  A topological space is reversible if each continuous bijection of it onto
itself is open. We introduce an analogue of this notion in the category of
topological groups: A topological group G is g-reversible if every continuous
automorphism of G (=continuous isomorphism of G onto itself) is open. The class
of g-reversible groups contains Polish groups, locally compact sigma-compact
groups, minimal groups, abelian groups with the Bohr topology, and reversible
topological groups. We prove that subgroups of R^n are g-reversible, for every
positive integer n. An example of a compact (so reversible) metric abelian
group having a countable dense non-g-reversible subgroup is given. We also
highlight the differences between reversible spaces and g-reversible
topological groups. Many open problems are scattered throughout the paper.
