On some realizable metabelian $5$-groups

TL;DR

This paper classifies certain 5-groups of maximal class with specific abelianization and transfer properties, showing they are uniquely determined and realizable as Galois groups over fields related to pure quintic extensions.

Contribution

It provides a complete classification of these metabelian 5-groups and demonstrates their realizability as Galois groups over particular number fields.

Findings

Groups are uniquely determined by their transfer properties.

These groups are realizable as Galois groups over fields from pure quintic extensions.

The classification links group structure with number field properties.

Abstract

Let $G$ be a $5$-group of maximal class and $\gamma_2(G) = [G, G]$ its derived group. Assume that the abelianization $G/\gamma_2(G)$ is of type $(5, 5)$ and the transfers $V_{H_1\to \gamma_2(G)}$ and $V_{H_2\to \gamma_2(G)}$ are trivial, where $H_1$ and $H_2$ are two maximal normal subgroups of $G$. Then $G$ is completely determined with the isomorphism class groups of maximal class. Moreover the group $G$ is realizable with some fields $k$, which is the normal closure of a pure quintic field.