The Transversality on locally pseudocompact groups

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.