# On a Question About Generalized Congruence Subgroups. I

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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*_{ir}*σ*_{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.

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