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Finite groups with an affine map of large order

  • Alexander Bors ORCID logo EMAIL logo
Published/Copyright: January 5, 2023
Journal of Group Theory
From the journal Journal of Group Theory Volume 26 Issue 4

Abstract

Let 𝐺 be a group. A function G → G of the form x ↦ x α ⁢ g for a fixed automorphism 𝛼 of 𝐺 and a fixed g ∈ G is called an affine map of 𝐺. The affine maps of 𝐺 form a group, called the holomorph of 𝐺. In this paper, we study finite groups 𝐺 with an affine map of large order. More precisely, we show that if 𝐺 admits an affine map of order larger than 1 2 ⁢ | G | , then 𝐺 is solvable of derived length at most 3. We also show that, more generally, for each ρ ∈ ( 0 , 1 ] , if 𝐺 admits an affine map of order at least ρ ⁢ | G | , then the largest solvable normal subgroup of 𝐺 has derived length at most 4 ⁢ ⌊ log 2 ⁡ ( ρ − 1 ) ⌋ + 3 .

Funding source: Austrian Science Fund

Award Identifier / Grant number: J4072-N32

Funding statement: The author was supported by the Austrian Science Fund (FWF), project J4072-N32 “Affine maps on finite groups”. The research for this paper was conducted at the Johann Radon Institute for Computational and Applied Mathematics (RICAM) in Linz, Austria.

Acknowledgements

The author would like to thank the anonymous referee for their careful reading of the manuscript and their helpful comments, which have improved the paper.

  1. Communicated by: Timothy C. Burness

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Received: 2022-05-30
Revised: 2022-09-19
Published Online: 2023-01-05
Published in Print: 2023-07-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Curvature distribution and hyperbolicity
  3. On Whitehead’s cut vertex lemma
  4. Abelian splittings and JSJ-decompositions of finitely presented Bestvina–Brady groups
  5. On endomorphisms of the direct product of two free groups
  6. The R ∞ property for nilpotent quotients of Generalized Solvable Baumslag–Solitar groups
  7. A note on torsion length and torsion subgroups
  8. On the separability of subgroups of nilpotent groups by root classes of groups
  9. The Addition Theorem for two-step nilpotent torsion groups
  10. A bound for the orders of centralizers of irreducible subgroups of algebraic groups
  11. The average character degree of finite groups and Gluck’s conjecture
  12. Character degrees of normally monomial 𝑝-groups of maximal class
  13. Finite groups with an affine map of large order
  14. Classification of non-solvable groups whose power graph is a cograph
https://doi.org/10.1515/jgth-2022-0096

Articles in the same Issue

  1. Frontmatter
  2. Curvature distribution and hyperbolicity
  3. On Whitehead’s cut vertex lemma
  4. Abelian splittings and JSJ-decompositions of finitely presented Bestvina–Brady groups
  5. On endomorphisms of the direct product of two free groups
  6. The R ∞ property for nilpotent quotients of Generalized Solvable Baumslag–Solitar groups
  7. A note on torsion length and torsion subgroups
  8. On the separability of subgroups of nilpotent groups by root classes of groups
  9. The Addition Theorem for two-step nilpotent torsion groups
  10. A bound for the orders of centralizers of irreducible subgroups of algebraic groups
  11. The average character degree of finite groups and Gluck’s conjecture
  12. Character degrees of normally monomial 𝑝-groups of maximal class
  13. Finite groups with an affine map of large order
  14. Classification of non-solvable groups whose power graph is a cograph
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