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Locally compact groups with closed subgroups open and p-adic

Published online by Cambridge University Press:  24 October 2008

Karl H. Hofmann
Affiliation:
Fachbereich Mathematik, Technische Hochschule Darmstadt, Schlonssgartenstrasse 7, D-6100 Darmstadt, Germany e-mail: [email protected]
Sidney A. Morris
Affiliation:
Faculty of Informatics, The University of Wollongong, Wollongong, NSW 2522, Australia, e-mail: [email protected]
Sheila Oates-Williams
Affiliation:
Department of Mathematics, The University of Queensland, Qld 4072, Australia e-mail: [email protected]
V. N. Obraztsov
Affiliation:
Department of Mathematics and Physics, Kostroma Teachers' Training Institute, Kostroma 156601, Russia

Extract

An open subgroup U of a topological group G is always closed, since U is the complement of the open set . An arbitrary closed subgroup C of G is almost never open, unless G belongs to a small family of exceptional groups. In fact, if G is a locally compact abelian group in which every non-trivial subgroup is open, then G is the additive group δp of p-adic integers or the additive group Ωp of p-adic rationale (cf. Robertson and Schreiber[5[, proposition 7). The fact that δp has interesting properties as a topological group has many roots. One is that its character group is the Prüfer group ℤp, which makes it unique inside the category of compact abelian groups. But even within the bigger class of not necessarily abelian compact groups the p-adic group δp is distinguished: it is the only one all of whose non-trivial subgroups are isomorphic (cf. Morris and Oates-Williams[2[), and it is also the only one all of whose non-trivial closed subgroups have finite index (cf. Morris, Oates-Williams and Thompson [3[).

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1995

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References

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