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.