τ-locally invariant groups

A class of τ-locally invariant topological groups is introduced; this class is a new one for τ a limit cardinal, and it coincides with Arhangelʹskiı̆ʹs class of λ-balanced groups for τ=λ+. It is proved that a topological group is τ-locally invariant if and only if this group is topologically isomorphic to a subgroup of a direct product of groups of topological character less than τ. Topological spaces whose free topological groups are τ-locally invariant are described.