|
A locally nilpotent group is a group in which every finitely generated subgroup is nilpotent.
All nilpotent groups are locally nilpotent, because subgroups of nilpotent groups are nilpotent.
An example of a locally nilpotent group that is not nilpotent is
 , the generalized dihedral group formed from the quasicyclic  -group
 .
The Fitting subgroup of any group is locally nilpotent.
All N-groups are locally nilpotent. More generally, all Gruenberg groups are locally nilpotent.
Any subgroup or quotient of a locally nilpotent group is locally nilpotent. Restricted direct products of locally nilpotent groups are locally nilpotent.
For each prime  , the elements of  -power order in a locally nilpotent group form a fully invariant subgroup (the maximal  -subgroup). The elements of finite order in a
locally nilpotent group also form a fully invariant subgroup (the torsion subgroup), which is the restricted direct product of the maximal  -subgroups. (This generalizes the fact that a finite nilpotent group is the direct product of its Sylow subgroups.)
Every group  has a unique maximal locally nilpotent normal subgroup. This subgroup is called the Hirsch-Plotkin radical, or locally nilpotent radical, and is often denoted  . If  is finite (or, more generally, satisfies the maximal condition), then the Hirsch-Plotkin radical is the same as the Fitting subgroup, and is nilpotent.
| 
