On a~question about generalized congruence subgroups.~I

V. A. Koibaev

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 33, Tome 470 (2018), pp. 105-110

Résumé

A system of additive subgroups $\sigma=(\sigma_{ij})$, $1\leq i,j\leq n$, of a field (or ring) $K$ is called a net of order $n$ over $K$ if $\sigma_{ir}\sigma_{rj}\subseteq{\sigma_{ij}}$ for all values of the indices $i,r,j$. The same system, but without the diagonal, is called an elementary net. A full or elementary net $\sigma=(\sigma_{ij})$ is called irreducible if all additive subgroups $\sigma_{ij}$ are different from zero. An elementary net $\sigma$ is closed if the subgroup $E(\sigma)$ does not contain new elementary transvections. This work is related to the question posed by Y. N. Nuzhin in connection with the question of V. M. Levchuk 15.46 from the Kourovka notebook about the admissibility (closedness) of the elementary net (carpet) $\sigma=(\sigma_{ij})$ over a field $K$. Let $J$ be an arbitrary a subset of the set $\{1,\dots,n\}$, $n\geq3$, we assume that the number $|J|=m$ of elements of the set $J$ satisfies the condition $2\leq m\leq n-1$. Let $R$ be a commutative integral domain (non-field) $1\in R$, $K$ be the quotient field of a $R$. We give an example of a net $\sigma=(\sigma_{ij})$ of order $n$ over a field $K$, for which the group $E(\sigma)\cap\langle t_{ij}(K)\colon i,j\in J\rangle$ is not contained in the group $\langle t_{ij}(\sigma_{ij})\colon i,j\in J\rangle$.