Finite Groups with K 4-Free Prime Graphs

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 dividing some degrees in cd(G). The prime graph Δ(G) has vertex set ρ(G) and there is an edge between two distinct vertices p and q if pq divides some degree a∈cd(G). In this paper we show that if Δ(G) is K 4-free, then |ρ(G)|≤7; and moreover, if G is solvable, then |ρ(G)|≤6. These bounds are best possible.