Locally compact groups with closed subgroups open and p-adic

Author:

Hofmann Karl H.,Morris Sidney A.,Oates-Williams Sheila,Obraztsov V. N.

Abstract

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[).

Publisher

Cambridge University Press (CUP)

Subject

General Mathematics

Reference5 articles.

1. The additive structure of integer groups and p–adic number fields;Robertson;Proc. Amer. Math. Soc.,1968

2. Some applications of graded diagrams in combinatorial group theory;Ivanov;Groups,1991

3. Locally Compact Groups with every Closed Subgroup of Finite Index

4. A Characterization of the Topological Group of p -Adic Integers

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2. Characterization of locally compact groups by closed totally disconnected subgroups;Monatshefte für Mathematik;2022-01-13

3. Hereditarily minimal topological groups;Forum Mathematicum;2019-05-01

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