Character degree graphs of solvable groups with diameter three

Abstract

Abstract Let G be a finite solvable group and cd ⁡ ( G ) ${\operatorname{cd}(G)}$ the set of character degrees of G. The character degree graph Δ ⁢ ( G ) ${\Delta(G)}$ is the graph whose vertices are the primes dividing the degrees in cd ⁡ ( G ) ${\operatorname{cd}(G)}$ and there is an edge between two distinct primes p and q if their product pq divides some degree in cd ⁡ ( G ) ${\operatorname{cd}(G)}$ . When Δ ⁢ ( G ) ${\Delta(G)}$ has diameter three, we can partition the vertices ρ ⁢ ( G ) ${\rho(G)}$ into four non-empty disjoint subsets ρ 1 ∪ ρ 2 ∪ ρ 3 ∪ ρ 4 ${\rho_{1}\cup\rho_{2}\cup\rho_{3}\cup\rho_{4}}$ where no prime in ρ 1 ${\rho_{1}}$ is adjacent to any prime in ρ 3 ∪ ρ 4 ${\rho_{3}\cup\rho_{4}}$ ; no prime in ρ 4 ${\rho_{4}}$ is adjacent to any prime in ρ 1 ∪ ρ 2 ${\rho_{1}\cup\rho_{2}}$ ; every prime in ρ 2 ${\rho_{2}}$ is adjacent to some prime in ρ 3 ${\rho_{3}}$ ; every prime in ρ 3 ${\rho_{3}}$ is adjacent to some prime in ρ 2 ${\rho_{2}}$ ; and | ρ 1 ∪ ρ 2 | ≤ | ρ 3 ∪ ρ 4 | ${|\rho_{1}\cup\rho_{2}|\leq|\rho_{3}\cup\rho_{4}|}$ . We will show the following: If G is a solvable group where Δ ⁢ ( G ) ${\Delta(G)}$ has diameter three, then ρ 3 ${\rho_{3}}$ has at least three vertices and G has a normal non-abelian Sylow p-subgroup where p ∈ ρ 3 ${p\in\rho_{3}}$ . If ρ 1 ∪ ρ 2 ${\rho_{1}\cup\rho_{2}}$ has n vertices, then ρ 3 ∪ ρ 4 ${\rho_{3}\cup\rho_{4}}$ must have at least 2 n ${2^{n}}$ vertices. The group G has Fitting height 3.