Groups whose prime graphs have no triangles

Abstract

Let G be a finite group and let cd(G) be the set of all complex irreducible character degrees of G Let ρ(G) be the set of all primes which divide some character degree of G. The prime graph Δ(G) attached to G is a graph whose vertex set is ρ(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 Δ(G) has no triangles, then Δ(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.