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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

References

[1] A. Bak, The stable structure of quadratic modules, Thesis, Columbia University, 1969. Search in Google Scholar

[2] A. Bak and R. Preusser, The E-normal structure of odd dimensional unitary groups, J. Pure Appl. Algebra 222 (2018), no. 9, 2823–2880. 10.1016/j.jpaa.2017.11.002Search in Google Scholar

[3] V. A. Petrov, Odd unitary groups, J. Math. Sci. 305 (2005), no. 3, 4752–4766. 10.1007/s10958-005-0372-zSearch in Google Scholar

[4] R. Preusser, Sandwich classification for O2⁢n+1⁢(R) and U2⁢n+1⁢(R,Δ) revisited, J. Group Theory 21 (2018), no. 4, 539–571. 10.1515/jgth-2018-0011Search in Google Scholar

[5] R. Preusser, Reverse decomposition of unipotents over noncommutative rings I: General linear groups, Linear Algebra Appl. 601 (2020), 285–300. 10.1016/j.laa.2020.05.013Search in Google Scholar

[6] L. N. Vaserstein, The subnormal structure of general linear groups, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 3, 425–431. 10.1017/S0305004100064367Search in Google Scholar

[7] L. N. Vaserstein, The subnormal structure of general linear groups over rings, Math. Proc. Cambridge Philos. Soc. 108 (1990), no. 2, 219–229. 10.1017/S0305004100069097Search in Google Scholar

[8] N. Vavilov, Towards the reverse decomposition of unipotents. II. The relative case, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 484 (2019), 5–22. 10.1007/s10958-021-05195-8Search in Google Scholar

[9] N. A. Vavilov, A note on the subnormal structure of general linear groups, Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 2, 193–196. 10.1017/S030500410006847XSearch in Google Scholar

[10] J. S. Wilson, The normal and subnormal structure of general linear groups, Proc. Cambridge Philos. Soc. 71 (1972), 163–177. 10.1017/S0305004100050416Search in Google Scholar

[11] H. You, Subgroups of classical groups normalized by relative elementary groups, J. Pure Appl. Algebra 216 (2012), no. 5, 1040–1051. 10.1016/j.jpaa.2011.12.003Search in Google Scholar

[12] Z. Zhang, Stable sandwich classification theorem for classical-like groups, Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 3, 607–619. 10.1017/S0305004107000527Search in Google Scholar

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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