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On neutral subgroups of topological groups. (English) Zbl 0627.22005

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.

MSC:

22D05 General properties and structure of locally compact groups
22A05 Structure of general topological groups
43A05 Measures on groups and semigroups, etc.
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References:

[1] Bourbaki, N.: General topology, Part 1. Paris: Hermann 1966 · Zbl 0145.19302
[2] Montgomery, D., Zippin, L.: Topological transformation groups. Huntington: Krieger 1974 · Zbl 0323.57023
[3] Poncet, J.: Une classe d’espaces homogènes possédant une mesure invariante. C.R. Acad. Sci. Paris238, 553-554 (1954) · Zbl 0057.02301
[4] Roelcke, W., Dierolf, S.: Uniform structures on topological groups and their quotients. New York: McGraw-Hill 1981 · Zbl 0489.22001
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