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The subnormal structure of classical-like groups over commutative rings

  • Raimund Preusser EMAIL logo
Published/Copyright: November 14, 2020
Journal of Group Theory
From the journal Journal of Group Theory Volume 24 Issue 3

Abstract

Let 𝑛 be an integer greater than or equal to 3, and let (R,Δ) be a Hermitian form ring, where 𝑅 is commutative. We prove that if 𝐻 is a subgroup of the odd-dimensional unitary group U2⁢n+1⁡(R,Δ) normalised by a relative elementary subgroup EU2⁢n+1⁡((R,Δ),(I,Ω)), then there is an odd form ideal (J,Σ) such that

EU2⁢n+1⁡((R,Δ),(J⁢Ik,ΩminJ⁢Ik∔Σ∘Ik))≤H≤CU2⁢n+1⁡((R,Δ),(J,Σ)),

where k=12 if n=3 respectively k=10 if n≥4. As a consequence of this result, we obtain a sandwich theorem for subnormal subgroups of odd-dimensional unitary groups.

Funding source: Russian Science Foundation

Award Identifier / Grant number: 19-71-30002

Funding statement: The work is supported by the Russian Science Foundation grant 19-71-30002.

  1. Communicated by: John S. Wilson

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Received: 2020-09-06
Revised: 2020-10-22
Published Online: 2020-11-14
Published in Print: 2021-05-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. The groups (2, 𝑚 | 𝑛, 𝑘 | 1, 𝑞): Finiteness and homotopy
  3. Obstruction to a Higman embedding theorem for residually finite groups with solvable word problem
  4. Metabelian groups: Full-rank presentations, randomness and Diophantine problems
  5. Nilpotence relations in products of groups
  6. The subnormal structure of classical-like groups over commutative rings
  7. A classification of the abelian minimal closed normal subgroups of locally compact second-countable groups
  8. On the stabilisers of points in groups with micro-supported actions
  9. Groups that have a partition by commuting subsets
  10. Commuting graph of 𝐴-orbits
  11. On the holomorph of finite semisimple groups
  12. TI subgroups and depth 3 subgroups in simple Suzuki groups
  13. On the minimal degree of a transitive permutation group with stabilizer a 2-group
  14. Exponent of a finite group admitting a coprime automorphism of prime order
https://doi.org/10.1515/jgth-2020-0136

Articles in the same Issue

  1. Frontmatter
  2. The groups (2, 𝑚 | 𝑛, 𝑘 | 1, 𝑞): Finiteness and homotopy
  3. Obstruction to a Higman embedding theorem for residually finite groups with solvable word problem
  4. Metabelian groups: Full-rank presentations, randomness and Diophantine problems
  5. Nilpotence relations in products of groups
  6. The subnormal structure of classical-like groups over commutative rings
  7. A classification of the abelian minimal closed normal subgroups of locally compact second-countable groups
  8. On the stabilisers of points in groups with micro-supported actions
  9. Groups that have a partition by commuting subsets
  10. Commuting graph of 𝐴-orbits
  11. On the holomorph of finite semisimple groups
  12. TI subgroups and depth 3 subgroups in simple Suzuki groups
  13. On the minimal degree of a transitive permutation group with stabilizer a 2-group
  14. Exponent of a finite group admitting a coprime automorphism of prime order
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