On some metabelian 2-groups and applications I

Abdelmalek Azizi¹ ; Abdelkader Zekhnini² ; Mohammed Taous³

¹ Department of Mathematics Faculty of Sciences Mohammed First University Oujda, Morocco
² Department of Mathematics Polydisciplinary Faculty of Nador Mohammed First University Nador, Morocco
³ Department of Mathematics Faculty of Sciences and Technology Moulay Ismail University Errachidia, Morocco

Colloquium Mathematicum, Tome 142 (2016) no. 1, pp. 99-113.

Résumé

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 for the $2$-ideal classes of some fields $\mathbf {k}$ satisfying the condition $\mathrm {Gal}(\mathbf {k}_2^{(2)}/\mathbf {k})\simeq G$, where $\mathbf {k}_2^{(2)}$ is the second Hilbert $2$-class field of $\mathbf {k}$.

Détail

DOI : 10.4064/cm142-1-5

Keywords: metabelian group satisfying condition simeq mathbb mathbb times mathbb mathbb times mathbb mathbb paper construct subgroups index abelianization types these subgroups compute kernel transfer map apply these results study capitulation problem ideal classes fields mathbf satisfying condition mathrm gal mathbf mathbf simeq where mathbf second hilbert class field mathbf