On restricting irreducible characters to normal subgroups

Abstract

This paper is about the situation where χ is an irreducible character of a finite group G and K is a normal subgroup. A construction of Serre's relating the characters of G with those of G/K is used to give a new proof of a well-known lemma concerning the case that χ \κ is irreducible and to generalize this lemma. It is seen that the irreducibility of χ I* is equivalent to the property that (1/| K\) ΣX^K I χ(x) I2 = 1 for each coset of G modulo K and also to the property that χ is not a component of λχ for any irreducible character λ of G/K except for λ — 1. The subgroup Jx = Ji(χ) is defined as the intersection of the kernels of the irreducible characters λ of G/K for which χ is a component of λχ. It is seen that an irreducible component σ of the restriction of % to if will extend to Ju eJχ(γ) = eκ(χ) and Jx is the maximal normal subgroup with these two properties.