Relative Centralizers of Relative Subgroups
Copyright policy )Relative Centralizers of Relative Subgroups
Let $R$ be an associative ring with 1, $G=GL(n, R)$ 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. We discuss also some further related problems, such as the interrelations of various birelative commutator subgroups, etc., and state several unsolved questions.
12 pages
- Beijing Institute of Technology China (People's Republic of)
- St Petersburg University Russian Federation
Unimodular groups, congruence subgroups (group-theoretic aspects), elementary subgroup, congruence subgroup, Rings and Algebras (math.RA), Structure theory for linear algebraic groups, FOS: Mathematics, relative centralizer, Mathematics - Rings and Algebras, Group Theory (math.GR), Linear algebraic groups over adèles and other rings and schemes, Mathematics - Group Theory
Unimodular groups, congruence subgroups (group-theoretic aspects), elementary subgroup, congruence subgroup, Rings and Algebras (math.RA), Structure theory for linear algebraic groups, FOS: Mathematics, relative centralizer, Mathematics - Rings and Algebras, Group Theory (math.GR), Linear algebraic groups over adèles and other rings and schemes, Mathematics - Group Theory
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