomorphic to ''P''<sub>1</sub>×''P''<sub>2</sub>×···×''P''<sub>''t''</sub>. {{paragraph}}Note first that each ''P''<sub>''i''</sub> is normal in ''G'' so ''P''<sub>1</sub>''P''<sub>2</sub>···''P''<sub>''t''</sub> is a subgroup of ''G''. Let ''H'' be the product ''P''<sub>1</sub>''P''<sub>2</sub>···''P''<sub>''t''−1</sub> and let ''K'' = ''P''<sub>''t''</sub>, so by induction ''H'' is isomorphic to ''P''<sub>1</sub>×''P''<sub>2</sub>×···×''P''<sub>''t''−1</sub>. In particular,|''H''| = |''P''<sub>1</sub>|&sdot;|''P''<sub>2</sub>|&sdot;···&sdot;|''P''<sub>''t''−1</sub>|. Since |''K''| = |''P''<sub>''t''</sub>|, the orders of ''H'' and ''K'' are relatively prime. Lagrange's Theorem implies the intersection of ''H'' and ''K'' is equal to 1. By definition,''P''<sub>1</sub>''P''<sub>2</sub>···''P''<sub>''t''</sub> = ''HK'', hence ''HK'' is isomorphic to ''H''×''K'' which is equal to ''P''<sub>1</sub>×''P''<sub>2</sub>×···×''P''<sub>''t''</sub>. This completes the induction. Now take ''t'' = ''s'' to obtain (d).
; (d)→(e): Note that a [[p-group]] of order ''p''<sup>''k''</sup> has a normal subgroup of order ''p''<sup>''m''</sup> for all 1≤''m''≤''k''. Since ''G'' is a direct product of its Sylow subgroups, and normality is preserved upon direct product of groups, ''G'' has a normal subgroup of order ''d'' for every divisor ''d'' of |''G''|.
; (e)→(a): For any prime ''p'' dividing |''G''|, the [[Sylow group|Sylow ''p''-subgroup]] is normal. Thus we can apply (c) (since we already proved (c)→(e)).

Statement (d) can be extended to infinite groups: if ''G'' is a nilpotent group, then every Sylow subgroup ''G''<sub>''p''</sub> of ''G'' is normal, and the direct product of these Sylow subgroups is the subgroup of all elements of finite order in ''G'' (see [[torsion subgroup]]).

Many properties of nilpotent groups are shared by [[hypercentral group]]s.

==Notes==
{{reflist}}

==References==
* {{cite book |author=Bechtell, Homer |title=The Theory of Groups |publisher= [[Addison-Wesley]] |year=1971 }}
* {{cite book |author-first=Friedrich | author-last= Von Haeseler |title=Automatic Sequences | series= De Gruyter Expositions in Mathematics | volume= 36 |publisher= [[Walter de Gruyter]] |location=Berlin |year=2002 |isbn=3-11-015629-6 }}
* {{cite book |author-last=Hungerford | author-first=Thomas W. | author-link=Thomas W. Hungerford |title=Algebra |publisher=Springer-Verlag |year=1974 |isbn=0-387-90518-9 }}
* {{cite book |last= Isaacs |first= I. Martin |author-link = Martin Isaacs|title= Finite Group Theory|year=2008|publisher=[[American Mathematical Society]]|isbn=978-0-8218-4344-4}}
* {{cite book |author=Palmer, Theodore W. |title=Banach Algebras and the General Theory of *-algebras |publisher=[[Cambridge University Press]] |year=1994 |isbn=0-521-36638-0 }}
* {{cite book| author-first=Urs | author-last= Stammbach | title= Homology in Group Theory | series= Lecture Notes in Mathematics | volume= 359 | publisher= Springer-Verlag | year= 1973 }} [http://projecteuclid.org/euclid.bams/1183537230 review]
* {{cite book |author=Suprunenko, D. A. |title=Matrix Groups |publisher= [[American Mathematical Society]] |location=Providence, Rhode Island |year=1976 |isbn=0-8218-1341-2 }}
* {{cite book |author1=Tabachnikova, Olga |author2=Smith, Geoff |title=Topics in Group Theory | series=Springer Undergraduate Mathematics Series |publisher=Springer  |year=2000 |isbn=1-85233-235-2 }}
* {{cite book |author-last=Zassenhaus | author-first= Hans | author-link= Hans Zassenhaus |title=The Theory of Groups |publisher= [[Dover Publications]] |location=New York |year=1999 |isbn=0-486-40922-8 }}

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{{DEFAULTSORT:Nilpotent Group}}
[[Category:Nilpotent groups]]
[[Category:Properties of groups]]