Annihilator ideals of two-generated metabelian p-groups

Abstract

For certain metabelian [Formula: see text]-groups [Formula: see text] with two generators [Formula: see text] and [Formula: see text], the annihilator [Formula: see text] of the main commutator [Formula: see text] of [Formula: see text], as an ideal of bivariate polynomials with integer coefficients, is determined by means of a presentation for [Formula: see text]. It is proved that together with Schreier’s polynomials [Formula: see text], the annihilator [Formula: see text] identifies the group [Formula: see text] uniquely, and the Furtwängler isomorphism of the additive group underlying the residue class ring [Formula: see text] to the commutator subgroup [Formula: see text] of [Formula: see text] admits the calculation of the abelian type invariants of [Formula: see text]. The results are underpinned by class field theoretic realizations of the groups [Formula: see text] as Galois groups [Formula: see text] of second Hilbert [Formula: see text]-class fields [Formula: see text] over algebraic number fields [Formula: see text].