Annihilator ideals of two-generated metabelian p-groups

TL;DR

This paper characterizes the annihilator ideals of the main commutator in two-generated metabelian p-groups, linking polynomial ideals to the structure of the group's commutator subgroup.

Contribution

It provides a method to determine the annihilator ideal using group presentations, connecting polynomial ideals with group structure.

Findings

Explicit description of the annihilator ideal A for G

Calculation of the abelian type of G' from the ideal A

Application of polynomial ideal theory to group structure analysis

Abstract

For a metabelian p-group G = <x,y> with two generators x and y, the annihilator A < Z[X,Y] of the main commutator [y,x] of G, as an ideal of bivariate polynomials with integer coefficients, is determined by means of a presentation for G. Furtw\"angler's isomorphism of the additive group underlying the residue class ring Z[X,Y]/A of Z[X,Y] modulo the annihilator A to the commutator subgroup G' of G admits the calculation of the abelian type of G'.