Nilpotent group: Difference between revisions

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, itsit has a [[central series]] is 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>

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 [[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_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 small non-abelian example, consider the [[quaternion group]] ''Q''₈, which is a smallest non-abelian ''p''-group. It has [[center (group theory)|center]] {1, &minus;1−1} of [[order of a group|order]] 2, and its upper central series is {1}, {1, &minus;1−1}, ''Q''₈; 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>

* Furthermore, every finite nilpotent group is the direct product of ''p''-groups.<ref>{{cite book|authorname="Zassenhaus|book-title=The theory of groups|year=1999|url={{Google books|plainurl=y|id=eCBK6tj7_vAC|page=143|text=Every finite nilpotent group is the direct product of its Sylow groups}}|page=143|title=Theorem 11}}<"/ref>

* The multiplicative group of upper [[Triangular matrix#Unitriangular matrix|unitriangular]] ''n'' x× ''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'' x× ''n'' matrices over a field ''F'' is not in general nilpotent, but is [[solvable group|solvable]].

Nilpotent groups are so 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 conjectured[[conjecture]]d to be nilpotent as long as they are [[Generating set offinitely agenerated group|finitely generated]]<!-- by Havas, Vaughan-Lee, Kappe, Nickel, etc. -->.

The following statements are equivalent for finite groups,<ref>Isaacs (2008), Thm. 1.26</ref> revealing some useful properties of nilpotency:{{ordered list

*(a) | ''G'' is a nilpotent group.

*(b) | If ''H'' is a proper subgroup of ''G'', then ''H'' is a proper [[normal subgroup]] of ''N''_''G''(''H'') (the [[normalizer]] of ''H'' in ''G''). This is called the '''normalizer property''' and can be phrased simply as "normalizers grow".

*(c) | Every [[Sylow subgroup]] of ''G'' is normal.

*(e) If | ''dG'' dividesis the [[Orderdirect product of a groupgroups|orderdirect product]] of ''G'',its thenSylow ''G'' has a [[normal subgroup]] of order ''d''subgroups.

; (a)→(b): By induction on |''G''|. If ''G'' is abelian, then for any ''H'', ''N''_''G''(''H'') = ''G''. If not, if ''Z''(''G'') is not contained in ''H'', then ''h''_''Z''''H''_''Z''⁻¹''h⁻¹'' = ''h''''H''''h⁻¹'' = ''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 ana subgroup of ''G''/''Z''(''G'') which normalizersnormalizes ''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.)

; (bc)→(cd): Let ''p''₁,''p''₂,...,''p''_''s'' be the distinct primes dividing its order and let ''P''_''i'' in ''Syl''_{''p''_''i''}(''G''),1≤ 1 ≤ ''i'' ≤ ''s''. LetFor any ''t'', 1 ≤ ''t'' ≤ ''s'' we show inductively that ''P''=₁''P''₂···''iP''_''t'' foris someisomorphic to ''iP''₁×''P''₂×···×''P''_''t''. and{{paragraph}}Note letfirst that each ''NP''=_''Ni'' is normal in ''G'' so ''P''₁(''P''). Since ₂···''P''_''t'' is a normal subgroup of ''NG'',. Let ''PH'' isbe characteristicthe inproduct ''NP''. Since ₁''P''₂···''P''_''t''−1 charand let ''NK'' and= ''NP''_''t'', isso aby normalinduction subgroup''H'' ofis isomorphic to ''NP''₁×''GP''₂(×···×''NP''),_''t''−1. weIn getparticular,|''H''| that= |''P''₁|&sdot;|''P''₂|&sdot;···&sdot;|''P''_''t''−1|. isSince a|''K''| normal subgroup of= |''NP''_''Gt''(|, the orders of ''NH'') and ''K'' are relatively prime. ThisLagrange's meansTheorem implies the intersection of ''NH'' and ''K'' is equal to 1. By definition,''P''₁''GP''₂(···''NP'')_''t'' is= a''HK'', subgrouphence of''HK'' is isomorphic to ''NH''×''K'' andwhich henceis equal to ''NP''₁×''GP''₂(×···×''NP'')=_''Nt''. ByThis (b)completes wethe mustinduction. thereforeNow havetake ''Nt'' = ''Gs'', whichto givesobtain (cd).

* {{cite book |last= Isaacs |first= I. Martin |author-link = Martin Isaacs|title= Finite Group Theory|year=2008|publisher=[[American Mathematical Society]]|isbn=978-0-8218-4344-34}}