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A classification of the prime graphs of pseudo-solvable groups

  • Ziyu Huang , Thomas Michael Keller ORCID logo EMAIL logo , Shane Kissinger , Wen Plotnick , Maya Roma and Yong Yang ORCID logo
Published/Copyright: July 19, 2023
Journal of Group Theory
From the journal Journal of Group Theory Volume 27 Issue 1

Abstract

The prime graph Γ ⁢ ( G ) of a finite group 𝐺 (also known as the Gruenberg–Kegel graph) has as its vertices the prime divisors of | G | , and p ⁢ - ⁢ q is an edge in Γ ⁢ ( G ) if and only if 𝐺 has an element of order p ⁢ q . Since their inception in the 1970s, these graphs have been studied extensively; however, completely classifying the possible prime graphs for larger families of groups remains a difficult problem. For solvable groups, such a classification was found in 2015. In this paper, we go beyond solvable groups for the first time and characterize the prime graphs of a more general class of groups we call pseudo-solvable. These are groups whose composition factors are either cyclic or isomorphic to A 5 . The classification is based on two conditions: the vertices { 2 , 3 , 5 } form a triangle in Γ ̄ ⁢ ( G ) or { p , 3 , 5 } form a triangle for some prime p ≠ 2 . The ideas developed in this paper also lay the groundwork for future work on classifying and analyzing prime graphs of more general classes of finite groups.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1757233

Funding source: National Security Agency

Award Identifier / Grant number: H98230-21-1-0333

Funding statement: This research was conducted at Texas State University under NSF-REU grant DMS-1757233 and NSA grant H98230-21-1-0333 during the summer of 2021.

Acknowledgements

The authors thank NSF and NSA for the financial support. The first, third, fourth, and fifth thank Texas State University for running the REU online during this difficult period of social distancing and providing a welcoming and supportive work environment. Those authors also thank their mentor, the second author Dr. Thomas Michael Keller, for his invaluable advice and guidance throughout this project. The sixth author, the director of the REU program, is recognized for conducting an inspired and successful research program. The results in this paper were mainly discovered by the first author under the guidance of the second author.

  1. Communicated by: Hung Tong-Viet

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Received: 2023-02-07
Revised: 2023-06-01
Published Online: 2023-07-19
Published in Print: 2024-01-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Elementary amenable groups of cohomological dimension 3
  3. The nilpotent genus of finitely generated residually nilpotent groups
  4. Quantifying lawlessness in finitely generated groups
  5. On multidimensional Schur rings of finite groups
  6. A classification of the prime graphs of pseudo-solvable groups
  7. On generations by conjugate elements in almost simple groups with socle 2𝐹4(𝑞2)′
  8. A characterization of the simple Ree groups 2𝐹4(𝑞2) by their character codegrees
  9. A characterization of finite groups having a single Galois conjugacy class of certain irreducible characters
  10. Class-two quotients of finite permutation groups
  11. A note on 𝑑-maximal 𝑝-groups
https://doi.org/10.1515/jgth-2023-0018

Articles in the same Issue

  1. Frontmatter
  2. Elementary amenable groups of cohomological dimension 3
  3. The nilpotent genus of finitely generated residually nilpotent groups
  4. Quantifying lawlessness in finitely generated groups
  5. On multidimensional Schur rings of finite groups
  6. A classification of the prime graphs of pseudo-solvable groups
  7. On generations by conjugate elements in almost simple groups with socle 2𝐹4(𝑞2)′
  8. A characterization of the simple Ree groups 2𝐹4(𝑞2) by their character codegrees
  9. A characterization of finite groups having a single Galois conjugacy class of certain irreducible characters
  10. Class-two quotients of finite permutation groups
  11. A note on 𝑑-maximal 𝑝-groups
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