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Sandwich classification for GLn(R), O2n(R) and U2n(R,Λ) revisited

  • Raimund Preusser EMAIL logo
Published/Copyright: September 2, 2017
Journal of Group Theory
From the journal Journal of Group Theory Volume 21 Issue 1

Abstract

Let n be a natural number greater than or equal to 3, R a commutative ring and σ∈GLn⁢(R). We show that tk⁢l⁢(σi⁢j) (resp. tk⁢l⁢(σi⁢i-σj⁢j)), where i≠j and k≠l can be expressed as a product of eight (resp. 24) matrices of the form σ±1ϵ, where ϵ∈En⁢(R). We prove similar results for the orthogonal groups O2⁢n⁢(R) and the hyperbolic unitary groups U2⁢n⁢(R,Λ) under the assumption that R is commutative and n≥3. This yields new, very short proofs of the Sandwich Classification Theorems for the groups GLn⁢(R), O2⁢n⁢(R) and U2⁢n⁢(R,Λ).

Communicated by John S. Wilson

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Received: 2017-5-5
Revised: 2017-7-21
Published Online: 2017-9-2
Published in Print: 2018-1-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with a single non-trivial Jordan block
  3. Sandwich classification for GLn(R), O2n(R) and U2n(R,Λ) revisited
  4. Critical maximal subgroups and conjugacy of supplements in finite soluble groups
  5. Finite $\smash{p}$\hbox{-}groups of conjugate type {1,p3}
  6. Normal subgroups in limit groups of prime index
  7. Uncountable locally free groups and their group rings
  8. Full box spaces of free groups
  9. Almost congruence extension property for subgroups of free groups
  10. Solving equations of length seven over torsion-free groups
  11. On the structure of virtually nilpotent compact p-adic analytic groups
https://doi.org/10.1515/jgth-2017-0028

Articles in the same Issue

  1. Frontmatter
  2. Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with a single non-trivial Jordan block
  3. Sandwich classification for GLn(R), O2n(R) and U2n(R,Λ) revisited
  4. Critical maximal subgroups and conjugacy of supplements in finite soluble groups
  5. Finite $\smash{p}$\hbox{-}groups of conjugate type {1,p3}
  6. Normal subgroups in limit groups of prime index
  7. Uncountable locally free groups and their group rings
  8. Full box spaces of free groups
  9. Almost congruence extension property for subgroups of free groups
  10. Solving equations of length seven over torsion-free groups
  11. On the structure of virtually nilpotent compact p-adic analytic groups
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