ON SOME METABELIAN 2-GROUPS AND APPLICATIONS I

Let G be some metabelian 2-group satisfying the condition G/G' similar or equal to Z/2Z x Z/2Z x Z/2Z. 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 k satisfying the condition Gal(k(2)((2))/k) similar or equal to G, where k(2)((2)) is the second Hilbert 2-class field of k.