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Gruenberg-Kegel graphs with four edges of finite solvable cut groups

Sara C. Debón, Diego García-Lucas, Ángel del Río

Abstract

We prove that there is a unique graph with four edges which is the Gruenberg-Kegel graph of a solvable cut group. This contributes to the classification of the Gruenberg-Kegel graphs of solvable cut groups initiated in [A. Bächle, A. Kiefer, S. Maheshwary, and A. del Río. "Gruenberg-Kegel graphs: Cut groups, rational groups and the prime graph question" Forum Mathematicum, vol. 35, no. 2, 2023, pp. 409-429. https://doi.org/10.1515/forum-2022-0086].

Gruenberg-Kegel graphs with four edges of finite solvable cut groups

Abstract

We prove that there is a unique graph with four edges which is the Gruenberg-Kegel graph of a solvable cut group. This contributes to the classification of the Gruenberg-Kegel graphs of solvable cut groups initiated in [A. Bächle, A. Kiefer, S. Maheshwary, and A. del Río. "Gruenberg-Kegel graphs: Cut groups, rational groups and the prime graph question" Forum Mathematicum, vol. 35, no. 2, 2023, pp. 409-429. https://doi.org/10.1515/forum-2022-0086].

Paper Structure

This paper contains 5 sections, 16 theorems, 39 equations.

Table of Contents

  1. Notation
  2. Technical results
  3. Technical results for cut groups
  4. Graph (s)
  5. Graph (t)

Key Result

Theorem 1

The only graph with four edges which is the GK-graph of a solvable cut group is $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (39)

  • Theorem 1
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 29 more