Products of locally minimal groups

Dikranjan, Dikran; Peng, Dekui; Xi, Wenfei; Xiao, Zhiqiang

doi:10.1016/j.topol.2022.108368

Cited by 1 publication

(2 citation statements)

References 32 publications

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“…In their recent paper [16], the authors call a minimal group G z-minimal if G/Z(G) is minimal. In view of the results obtained in [2,8], it holds that in case F is a local field, then SL(n, F) is z-minimal.…”

Section: Remarkmentioning

confidence: 99%

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Minimality Conditions Equivalent to the Finitude of Fermat and Mersenne Primes

Shlossberg¹

2023

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The question is still open as to whether there exist infinitely many Fermat primes or infinitely many composite Fermat numbers. The same question concerning Mersenne numbers is also unanswered. Extending some recent results of Megrelishvili and the author, we characterize the Fermat primes and the Mersenne primes in terms of the topological minimality of some matrix groups. This is achieved by showing, among other things, that if F is a subfield of a local field of characteristic ≠2, then the special upper triangular group ST+(n,F) is minimal precisely when the special linear group SL(n,F) is. We provide criteria for the minimality (and total minimality) of SL(n,F) and ST+(n,F), where F is a subfield of C. Let Fπ and Fc be the set of Fermat primes and the set of composite Fermat numbers, respectively. As our main result, we prove that the following conditions are equivalent for A∈{Fπ,Fc}: A is finite; ∏Fn∈ASL(Fn−1,Q(i)) is minimal, where Q(i) is the Gaussian rational field; and ∏Fn∈AST+(Fn−1,Q(i)) is minimal. Similarly, denote by Mπ and Mc the set of Mersenne primes and the set of composite Mersenne numbers, respectively, and let B∈{Mπ,Mc}. Then the following conditions are equivalent: B is finite; ∏Mp∈BSL(Mp+1,Q(i)) is minimal; and ∏Mp∈BST+(Mp+1,Q(i)) is minimal.

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Section: Remarkmentioning

confidence: 99%

“…Moreover, by Lemma 1 and [2] (Theorem 3.17) also ST + (n, F) is z-minimal in case the local field F has a characteristic distinct from 2. By [16] (Corollary 4.9), a product of complete z-minimal groups is minimal. This provides an alternative proof for Theorem 6.…”

Section: Remarkmentioning

confidence: 99%

Minimality Conditions Equivalent to the Finitude of Fermat and Mersenne Primes

Shlossberg¹

2023

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Products of locally minimal groups

Cited by 1 publication

References 32 publications

Minimality Conditions Equivalent to the Finitude of Fermat and Mersenne Primes

Minimality Conditions Equivalent to the Finitude of Fermat and Mersenne Primes

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