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 Zap. Nauchn. Sem. POMI, 2018, Volume 470, Pages 105–110 (Mi znsl6613)

On a question about generalized congruence subgroups. I

V. A. Koibaevab

a North Ossetian State University after Kosta Levanovich Khetagurov, Vladikavkaz, Russia
b Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences, Vladikavkaz, Russia

Abstract: 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 $\{1,…,n\}$, $n\geq3$, 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 $\langle t_{ij}(\sigma_{ij})\colon i,j\in J\rangle$.

Key words and phrases: nets, elementary nets, closed elementary nets, elementary net group, carpets, carpet groups, admissible elementary nets, transvection.

 Funding Agency Grant Number Russian Academy of Sciences - Federal Agency for Scientific Organizations

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English version:
Journal of Mathematical Sciences (New York), 2019, 243:4, 573–576

UDC: 512.5

Citation: V. A. Koibaev, “On a question about generalized congruence subgroups. I”, Problems in the theory of representations of algebras and groups. Part 33, Zap. Nauchn. Sem. POMI, 470, POMI, St. Petersburg, 2018, 105–110; J. Math. Sci. (N. Y.), 243:4 (2019), 573–576

Citation in format AMSBIB
\Bibitem{Koi18} \by V.~A.~Koibaev \paper On a~question about generalized congruence subgroups.~I \inbook Problems in the theory of representations of algebras and groups. Part~33 \serial Zap. Nauchn. Sem. POMI \yr 2018 \vol 470 \pages 105--110 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6613} \transl \jour J. Math. Sci. (N. Y.) \yr 2019 \vol 243 \issue 4 \pages 573--576 \crossref{https://doi.org/10.1007/s10958-019-04557-7} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85074828081} 

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