  For a group $G$ and a subgroup $H$ of $G$ this article discusses the
normalizer of $H$ in the units of a group ring $RG$. We prove that $H$ is only
normalized by the `obvious' units, namely products of elements of $G$
normalizing $H$ and units of $RG$ centralizing $H$, provided $H$ is cyclic.
Moreover we show that the normalizers of all subgroups of certain nilpotent and
metacyclic groups in the corresponding group rings are as small as possible.
These classes contain all dihedral groups, all finite nilpotent groups and all
finite groups with all Sylow subgroups being cyclic.
