Summary: For a group \(G\), let \(\mathcal{G} (G)\) be the lattice of all topological group topologies on \(G\). We prove that if \(G\) is abelian, \(\tau ,\sigma \in \mathcal{G} (G)\) and \(\sigma\) is a successor of \(\tau\) in \(\mathcal{G} (G)\), then \(\sigma\) is precompact iff \(\tau\) is precompact. This fact is used to show that if a divisible or connected topological abelian group \((G,\tau)\) contains a discrete subgroup \(N\) such that \(G/N\) is compact, then \(\tau\) does not have successors in \(\mathcal{G}(G)\). In particular, no compact Hausdorff topological group topology on a divisible abelian group \(G\) has successors in \(\mathcal{G}(G)\) and the usual interval topology on \(\mathbb{R}\) has no successors in \(\mathcal{G}(\mathbb{R})\). We also prove that a compact Hausdorff topological group topology \(\tau\) on an abelian group \(G\) has a successor in \(\mathcal{G} (G)\) if and only if there exists a prime number \(p\) such that \(G/pG\) is infinite. Therefore, the usual compact topological group topology of the group \(\mathbb{Z}_p\) of \(p\)-adic integers does not have successors in \(\mathcal{G} (\mathbb{Z}_p)\). Our results solve two problems posed by different authors in the years 2006-2018.