Notes on normal p-complements and monomial characters

Abstract

Let G be a finite group and Irr(G) be the set of irreducible (complex) characters of G. Let χ ∈ Irr(G) and write cod(χ) = |G : ker χ|/χ(1) as the codegree of χ, where ker χ is the kernel of χ. Denote by the intersection of the kernels of all the monolithic, monomial irreducible characters of a p-solvable group G with codegrees divisible by a prime p. We prove in this note that has a normal p-complement, and then the theorem in [2] is generalized. Also, we prove that about the theorem in [2] more is true.