  For an infinite group $G$, the set of group topologies, $\mathcal{L}_G$,
forms a complete lattice. It is known that $\mathcal{L}_G$ is modular if $G$ is
abelian but the same result does not hold for nilpotent groups. We prove that
the lattice $\mathcal{L}_G$ is semi-modular if and only if $\mathcal{L}_{G/Z}$
is semi-modular, where $Z$ is the centre. As a corollary, for every nilpotent
group $G$, the lattice $\mathcal{L}_G$ is semi-modular.
  Moreover, in the famous Kourovka Notebook, several questions about
$\mathcal{L}_G$ were formalized by Arnautov. We answer two of them in this
note.
