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On some metabelian 2-groups and applications I

Volume 142 / 2016

Abdelmalek Azizi, Abdelkader Zekhnini, Mohammed Taous Colloquium Mathematicum 142 (2016), 99-113 MSC: 11R11, 11R20, 11R29, 11R32, 11R37, 20D15, 20D20, 20E28. DOI: 10.4064/cm142-1-5

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 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 $\mathbf {k}$.

Authors

  • Abdelmalek AziziDepartment of Mathematics
    Faculty of Sciences
    Mohammed First University
    Oujda, Morocco
    e-mail
  • Abdelkader ZekhniniDepartment of Mathematics
    Polydisciplinary Faculty of Nador
    Mohammed First University
    Nador, Morocco
    e-mail
  • Mohammed TaousDepartment of Mathematics
    Faculty of Sciences and Technology
    Moulay Ismail University
    Errachidia, Morocco
    e-mail

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