# locally nilpotent group

## Definition

A *locally nilpotent group* is
a 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 nilpotent is $\mathrm{Dih}(\mathbb{Z}({2}^{\mathrm{\infty}}))$, the generalized dihedral group formed 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 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 $\mathrm{HP}(G)$.
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 |