locally nilpotent group



All nilpotent groups are locally nilpotent, because subgroupsMathworldPlanetmathPlanetmath of nilpotent groups are nilpotent.

An example of a locally nilpotent group that is not nilpotent is Dih((2)), the generalized dihedral group formed from the quasicyclic 2-group (http://planetmath.org/PGroup4) (2).

The Fitting subgroupMathworldPlanetmath of any group is locally nilpotent.

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


Any subgroup or quotient (http://planetmath.org/QuotientGroup) of a locally nilpotent group is locally nilpotent. Restricted direct productsPlanetmathPlanetmath 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 productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of its Sylow subgroups.)

Every group G has a unique maximal locally nilpotent normal subgroupMathworldPlanetmath. 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.

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