Successors of topologies of connected locally compact groups

Let G be a group and, be topological group topologies on G. We say that is a successor of if is strictly finer than and there is not a group topology properly between them. In this note, we explore the existence of successor topologies in topological groups, particularly focusing on non-abelian connected locally compact groups. Our main contributions are twofold: for a connected locally compact group (G, ), we show that (1) if (G, ) is compact, then has a precompact successor if and only if there exists a discontinuous homomorphism from G into a simple connected compact group with dense image, and (2) if G is solvable, then has no successors. Our work relies on the previous characterization of locally compact group topologies on abelian groups processing successors.