Finite Groups With Some Special Degrees Of Irreducible Characters

Let G be a finite nonabelian group and X be a nonlinear irreducible character of G.It is well known that |G/kerχ|=tχ·χ(1)for some positive integer tχ,andχ(1)2||G/kerχ| for each χ∈Irr(G)if and only if the group G is nilpotent.The main objective of this thesis is to study the influence of |G/kerχ/χ(1)on the structure of G.Firstly,we consider a general situation,that is |G/kerχ|≤pmχ(1)2 for χ∈Irr1(G),pm is the largest prime divisor of |G/kerx|.We show that this class of groups are not simple by using classification of finite simple groups.Furthermore,we discuss the solvability of this class of groups.Our main results are:Theorem 3.4 Let G be a nonabelian finite group.If |G/kerχ|≤pmχ(1)2 for each χ∈Irr1(G),where Pm is the largest prime divisor of |G/kerχ|,then G is not simple.Theorem 3.5 Let G be a nonabelian finite group with |G/kerχ|≤pmx(1)2 for each χ∈ Irr1(G),where pm is the largest prime divisor of |G/kerχ|.If G is not a solvable group,then the minimal normal subgroup of G is a finite simple group of Lie type.