## Abstract

Our aim is to study topological minimality of some natural matrix groups. We show that the special upper triangular group ST^{+}(n,F) is minimal for every local field F of characteristic ≠2. This result is new even for the field R of reals and it leads to some important consequences. We prove criteria for the minimality and total minimality of the special linear group SL(n,F), where F is a subfield of a local field. This extends some known results of Remus–Stoyanov (1991) and Bader–Gelander (2017). One of our main applications is a characterization of Fermat primes, which asserts that for an odd prime p the following conditions are equivalent: (1) p is a Fermat prime; (2) SL(p−1,Q) is minimal, where Q is the field of rationals equipped with the p-adic topology; (3) SL(p−1,Q(i)) is minimal, where Q(i)⊂C is the Gaussian rational field.

Original language | English |
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Article number | 108272 |

Journal | Topology and its Applications |

Volume | 322 |

DOIs | |

State | Published - 1 Dec 2022 |

### Bibliographical note

Publisher Copyright:© 2022 Elsevier B.V.

### Funding

This research was supported by a grant of the Israel Science Foundation ( ISF 1194/19 ) and also by the Gelbart Research Institute at the Department of Mathematics, Bar-Ilan University .

Funders | Funder number |
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Gelbart Research Institute at the Department of Mathematics, Bar-Ilan University | |

Israel Science Foundation | ISF 1194/19 |

## Keywords

- Fermat primes
- Iwasawa decomposition
- Local field
- Matrix group
- Minimal topological group
- Projective linear group
- Special linear group