Extending characters and lattices
The main result of this paper is: Theorem 1. Let \(N\) be a normal subgroup of a finite group and let \(\pi\) be a set of primes such that \(G/N\) is a \(\pi\)-group. Let \(\mu\) be a \(G\)-stable irreducible character of \(N\) whose degree is divisible by \(|N|_\pi\). Let \(h\) be the \(\pi'\)-part of the exponent of \(G\), and let \(\omega\) be a primitive complex \(h\)-th root of unity. Let \(R=\{\tfrac\alpha\beta\mid\alpha\in\mathbb{Z}[\omega]\) and \(\beta\in\mathbb{Z}[\omega]\setminus\rho\) for each prime ideal divisor \(\rho\) of \(|G|_\pi\}\). Then: (a) \(\mu\) extends to a \(\pi\)-rational irreducible character of \(G\) and furthermore, this extension is unique up to multiplication by linear characters of order 2. (b) Each extension of \(\mu\) in part (a) may be afforded by an \(R\)-free \(RG\)-module which is determined up to isomorphism by its character.
scientific article; zbMATH DE number 1429471
scientific article