## Abstract

In this paper, we study topological groups having all closed subgroups (totally) minimal and we call such groups c-(totally) minimal. We show that a locally compact c-minimal connected group is compact. Using a well-known theorem of [P. Hall and C. R. Kulatilaka, A property of locally finite groups, J. Lond. Math. Soc. 39 1964, 235-239] and a characterization of a certain class of Lie groups, due to [S. K. Grosser and W. N. Herfort, Abelian subgroups of topological groups, Trans. Amer. Math. Soc. 283 1984, 1, 211-223], we prove that a c-minimal locally solvable Lie group is compact. It is shown that a topological group G is c-(totally) minimal if and only if G has a compact normal subgroup N such that G/NG/N is c-(totally) minimal. Applying this result, we prove that a locally compact group G is c-totally minimal if and only if its connected component c(G) is compact and G/c(G) is c-totally minimal. Moreover, a c-totally minimal group that is either complete solvable or strongly compactly covered must be compact. Negatively answering [D. Dikranjan and M. Megrelishvili, Minimality conditions in topological groups, Recent Progress in General Topology. III, Atlantis Press, Paris 2014, 229-327, Question 3.10 (b)], we find, in contrast, a totally minimal solvable (even metabelian) Lie group that is not compact.

Original language | English |
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Pages (from-to) | 1641-1650 |

Number of pages | 10 |

Journal | Forum Mathematicum |

Volume | 33 |

Issue number | 6 |

DOIs | |

State | Published - 1 Nov 2021 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2021 Walter de Gruyter GmbH, Berlin/Boston.

### Funding

The first author is supported by the National Natural Science Foundation of China (Grant No. 12001264)

Funders | Funder number |
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National Natural Science Foundation of China | 12001264 |

## Keywords

- C-minimal group
- Hereditarily non-topologizable group
- Lie group
- Locally solvable group
- Minimal group