Nilpotent metacyclic irreducible linear groups of odd order
Let \(\mathbb{F}\) be an arbitrary field, \(G\) a nilpotent metacyclic group of odd order, and \(G^{\text{Aut }G}\) the subgroup consisting of those elements of \(G\) which are fixed by every automorphism of \(G\). We show that the images of two faithful irreducible \(\mathbb{F}\)-representations of \(G\) are linearly isomorphic if and only if the restrictions of the representations to \(G^{\text{Aut }G}\) are equivalent. If the centre of \(G\) is cyclic and the characteristic of \(\mathbb{F}\) does not divide the order of \(G\), then the number of linear isomorphism types of the irreducible \(\mathbb{F}\)-linear groups that are abstractly isomorphic to \(G\) is precisely the number of equivalence types of faithful irreducible \(\mathbb{F}\)- representations of \(G^{\text{Aut }G}\). For groups of this kind, we also show how to calculate the order of \(G^{\text{Aut }G}\).
faithful irreducible \(\mathbb{F}\)-representations
irreducible \(\mathbb{F}\)-linear groups
scientific article; zbMATH DE number 815691