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Classification of non-solvable groups whose power graph is a cograph

  • Jendrik Brachter and Eda Kaja ORCID logo EMAIL logo
Published/Copyright: January 20, 2023
Journal of Group Theory
From the journal Journal of Group Theory Volume 26 Issue 4

Abstract

In recent work, Cameron, Manna and Mehatari have studied the finite groups whose power graph is a cograph, which we refer to as power-cograph groups. They classify the nilpotent groups with this property, and they establish partial results in the general setting, highlighting certain number-theoretic difficulties that arise for the simple groups of the form PSL 2 ⁡ ( q ) or Sz ⁡ ( 2 2 ⁢ e + 1 ) . In this paper, we prove that these number-theoretic problems are in fact the only obstacles to the classification of non-solvable power-cograph groups. Specifically, for the non-solvable case, we give a classification of power-cograph groups in terms of such groups isomorphic to PSL 2 ⁡ ( q ) or Sz ⁡ ( 2 2 ⁢ e + 1 ) . For the solvable case, we are able to precisely describe the structure of solvable power-cograph groups. We obtain a complete classification of solvable power-cograph groups whose Gruenberg–Kegel graph is connected. Moreover, we reduce the case where the Gruenberg–Kegel graph is disconnected to the classification of 𝑝-groups admitting fixed-point-free automorphisms of prime power order, which is in general an open problem.

Funding source: H2020 European Research Council

Award Identifier / Grant number: 820148

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: SFB-TRR 195

Funding statement: The research leading to these results has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (EngageS: grant agreement No. 820148) and from the German Research Foundation DFG (SFB-TRR 195 “Symbolic Tools in Mathematics and their Application”).

Acknowledgements

We thank Pascal Schweitzer for the helpful comments on an earlier draft of this paper. We thank an anonymous referee for pointing out a gap in a previous proof of Lemma 6.3 which led to the inclusion of the group M 10 into Theorem 1.1. We are grateful to a potentially different anonymous referee for the helpful comments which have improved the presentation of this paper and provided us with shorter arguments for Lemma 6.3 and Lemma 6.5.

  1. Communicated by: Christopher W. Parker

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Received: 2022-05-03
Revised: 2022-09-30
Published Online: 2023-01-20
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-2022-0081

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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