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Subgroups of $GL_{n}(R)$ for local rings $R$
Let $ R $ be a local ring, with maximal ideal $ {\bf m} $, and residue class division ring $ R/{\bf m} = D $. Put $ A = M_n(R), n \geq 1 $, and denote by $ A^* = GL_n(R) $ the group of units of $ A $. Here we investigate some algebraic structure of subnormal and maximal subgroups of $ A^* $. For ins...
Autores principales: | , |
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Lenguaje: | eng |
Publicado: |
2002
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Materias: | |
Acceso en línea: | http://cds.cern.ch/record/645825 |
Sumario: | Let $ R $ be a local ring, with maximal ideal $ {\bf m} $, and residue class division ring $ R/{\bf m} = D $. Put $ A = M_n(R), n \geq 1 $, and denote by $ A^* = GL_n(R) $ the group of units of $ A $. Here we investigate some algebraic structure of subnormal and maximal subgroups of $ A^* $. For instance, when $ D $ is of finite dimension over its center, it is shown that finitely generated subnormal subgroups of $ A^* $ are central. It is also proved that maximal subgroups of $ A^* $ are not finitely generated. Furthermore, assume that $ P $ is a nonabelian maximal subgroup of $ GL_1(R) $ such that $ P $ contains a noncentral soluble normal subgroup of finite index, it is shown that $ D $ is a crossed product division algebra. |
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