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