# locally nilpotent group

## Examples

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 $\operatorname{Dih}(\mathbb{Z}(2^{\infty}))$, the generalized dihedral group formed from the quasicyclic $2$-group (http://planetmath.org/PGroup4) $\mathbb{Z}(2^{\infty})$.

The Fitting subgroup of any group is locally nilpotent.

All N-groups are locally nilpotent. More generally, all Gruenberg groups are locally nilpotent.

## Properties

Any subgroup or quotient (http://planetmath.org/QuotientGroup) 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 (http://planetmath.org/PGroup4)). 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 $\operatorname{HP}(G)$. 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.

 Title locally nilpotent group Canonical name LocallyNilpotentGroup Date of creation 2013-03-22 15:40:42 Last modified on 2013-03-22 15:40:42 Owner yark (2760) Last modified by yark (2760) Numerical id 7 Author yark (2760) Entry type Definition Classification msc 20F19 Related topic LocallyCalP Related topic NilpotentGroup Related topic NormalizerCondition Defines locally nilpotent Defines Hirsch-Plotkin radical Defines locally nilpotent radical