A survey on strong reflexivity of abelian topological groups

An Abelian topological group is called strongly reflexive if every closed subgroup and every Hausdorff quotient of the group and of its dual group are reflexive. In the class of locally compact Abelian groups (LCA) there is no need to define "strong reflexivity": it does not add anything new to reflexivity, which by the Pontryagin - van Kampen Theorem is known to hold for every member of the class. In this survey we collect how much of "reflexivity" holds for diferent classes of groups, with especial emphasis in the classes of pseudocompact groups, !-groups and P-groups, in which some reexive groups have been recently detected. In section 3.5 we complete the duality relationship between the classes of P-groups and !-bounded groups, already outlined in [26]. By no means we can claim completeness of the survey: just an ordered view of the topic, with some small new results indicated in the text