TY - JOUR

T1 - Subgroups of the full linear group over a Dedekind ring

AU - Borevich, Z. I.

AU - Vavilov, N. A.

AU - Narkiewicz, W.

N1 - Copyright: Copyright 2007 Elsevier B.V., All rights reserved.

PY - 1982/5

Y1 - 1982/5

N2 - We study the subgroups of the full linear group GL(n, R) over a Dedekind ring R that contain the group of quasidiagonal matrices of fixed type with diagonal blocks of at least third order, each of which is generated by elementary matrices. For any such subgroup H there exists a unique D-net σ of ideals of R such that[Figure not available: see fulltext.], where E(σ) is the subgroup generated by all transvections of the net subgroup G(σ). and[Figure not available: see fulltext.] is the normalizer of G(σ). The subgroup E(σ) is normal in[Figure not available: see fulltext.]. To study the factor group[Figure not available: see fulltext.] we introduce an intermediate subgroup F(σ), E(σ) ≤ F(σ) ≤ G(σ). The group[Figure not available: see fulltext.] is finite and is connected with permutations in the symmetric group. The factor group G(σ)/F(σ) is Abelian - these are the values of a certain "determinant." In the calculation of F(σ)/E(σ) appears the SK1-functor. Results are stated without proof.

AB - We study the subgroups of the full linear group GL(n, R) over a Dedekind ring R that contain the group of quasidiagonal matrices of fixed type with diagonal blocks of at least third order, each of which is generated by elementary matrices. For any such subgroup H there exists a unique D-net σ of ideals of R such that[Figure not available: see fulltext.], where E(σ) is the subgroup generated by all transvections of the net subgroup G(σ). and[Figure not available: see fulltext.] is the normalizer of G(σ). The subgroup E(σ) is normal in[Figure not available: see fulltext.]. To study the factor group[Figure not available: see fulltext.] we introduce an intermediate subgroup F(σ), E(σ) ≤ F(σ) ≤ G(σ). The group[Figure not available: see fulltext.] is finite and is connected with permutations in the symmetric group. The factor group G(σ)/F(σ) is Abelian - these are the values of a certain "determinant." In the calculation of F(σ)/E(σ) appears the SK1-functor. Results are stated without proof.

UR - http://www.scopus.com/inward/record.url?scp=34250225169&partnerID=8YFLogxK

U2 - 10.1007/BF01476109

DO - 10.1007/BF01476109

M3 - Article

AN - SCOPUS:34250225169

VL - 19

SP - 982

EP - 987

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 1

ER -