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

Abstract

Let $G$ be some metabelian $2$-group satisfying the condition $G/G^{\prime }\simeq \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$. 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 $\mathbf{k}$.

in Journal of Taibah University for Science (2015)

References in corpus (1)

• Structure of $Gal(k_2^{(2)}/k)$ for some fields $k=Q(\sqrt{2p_1{p}_2},\sqrt{-1})$