locally nilpotent group LocallyNilpotentGroup 2006-02-14 13:07:26 2007-06-13 14:25:14 Definition locally nilpotent Hirsch-Plotkin radical locally nilpotent radical \usepackage{amsmath} \usepackage{amsfonts} \DeclareMathOperator{\Dih}{Dih} \DeclareMathOperator{\HP}{HP} \def\Z{\mathbb{Z}} \PMlinkescapeword{finite} \PMlinkescapeword{satisfies} \section*{Definition} A \emph{locally nilpotent group} is a group in which every finitely generated subgroup is nilpotent. \section*{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 $\Dih(\Z(2^\infty))$, the generalized dihedral group formed from the quasicyclic \PMlinkname{$2$-group}{PGroup4} $\Z(2^\infty)$. The Fitting subgroup of any group is locally nilpotent. All N-groups are locally nilpotent. More generally, all Gruenberg groups are locally nilpotent. \section*{Properties} Any subgroup or \PMlinkname{quotient}{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 \PMlinkname{$p$-subgroup}{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 \emph{Hirsch-Plotkin radical}, or \emph{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.