Groups whose prime graphs have no triangles

TL;DR

This paper investigates the structure of prime graphs of finite groups with no triangles, establishing an upper vertex limit and classifying possible graphs, while ruling out certain cycle and tree configurations.

Contribution

It proves that prime graphs without triangles have at most five vertices and classifies all such graphs, also excluding cycle and tree structures with five or more vertices.

Findings

Prime graphs with no triangles have at most 5 vertices.

Complete classification of triangle-free prime graphs with 5 vertices.

Prime graphs cannot be cycles or trees with at least 5 vertices.

Abstract

Let G be a finite group and let cd(G) be the set of all complex irreducible character degrees of G Let \rho(G) be the set of all primes which divide some character degree of G. The prime graph \Delta(G) attached to G is a graph whose vertex set is \rho(G) and there is an edge between two distinct primes u and v if and only if the product uv divides some character degree of G. In this paper, we show that if G is a finite group whose prime graph \Delta(G) has no triangles, then \Delta(G) has at most 5 vertices. We also obtain a classification of all finite graphs with 5 vertices and having no triangles which can occur as prime graphs of some finite groups. Finally, we show that the prime graph of a finite group can never be a cycle nor a tree with at least 5 vertices.