Prime Character Degree Graphs of Solvable Groups having Diameter Thre

Let G be a finite solvable group and cd(G) the set of character degrees of G. The prime character degree graph Delta(G) is the graph whose vertices are the primes dividing the degrees in cd(G) and there is an edge between two distinct primes p and q if their product pq divides some degree in cd(G). When Delta( G) has diameter three we can partition the vertices rho( G) into four non-empty disjoint subsets rho1 ∪ rho 2 ∪ rho3 ∪ rho4 where no prime in rho 1 is adjacent to any prime in rho3 ∪ rho4; no prime in rho4 is adjacent to any prime in rho1 ∪ rho 2; every prime in rho2 is adjacent to some prime in rho 3; every prime in rho3 is adjacent to some prime in rho 2; and |rho1 ∪ rho2| ≤ |rho 3 ∪ rho4|. We will show that the subset rho 3 must contain at least three vertices. The solvable group G has exactly one normal nonabelian Sylow p-subgroup P, and the prime p is contained in the subset rho3. The factor group G/P' cannot have a normal nonabelian Sylow subgroup where P' is the derived subgroup of P and so Delta(G/P') must be disconnected. If the subset rho 1 ∪ rho2 has n vertices, then the subset rho3 ∪ rho4 must have 2 n vertices. The group G must have Fitting height three.