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Let G be a group and σ, τ be topological group topologies on G. We say that σ is a successor of τ if σ is strictly finer than τ 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, τ ), we show that (1) if (G, τ ) is compact, then τ 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 τ has no successors. Our work relies on the previous characterization of locally compact group topologies on abelian groups processing successors.

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