locally nilpotent group

Definition

A locally nilpotent group isa group in which every finitely generated subgroup is nilpotent.

Examples

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

An example of a locally nilpotent group that is not nilpotentis $\mathrm{Dih}(\mathbb{Z}(2^{\mathrm{\infty }}))$, the generalized dihedral groupformed from the quasicyclic $2$-group (http://planetmath.org/PGroup4) $\mathbb{Z}(2^{\mathrm{\infty }})$.

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 groupform a fully invariant subgroup(the maximal $p$-subgroup (http://planetmath.org/PGroup4)).The elements of finite order in a locally nilpotent groupalso 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 groupis 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 $\mathrm{HP}(G)$.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