Complete lattices of subgroups in compact groups |
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Authors: | Helmut Pommer |
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Affiliation: | (1) Present address: Mathematisches Institut der Universität, 74 Tübingen |
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Abstract: | Summary LetG be a compact group and a sublattice of the lattice of all closed subgroups ofG. In Proposition 1 it is shown that is a complete lattice if it is a closed subset of the spaceGc 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): If is complete and if each element of is normalized by the connected component of the identity ofG, then is a closed, totally disconnected subset ofGc. We mention the following corollary: IfG is totally disconnected or abelian, then is complete if and only if it is a closed subset ofGc.While writing this paper the author was a fellow of the National Research Council (A 7171). |
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