Nilpotent group

Inmathematics,specificallygroup theory,anilpotent groupGis agroupthat has anupper central seriesthat terminates withG.Equivalently, it has acentral seriesof finite length or itslower central seriesterminates with {1}.

Intuitively, a nilpotent group is a group that is "almostabelian".This idea is motivated by the fact that nilpotent groups aresolvable,and forfinitenilpotent groups, two elements havingrelatively primeordersmustcommute.It is also true that finite nilpotent groups aresupersolvable.The concept is credited to work in the 1930s by Russian mathematicianSergei Chernikov.

Nilpotent groups arise inGalois theory,as well as in the classification of groups. They also appear prominently in the classification ofLie groups.

Analogous terms are used forLie algebras(using theLie bracket) includingnilpotent,lower central series,andupper central series.

Definition

The definition uses the idea of acentral seriesfor a group. The following are equivalent definitions for a nilpotent groupG:

• Ghas acentral seriesof finite length. That is, a series ofnormal subgroups

  $\{1\}=G_{0}\triangleleft G_{1}\triangleleft \dots \triangleleft G_{n}=G$

  where$G_{i+1}/G_{i}\leq Z(G/G_{i})$,or equivalently$[G,G_{i+1}]\leq G_{i}$.
• Ghas alower central seriesterminating in thetrivialsubgroupafter finitely many steps. That is, a series of normal subgroups

  $G=G_{0}\triangleright G_{1}\triangleright \dots \triangleright G_{n}=\{1\}$

  where$G_{i+1}=[G_{i},G]$.
• Ghas anupper central seriesterminating in the whole group after finitely many steps. That is, a series of normal subgroups

  $\{1\}=Z_{0}\triangleleft Z_{1}\triangleleft \dots \triangleleft Z_{n}=G$

  where$Z_{1}=Z(G)$and$Z_{i+1}$is the subgroup such that$Z_{i+1}/Z_{i}=Z(G/Z_{i})$.

For a nilpotent group, the smallestnsuch thatGhas a central series of lengthnis called thenilpotency classofG;andGis said to benilpotent of classn.(By definition, the length isnif there are$n+1$different subgroups in the series, including the trivial subgroup and the whole group.)

Equivalently, the nilpotency class ofGequals the length of the lower central series or upper central series. If a group has nilpotency class at mostn,then it is sometimes called anil-ngroup.

It follows immediately from any of the above forms of the definition of nilpotency, that the trivial group is the unique group of nilpotency class0,and groups of nilpotency class1are exactly the non-trivial abelian groups.

Examples

A portion of theCayley graphof the discreteHeisenberg group,a well-known nilpotent group.

• As noted above, every abelian group is nilpotent.
• For a small non-abelian example, consider thequaternion groupQ₈,which is a smallest non-abelianp-group. It hascenter{1, −1} oforder2, and its upper central series is {1}, {1, −1},Q₈;so it is nilpotent of class 2.
• Thedirect productof two nilpotent groups is nilpotent.
• All finitep-groupsare in fact nilpotent (proof). The maximal class of a group of orderpⁿisn(for example, any group of order 2 is nilpotent of class 1). The 2-groups of maximal class are the generalisedquaternion groups,thedihedral groups,and thesemidihedral groups.
• Furthermore, every finite nilpotent group is the direct product ofp-groups.
• The multiplicative group of upperunitriangularn×nmatrices over any fieldFis anilpotent groupof nilpotency classn− 1. In particular, takingn= 3 yields theHeisenberg groupH,an example of a non-abelianinfinite nilpotent group.It has nilpotency class 2 with central series 1,Z(H),H.
• The multiplicative group ofinvertible upper triangularn×nmatrices over a fieldFis not in general nilpotent, but issolvable.
• Any nonabelian groupGsuch thatG/Z(G) is abelian has nilpotency class 2, with central series {1},Z(G),G.

Thenatural numberskfor which any group of orderkis nilpotent have been characterized (sequenceA056867in theOEIS).

Explanation of term

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

This is not a defining characteristic of nilpotent groups: groups for which$\operatorname {ad} _{g}$is nilpotent of degree$n$(in the sense above) are called$n$-Engel groups,and need not be nilpotent in general. They are proven to be nilpotent if they have finiteorder,and areconjecturedto be nilpotent as long as they arefinitely generated.

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

Properties

Since each successivefactor groupZ_i+1/Z_iin theupper central seriesis abelian, and the series is finite, every nilpotent group is asolvable groupwith a relatively simple structure.

Every subgroup of a nilpotent group of classnis nilpotent of class at mostn;in addition, iffis ahomomorphismof a nilpotent group of classn,then the image offis nilpotentof class at mostn.

The following statements are equivalent for finite groups,revealing some useful properties of nilpotency:

1. Gis a nilpotent group.
2. IfHis a proper subgroup ofG,thenHis a propernormal subgroupofN_G(H) (thenormalizerofHinG). This is called thenormalizer propertyand can be phrased simply as "normalizers grow".
3. EverySylow subgroupofGis normal.
4. Gis thedirect productof its Sylow subgroups.
5. Ifddivides theorderofG,thenGhas anormal subgroupof orderd.

Proof:

(a)→(b)

By induction on |G|. IfGis abelian, then for anyH,N_G(H) =G.If not, ifZ(G) is not contained inH,thenh_ZH_Z⁻¹h⁻¹=h'H'h⁻¹=H,soH·Z(G) normalizersH.IfZ(G) is contained inH,thenH/Z(G) is contained inG/Z(G). Note,G/Z(G) is a nilpotent group. Thus, there exists a subgroup ofG/Z(G) which normalizesH/Z(G) andH/Z(G) is a proper subgroup of it. Therefore, pullback this subgroup to the subgroup inGand it normalizesH.(This proof is the same argument as forp-groups – the only fact we needed was ifGis nilpotent then so isG/Z(G) – so the details are omitted.)

(b)→(c)

Letp₁,p₂,...,p_sbe the distinct primes dividing its order and letP_iinSyl_pi(G), 1 ≤i≤s.LetP=P_ifor someiand letN=N_G(P). SincePis a normal Sylow subgroup ofN,PischaracteristicinN.SincePcharNandNis a normal subgroup ofN_G(N), we get thatPis a normal subgroup ofN_G(N). This meansN_G(N) is a subgroup ofNand henceN_G(N) =N.By (b) we must therefore haveN=G,which gives (c).

(c)→(d)

Letp₁,p₂,...,p_sbe the distinct primes dividing its order and letP_iinSyl_pi(G), 1 ≤i≤s.For anyt,1 ≤t≤swe show inductively thatP₁P₂···P_tis isomorphic toP₁×P₂×···×P_t.

Note first that eachP_iis normal inGsoP₁P₂···P_tis a subgroup ofG.LetHbe the productP₁P₂···P_t−1and letK=P_t,so by inductionHis isomorphic toP₁×P₂×···×P_t−1.In particular,|H| = |P₁|⋅|P₂|⋅···⋅|P_t−1|. Since |K| = |P_t|, the orders ofHandKare relatively prime. Lagrange's Theorem implies the intersection ofHandKis equal to 1. By definition,P₁P₂···P_t=HK,henceHKis isomorphic toH×Kwhich is equal toP₁×P₂×···×P_t.This completes the induction. Now taket=sto obtain (d).

(d)→(e)

Note that ap-groupof orderp^khas a normal subgroup of orderp^mfor all 1≤m≤k.SinceGis a direct product of its Sylow subgroups, and normality is preserved upon direct product of groups,Ghas a normal subgroup of orderdfor every divisordof |G|.

(e)→(a)

For any primepdividing |G|, theSylowp-subgroupis normal. Thus we can apply (c) (since we already proved (c)→(e)).

Statement (d) can be extended to infinite groups: ifGis a nilpotent group, then every Sylow subgroupG_pofGis normal, and the direct product of these Sylow subgroups is the subgroup of all elements of finite order inG(seetorsion subgroup).

Many properties of nilpotent groups are shared byhypercentral groups.

References

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