On Hewitt groups and finer locally compact group topologies

We study the class HH of all topological abelian groups (G,τ)(G,τ) with the property that the only locally compact group topology on G strictly finer than τ   is the discrete topology. Hewitt showed that the reals RR and the circle group TT belong to HH and Rickert and Rajagopalan characterized the non-discrete locally compact groups in the class HH as those that contain either T,RT,R, or the compact group JpJp of p  -adic integers as an open subgroup. We describe the entire class HH, providing many examples of non-locally compact groups with this property. This allows us to obtain as a by-product the results of Hewitt, Rajagopalan and Rickert and give some applications to characterized subgroups.