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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 $\Dih(\Z(2^\infty))$ , the generalized dihedral group formed from the quasicyclic $2$ -group $\Z(2^\infty)$ .
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 $p$ , the elements of $p$ -power order in a locally nilpotent group form a fully invariant subgroup (the maximal $p$ -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 $p$ -subgroups. (This generalizes the fact that a finite nilpotent group is the direct product of its Sylow subgroups.)
Every group $G$ has a unique maximal locally nilpotent normal subgroup. This subgroup is called the Hirsch-Plotkin radical, or locally nilpotent radical, and is often denoted $\HP(G)$ . If $G$ is finite (or, more generally, satisfies the maximal condition), then the Hirsch-Plotkin radical is the same as the Fitting subgroup, and is nilpotent.
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