The Lattice of Group Topologies

For an infinite group $G$, the poset $\mathcal{L}_G$ of group topologies constitutes a complete lattice. Although $\mathcal{L}_G$ is modular when $G$ is abelian, this property fails to persist for nilpotent groups. Extending Arnautov's 2010 work on the semi-modularity of $\mathcal{L}_G$ for nilpotent groups, we present an alternative proof with enhanced structural clarity. Additionally, we resolve two open questions from the Kourovka Notebook regarding lattice-theoretic properties of $\mathcal{L}_G$: (1) explicit construction of a countably infinite non-abelian nilpotent group with modular topology lattice, and (2) establishing the absence of property $P_2$ in infinite abelian groups.