Bounding Fitting Heights of Several Classes of Character Degree Graphs
|Keywords||Fitting height character degree graph representations of solvable groups|
By a definition of Lewis in , a degree graphΔhas bounded Fitting height if there is a bound on the Fitting height for the solvable group G withΔ（G） =Δ. Lewis proved in : a degree graphΔwith n vertices has bounded Fitting height if and only ifΔhas at most one vertex of degree n -1. Lewis also obtained in  that ifΔhas bounded Fitting height, the bound is linear in the number of vertices of the graph. He observed that no graph with bounded Fitting height is found for a solvable group where the bound is bigger than 4. Thus M.L.Lewis has a conjecture（conjecture 5.5 of ） that if G be a solvable group whereΔ（G） is a graph with bounded Fitting height. Then G has Fitting height at most 4. Lewis has proved that the conjecture is right at least in the following situations:Theorem A. Let G be a solvable group and suppose thatρ（G） =π1∪π2∪p is a disjoint union where |πi|≥1 for i=1,2. Assume that no prime inπ1 is adjacent inΔ（G） to any prime inπ2. then G has Fitting height at most 4.Theorem B. Let G be a solvable group. Suppose thatΔ（G） is the graph having four vertices where every vertex has degree 2. Then the Fitting height of G is at most 4.In this paper, we proved that the conjecture is right in other situations, and some main theorems as following.Theorem 2.2. Let G be a solvable group with |ρ（G）|≥4. If the total sum of degrees of each derived subgraph with four vertices inΔ（G） is not more than 8, then the Fitting height of G is at most 4. Theorem 3.4. Let G be a solvable group. If there is a fif-order circle inΔ（G）, and the total sum of degrees of each derived subgraph with a fif-order circle inΔ（G） is 10, then the Fitting height of G is at most 4.Theorem 4.1. Let G be a solvable group and suppose thatρ（G） =π1∪π2 is a disjoint union where |πi|≥2, pi,qi∈πi, for i = 1,2. Assume that no prime inπ1 is adjacent inΔ（G） to any prime inπ2, except for p1p2 and q1q2. Then the Fitting height of G is at most 4.