On a Question About Generalized Congruence Subgroups. I

A set of additive subgroups σ = (σ_ij), 1 ≤ i, j ≤ n, of a field (or ring) K is called a net of order n over K if σ_irσ_rj ⊆ σ_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_ij) is said to be irreducible if all the additive subgroups σ_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_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_ij) of order n over K, for which the group E(σ) ∩ 〈t_ij(K) : i, j ∈ J〉 is not contained in the group 〈t_ij(σ_ij) : i, j ∈ J〉, is constructed.