On some metabelian 2-groups and applications III
Let G be some metabelian 2-group satisfying the condition G/G′≃Z/2Z×Z/2Z×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 for the 2-ideal classes of some fields k satisfying the condition Gal(k(2)2/k)≃G, where k(2)2 is the second Hilbert 2-class field of k.
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