Representations of metabelian groups

Abstract

A knowledge of the simple representation theory of finite abelian groups is useful for understanding the representations of solvable groups, since these provide the one-dimensional representations. The representation theory of metabelian groups (those G with abelian commutator subgroup G′) would seem to be a natural next level. In this paper we shall show that these representations, too, may be simply described in several ways: they are induced from linear representations of some explicity defined subgroups; their degrees may be calculated from a knowledge of the subgroups of G; these degrees depend only on the kernel of the representation (in fact, only on the intersection of this kernel with G′). As an application of these results, we can calculate for metabelian groups a certain measure of group-commutativity studied in an earlier paper [4].