Vol. 59, No. 1, 1975

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On restricting irreducible characters to normal subgroups

Richard Lewis Roth

Vol. 59 (1975), No. 1, 229–235

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|) xhK|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.

Mathematical Subject Classification 2000
Primary: 20C15
Received: 26 December 1974
Published: 1 July 1975
Richard Lewis Roth