On some metabelian 2-group and applications II

Abdelmalek Azizi,Mohammed Taous,Abdelkader Zekhnini

doi:10.5269/bspm.v34i2.27016

Abdelmalek Azizi, Mohammed Taous + Show 1 more

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https://doi.org/10.5269/bspm.v34i2.27016

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Abstract

Let G be some metabelian 2-group satisfying the condition G/G' is of type (2, 2, 2). 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) is isomorphic to G, where k_2^{(2)} is the second Hilbert 2-class field of k.

Highlights

In this paper, we construct all the subgroups of Gn,m of index 2 or 4, we give the abelianization types of these subgroups and we compute the kernel of the transfer map VG→H : Gn,m/G′n,m → H/H′, for any subgroup H of Gn,m, defined by the Artin map
We illustrate our results by some examples which show that our group is realizable i.e. there is a field k such that Gal(k(22)/k) ≃ Gn,m
Let M be an unramified extension of k and CM be the subgroup of Ck associated to M by Class Field Theory

Summary

Introduction

We construct all the subgroups of Gn,m of index 2 or 4, we give the abelianization types of these subgroups and we compute the kernel of the transfer map VG→H : Gn,m/G′n,m → H/H′, for any subgroup H of Gn,m, defined by the Artin map. These subgroups, their derived groups and the types of their abelianizations are given in Tables 1 and 2 below, where b = max(m, n + 1). Let us prove some entries of the Tables, using Lemmas 2.1 and 2.2. Let H be a normal subgroup of a group G.

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