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Hall classes in linear groups

  • Francesco de Giovanni EMAIL logo , Marco Trombetti ORCID logo and Bertram A. F. Wehrfritz
Published/Copyright: September 21, 2023
Journal of Group Theory
From the journal Journal of Group Theory Volume 27 Issue 2

Abstract

A well-known theorem of Philip Hall states that if a group 𝐺 has a nilpotent normal subgroup 𝑁 such that G/Nβ€² is nilpotent, then 𝐺 itself is nilpotent. We say that a group class 𝔛 is a Hall class if it contains every group 𝐺 admitting a nilpotent normal subgroup 𝑁 such that G/Nβ€² belongs to 𝔛. Examples have been given in [F. de Giovanni, M. Trombetti and B. A. F. Wehfritz, Hall classes of groups, to appear] to show that finite-by-𝔛 groups do not form a Hall class for many natural choices of the Hall class 𝔛. Although these examples are often linear, our aim here is to prove that the situation is much better within certain natural subclasses of the universe of linear groups.

Acknowledgements

The first two authors are supported by GNSAGA (INdAM) and are members of AGTA – Advances in Group Theory and Applications (www.advgrouptheory.com).

  1. Communicated by: Evgenii I. Khukhro

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Received: 2023-04-25
Revised: 2023-08-17
Published Online: 2023-09-21
Published in Print: 2024-03-01

Β© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Presentations of Schur covers of braid groups
  3. Graphical complexes of groups
  4. Narrow normal subgroups of Coxeter groups and of automorphism groups of Coxeter groups
  5. A closure operator on the subgroup lattice of GL(𝑛,π‘ž) and PGL(𝑛,π‘ž) in relation to the zeros of the MΓΆbius function
  6. On the strong connectivity of the 2-Engel graphs of almost simple groups
  7. Finitely generated metabelian groups arising from integer polynomials
  8. Finite 𝑝-groups of class two with a small multiple holomorph
  9. Hall classes in linear groups
  10. Projective representations of Heisenberg groups over the rings of order 𝑝2
https://doi.org/10.1515/jgth-2023-0063

Articles in the same Issue

  1. Frontmatter
  2. Presentations of Schur covers of braid groups
  3. Graphical complexes of groups
  4. Narrow normal subgroups of Coxeter groups and of automorphism groups of Coxeter groups
  5. A closure operator on the subgroup lattice of GL(𝑛,π‘ž) and PGL(𝑛,π‘ž) in relation to the zeros of the MΓΆbius function
  6. On the strong connectivity of the 2-Engel graphs of almost simple groups
  7. Finitely generated metabelian groups arising from integer polynomials
  8. Finite 𝑝-groups of class two with a small multiple holomorph
  9. Hall classes in linear groups
  10. Projective representations of Heisenberg groups over the rings of order 𝑝2
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