  Two non-discrete Hausdorff group topologies $\tau, \delta$ on a group $G$ are
called {\it transversal} if the least upper bound $\tau\vee \delta$ of $\tau$
and $\delta$ is the discrete topology. In this paper, we discuss the existence
of transversal group topologies on locally pseudocompact, locally precompact or
locally compact groups. We prove that each locally pseudocompact, connected
topological group satisfies CSP, which gives an affirmative answer to a problem
posed by Dikranjan, Tkachenko and Yaschenko in 2006. For a compact normal
subgroup $K$ of a locally compact totally disconnected group $G$, if $G$ admits
a transversal group topology then $G/K$ admits a transversal group topology,
which give a partial answer again to a problem posed by Dikranjan, Tkachenko
and Yaschenko in 2006. Moreover, we characterize some classes of locally
compact groups that admit transversal group topologies.
