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|K is irreducible and to generalize this lemma.
It is seen that the irreducibility of χ|K is equivalent to the property that
(1∕|K|)∑x∈hK|χ(x)|2= 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 J1= J1(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 χ to K will extend to
J1,eJ1(χ) = eK(χ) and J1 is the maximal normal subgroup with these two
properties.
