2020/04/29 by N. A. Vavilov, Vavilov, Nikolai, Zuhong Zhang +1
Computer Science · Mathematics · #Advanced Topics in Algebra #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2004.14285
openalex publication_date 2020/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let R be an associative ring with 1, G=GL(n, R) be the general linear group of degree n≥ 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. We discuss also some further related problems, such as the interrelations of various birelative commutator subgroups, etc., and state several unsolved questions.