On some metabelian 2-groups and applications I
Tom 142 / 2016
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 $\mathbf {k}$.