Editing Nilpotent group (section)

==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}}.