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Zap. Nauchn. Sem. POMI, 2020, Volume 492, Pages 10–24 (Mi znsl6953)  

Relative centralisers of relative subgroups

N. A. Vavilova, Z. Zhangb

a Department of Mathematics and Computer Science, St. Petersburg State University, St. Petersburg, Russia
b Department of Mathematics, Beijing Institute of Technology, Beijing, China

Abstract: Let $R$ be an associative ring with $1$, $G=\mathrm{GL}(n, R)$ be the general linear group of degree $n\ge 3$ over $R$. In this paper we calculate the relative centralisers of the relative elementary subgroups or the principal congruence subgroups, corresponding to an ideal $A\unlhd R$ modulo the relative elementary subgroups or the principal congruence subgroups, corresponding to another ideal $B\unlhd R$. Modulo congruence subgroups the results are essentially easy exercises in linear algebra. But modulo the elementary subgroups they turned out to be quite tricky, and we could get definitive answers only over commutative rings, or, in some cases, only over Dedekind rings/Dedekind rings of arithmetic type. Bibliography: 43 titles.

Key words and phrases: General linear groups, elementary subgroups, congruence subgroups, standard commutator formula, unrelativised commutator formula, elementary generators.

Funding Agency Grant Number
Russian Science Foundation 17-11-01261
This publication is supported by the Russian Science Foundation grant 17-11-01261.


Full text: PDF file (188 kB)
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UDC: 512.5
Received: 10.03.2020
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Citation: N. A. Vavilov, Z. Zhang, “Relative centralisers of relative subgroups”, Problems in the theory of representations of algebras and groups. Part 35, Zap. Nauchn. Sem. POMI, 492, POMI, St. Petersburg, 2020, 10–24

Citation in format AMSBIB
\Bibitem{VavZha20}
\by N.~A.~Vavilov, Z.~Zhang
\paper Relative centralisers of relative subgroups
\inbook Problems in the theory of representations of algebras and groups. Part~35
\serial Zap. Nauchn. Sem. POMI
\yr 2020
\vol 492
\pages 10--24
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6953}


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