On a Question About Generalized Congruence Subgroups. I
- North-Ossetian State University, SMI VSC RAS (Russian Federation)
A set of additive subgroups σ = (σ{sub ij}), 1 ≤ i, j ≤ n, of a field (or ring) K is called a net of order n over K if σ{sub ir}σ{sub rj} ⊆ σ{sub ij} for all values of the indices i, r, j. The same system, but without diagonal, is called an elementary net. A full or elementary net σ = (σ{sub ij}) is said to be irreducible if all the additive subgroups σ{sub ij} are different from zero. An elementary net σ is closed if the subgroup E(σ) does not contain new elementary transvections. The present paper is related to a question posed by Y. N. Nuzhin in connection with V. M. Levchuk’s question No. 15.46 from the Kourovka notebook about the admissibility (closure) of elementary net (carpet) σ = (σ{sub ij}) over a field K. Let J be an arbitrary subset of {1, . . . , n}, n ≥ 3, and the cardinality m of J satisfies the condition 2 ≤ m ≤ n − 1. Let R be a commutative integral domain (non-field) with identity, and let K be the quotient field of R. An example of a net σ = (σ{sub ij}) of order n over K, for which the group E(σ) ∩ 〈t{sub ij}(K) : i, j ∈ J〉 is not contained in the group 〈t{sub ij}(σ{sub ij}) : i, j ∈ J〉, is constructed.
- OSTI ID:
- 22921136
- Journal Information:
- Journal of Mathematical Sciences, Vol. 243, Issue 4; Other Information: Copyright (c) 2019 Springer Science+Business Media, LLC, part of Springer Nature; Country of input: International Atomic Energy Agency (IAEA); ISSN 1072-3374
- Country of Publication:
- United States
- Language:
- English
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