locally nilpotent group
The Fitting subgroup of any group is locally nilpotent.
For each prime , the elements of -power order in a locally nilpotent group form a fully invariant subgroup (the maximal -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 -subgroups. (This generalizes the fact that a finite nilpotent group is the direct product of its Sylow subgroups.)
Every group has a unique maximal locally nilpotent normal subgroup. This subgroup is called the Hirsch-Plotkin radical, or locally nilpotent radical, and is often denoted . If 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|
|Date of creation||2013-03-22 15:40:42|
|Last modified on||2013-03-22 15:40:42|
|Last modified by||yark (2760)|
|Defines||locally nilpotent radical|