Groups whose Bipartite Divisor Graph for Character Degrees Has Five Vertices

Abstract

. Let G be a finite group and cd ∗ ( G ) be the set of nonlinear irreducible character degrees of G . Suppose that ρ ( G ) denotes the set of primes dividing some element of cd ∗ ( G ). The bipartite divisor graph for the set of character degrees which is denoted by B ( G ), is a bipartite graph whose vertices are the disjoint union of ρ ( G ) and cd ∗ ( G ), and a vertex p ∈ ρ ( G ) is connected to a vertex a ∈ cd ∗ ( G ) if and only if p | a . In this paper, we investigate the structure of a group G whose graph B ( G ) has five vertices. Especially we show that all these groups are solvable.