Bounding Fitting Heights of Several Classes of Character Degree Graphs 

Author  ZhangXianXiu 
Tutor  ZhangGuangXiang 
School  Southwestern University 
Course  Basic mathematics 
Keywords  Fitting height character degree graph representations of solvable groups 
CLC  O152 
Type  Master's thesis 
Year  2009 
Downloads  3 
Quotes  0 
By a definition of Lewis in [3], 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 [3]: 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 [3] 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 [5]） 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） =π_{1}∪π_{2}∪p is a disjoint union where π_{i}≥1 for i=1,2. Assume that no prime inπ_{1} is adjacent inΔ（G） to any prime inπ_{2}. 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 fiforder circle inΔ（G）, and the total sum of degrees of each derived subgraph with a fiforder 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） =π_{1}∪π_{2} is a disjoint union where π_{i}≥2, p_{i},q_{i}∈π_{i}, for i = 1,2. Assume that no prime inπ_{1} is adjacent inΔ（G） to any prime inπ_{2}, except for p_{1}p_{2} and q_{1}q_{2}. Then the Fitting height of G is at most 4.