## Abstract

We study locally compact groups having all dense subgroups (locally) minimal. We call such groups densely (locally) minimal. In 1972 Prodanov proved that the infinite compact abelian groups having all subgroups minimal are precisely the groups Z_{p} of p-adic integers. In [30], we extended Prodanov's theorem to the non-abelian case at several levels. In this paper, we focus on the densely (locally) minimal abelian groups. We prove that in case that a topological abelian group G is either compact or connected locally compact, then G is densely locally minimal if and only if G either is a Lie group or has an open subgroup isomorphic to Z_{p} for some prime p. This should be compared with the main result of [9]. Our Theorem C provides another extension of Prodanov's theorem: an infinite locally compact group is densely minimal if and only if it is isomorphic to Z_{p}. In contrast, we show that there exists a densely minimal, compact, two-step nilpotent group that neither is a Lie group nor it has an open subgroup isomorphic to Z_{p}.

Original language | English |
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Article number | 106846 |

Journal | Topology and its Applications |

Volume | 266 |

DOIs | |

State | Published - 1 Oct 2019 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2019 Elsevier B.V.

### Funding

The first-named author is supported by grant NSFC , number 11571175 , and takes this opportunity to thank Dikran Dikranjan for his generous hospitality and support. The second-named author is partially supported by grant PSD-2015-2017-DIMA - progetto PRID TokaDyMA of Udine University . The third and fourth-named authors are supported by Programma SIR 2014 by MIUR , project GADYGR, number RBSI14V2LI , cup G22I15000160008. The first-named author is supported by grant NSFC, number 11571175, and takes this opportunity to thank Dikran Dikranjan for his generous hospitality and support. The second-named author is partially supported by grant PSD-2015-2017-DIMA - progetto PRID TokaDyMA of Udine University. The third and fourth-named authors are supported by Programma SIR 2014 by MIUR, project GADYGR, number RBSI14V2LI, cup G22I15000160008.

Funders | Funder number |
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Udine University | |

National Natural Science Foundation of China | PSD-2015-2017-DIMA, 11571175 |

Ministero dell’Istruzione, dell’Università e della Ricerca | G22I15000160008 |

Università degli Studi di Udine |

## Keywords

- Hereditarily (locally) minimal
- Hilbert-Smith Conjecture
- Lie group
- Locally minimal group
- Non-topologizable group
- p-adic integer