On some metabelian 2-group whose abelianization is of type (2, 2, 2) and applications

TL;DR

This paper investigates the structure of certain metabelian 2-groups with a specific abelianization, classifies their subgroups, and applies these findings to problems in algebraic number theory related to ideal class capitulation.

Contribution

It constructs all subgroups of index 2 or 4 in these groups, determines their abelianization types, and applies the results to capitulation problems in number fields.

Findings

Classified subgroups of the specified metabelian 2-group

Determined abelianization types of these subgroups

Applied results to ideal class capitulation in number fields

Abstract

Let $G$ be some metabelian $2$-group satisfying the condition $G/G'\simeq \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$. In this paper, we construct all the subgroups of $G$ of index $2$ or $4$, we give the abelianization types of these subgroups and we compute the kernel of the transfer map. Then we apply these results to study the capitulation problem of the $2$-ideal classes of some fields $\mathbf{k}$ satisfying the condition $\mathrm{G}al(\mathbf{k}_2^{(2)}/\mathbf{k})\simeq G$, where $\mathbf{k}_2^{(2)}$ is the second Hilbert $2$-class field of $\mathbf{k}$.