Summary
LetG be a compact group andℒ a sublattice of the lattice of all closed subgroups ofG. In Proposition 1 it is shown thatℒ is a complete lattice if it is a closed subset of the spaceG c of all closed non empty subsets ofG. In general the converse of this fact is not true (Example 3), but the following result can be obtained (Theorem 5): Ifℒ is complete and if each element ofℒ is normalized by the connected component of the identity ofG, thenℒ is a closed, totally disconnected subset ofG c. We mention the following corollary: IfG is totally disconnected or abelian, thenℒ is complete if and only if it is a closed subset ofG c.
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References
A. M. Gleason, Groups without small subgroups. Ann. of Math.56, 193–212 (1952).
S. Grosser andM. Moskowitz, On central topological groups. Trans. Amer. Math. Soc.127, 317–339 (1967).
E.Hewitt and K. H.Ross, Abstract Harmonic Analysis. Berlin-Heidelberg 1963.
K. H. Hofmann undP. S. Mostert, Die topologische Struktur des Raumes der Epimorphismen kompakter Gruppen. Arch. Math.16, 191–196 (1965).
D.Montgomery and L.Zippin, Topological Transformation Groups. New York 1955.
H. Pommer, Subnormalität in topologischen Gruppen. Math. Z.114, 194–200 (1970).
L. S.Pontrjagin, Topologische Gruppen I. Leipzig 1957.
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While writing this paper the author was a fellow of the National Research Council (A 7171).
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Pommer, H. Complete lattices of subgroups in compact groups. Arch. Math 22, 205–208 (1971). https://doi.org/10.1007/BF01222563
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DOI: https://doi.org/10.1007/BF01222563