On some realizable metabelian $5$-groups

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.

9 pages, 1 figures, 1 table