Remarks on g-reversible topological groups

Abstract

Our main objective is to study g-reversible groups introduced in Chatyrko and Shakhmatov (2020) [3]. We first give a necessary and sufficient condition for a locally compact group with an open compact subgroup to be g-reversible, it can be applied to give complete solutions of Questions 8.5, 9.7, 9.8, and Problem 9.6 in the previous article [3]. We also investigate g-reversibility of some precompact abelian groups, in particular, we give a pseudocompact g-reversible abelian group which is not minimal, which answers Question 11.5 in the same article [3] negatively. We give descriptions of connected (or totally disconnected) compact abelian groups of which all subgroups are g-reversible. Finally, we give a characterization of the abelian groups which admit hereditarily g-reversible compact group topologies.