Classification of metabelian 2-groups \(G\) with \(\mathbf{G}^{\mathrm{ab}} (\mathbf{2},\mathbf{2}^{\mathbf n})\), \(\mathbf n\geq \mathbf 2\), and rank \(\mathbf d(\mathbf G^{\prime})=\mathbf 2\). Applications to real quadratic number fields
The authors continue their investigations whether an algebraic number field $k$ has a finite $2$-class field tower. This holds when the rank of $Cl_2(k^1)$, the $2$-part of the class group of the Hilbert class field of $k$, is at most $2$. If it equals $2$, this means that the Galois group of $k^2/k$ should be as described in the title. In Theorem 1 the authors give a complete description of such groups $G$, which depends only on (the abelianizations of) the 3 maximal subgroups and the 3 normal subgroups of index $4$ of $G$. Theorem 2 states how the rank of $Cl_2(k^1)$ depends on the different cases. Finally, these results are applied to real quadratic number fields, whose $2$-class group has rank $2$, and concludes with some numerical examples.
scientific article; zbMATH DE number 6934111
scientific article