A topological group is locally pseudocompact if it contains a nonempty open set with pseudocompact closure. In this paper, we prove that if G is a group with the property that every closed subgroup of G is locally pseudocompact,
then G0 is dense in the component of the completion of G, and G/G_0 is zero-dimensional. We also provide examples of hereditarily disconnected pseudocompact groups with strong minimality properties of arbitrarily large dimension, and thus show that G/G_0 may fail to be zero-dimensional even for totally minimal pseudocompact groups.
