Certain monomial characters and their normal constituents

Abstract

Abstract Suppose that G is a finite p -solvable group such that N G ( P ) /P has odd order, where P ∈ Syl p ( G ). If χ is an irreducible complex character with degree not divisible by p and field of values contained in a cyclotomic field Q p a , then every subnormal constituent of χ is monomial. Also, the number of such irreducible characters is the number of N G ( P )-orbits on P/P’ . Introduction There are few results guaranteeing that a single irreducible complex character χ ∈ Irr( G ) of a finite group G is monomial. Recall that χ ∈ Irr( G ) is monomial if there is ƛ ∈ Irr( U ) linear such that ƛG = χ . It is known that every irreducible character of a supersolvable group is monomial, for instance, but this result depends more on the structure of the group rather than on the properties of the characters themselves. An exception is a theorem by R. Gow of 1975 ([3]): an odd degree real valued irreducible character of a solvable group is monomial. Recently, we gave in [8] an extension of this theorem which also dealt with the degree and the field of values of the character. (Yet another similar monomiality criterium was given in [9]: if the field of values Q( χ ) of χ is contained in the cyclotomic field Q n and ( χ (1) , 2 n ) = 1, then χ is monomial whenever G is solvable.) In this note, we apply non-trivial Isaacs π -theory of solvable groups to give a shorter proof of the above result at the same time that we gain some new information about the subnormal constituents of the characters, among other things. It does not seem easy at all to prove these new facts without using this deep theory. Recall that for every solvable group and any set of primes π , M. Isaacs defined a canonical subset B π ( G ) of Irr( G ) with remarkable properties ([4]). Since, by definition, every χ ∈ B π ( G ) is induced from a character of π -degree, it is clear that B π -characters of π’ -degree are monomial.