On neutral subgroups of topological groups (Q581562)

A subgroup G of a topological group X is called neutral if, for every neighourhood U of the neutral element e in X, there is a neighbourhood V of e such that GV\(\subset UG\). Neutral subgroups are related to the existence of invariant measures on X/G and to the completeness of (X/G, \({\mathcal L}/G)\) if the left uniformity \({\mathcal L}\) of X is complete. Every normal, every open, and every compact subgroup is neutral. Under certain compactness and connectivity conditions the authors prove: (i) some characterizations of neutrality, (ii) the intersection of a family of closed neutral subgroups of X is again neutral, (iii) for every subgroup G of X there exists a smallest closed neutral subgroup of X containing G. - To include the case of (almost) connected locally compact groups X, the main approximation theorem of Montgomery-Zippin is used. The paper contains many examples.