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 _X ∈  Irr(G) and write cod(_X) = |G : ker _X |/x(1) as the codegree of _X , where ker _X is the kernel of _X. Denote by M(G) 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 M(G) 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.