A Study On the Kernels of Irreducible Characters of Finite Groups

Abstract

Let G be a finite group and χ∈Irr(G), where Irr(G) denotes the set of all irreducible characters of G. The kernel of χ is defined by ker⁡(χ)={ g∈G ┤| χ(g)=χ(1)}, where χ(1) is the character degree of χ. The irreducible character χ of G is called as monolithic when the factor group G/ker⁡(χ) has only one minimal normal subgroup. In this study, we have proven some results by concentrating on the kernels of nonlinear irreducible characters of G. First, we have provided an alternative proof for the classification of finite groups possessing two nonlinear irreducible characters by using their kernels. Also, we have presented the structure the solvable group G in which every nonlinear monolithic characters has same kernel