The Gruenberg-Kegel graph of finite solvable rational groups

TL;DR

This paper classifies the Gruenberg-Kegel graphs of finite solvable rational groups, completing previous work and providing a comprehensive understanding of their prime graph structures.

Contribution

It completes the classification of Gruenberg-Kegel graphs for finite solvable rational groups, advancing the understanding of their structure.

Findings

Complete classification of Gruenberg-Kegel graphs for these groups

Identification of possible prime graph configurations

Extension of previous partial results

Abstract

A finite group $G$ is said to be rational if every character of $G$ is rational-valued. The Gruenberg-Kegel graph of a finite group $G$ is the undirected graph whose vertices are the primes dividing the order of $G$ and the edges join different primes $p$ and $q$ whenever $G$ contains an element of order $pq$. In this paper, we complete the classification of the Gruenberg-Kegel graphs of finite solvable rational groups initiated in \cite{BKMdR}.