All topological groups appearing in this paper shall be considered Hausdorff. A topological group is minimally almost periodic ( \({\textrm{MinAP}}\) ) if it admits no non-trivial continuous homomorphism to a compact group. A topological group G is said to have no small normal subgroups \((\textrm{NSnS})\) if it admits an open neighbourhood of the identity containing no non-trivial normal subgroups of G. This property is a generalization of the classical \(\textrm{NSS}\) (:= no small subgroup) property involved in the literature of the historical fifth problem of Hilbert. In this paper we compare three different, but natural generalizations of the \({\textrm{MinAP}}\) groups: the topological groups G which admit no non-trivial continuous homomorphisms to locally compact groups, to Lie groups and to \(\textrm{NSS}\) groups respectively; we prove that they differ in general. In addition, we give a complete characterization of the topological groups G with the following property: the only \(\textrm{NSnS}\) topological group quotient of G is the trivial group. From this result, we deduce a complete description of the Abelian groups which admit no non-trivial continuous homomorphism to an \(\textrm{NSS}\) group, and prove that the family of these groups is the union of all the classes \(\texttt{SSGP}(\alpha )\) (where \(\alpha \) is an ordinal), which were invented by Dikranjan and Shakhmatov in 2016. As a consequence, we prove that an Abelian group G admits a group topology with no non-trivial continuous homomorphism to an \(\textrm{NSS}\) group if and only if G admits a group topology with the so-called small subgroup generating property of Gould.
