On some metabelian 2-groups and applications I
Tom 142 / 2016
Colloquium Mathematicum 142 (2016), 99-113
MSC: 11R11, 11R20, 11R29, 11R32, 11R37, 20D15, 20D20, 20E28.
DOI: 10.4064/cm142-1-5
Streszczenie
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 $\mathbf {k}$.
