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

TL;DR

This paper establishes the uniqueness of a specific Gruenberg-Kegel graph with four edges for finite solvable cut groups, advancing their classification.

Contribution

It proves the existence of a unique four-edge Gruenberg-Kegel graph for solvable cut groups, completing part of their classification effort.

Findings

Identified the unique four-edge Gruenberg-Kegel graph for solvable cut groups.

Contributed to the classification of Gruenberg-Kegel graphs in solvable 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. B\"achle, A. Kiefer, S. Maheshwary, and A. del R\'io. "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].