A group theoretical version of Hilbert's theorem 90

TL;DR

This paper extends Hilbert's Theorem 90 to a group-theoretic context, revealing new properties of kernels and modules related to group quotients and their implications in number theory.

Contribution

It introduces a group-theoretic version of Hilbert's Theorem 90, establishing new relations between transfer kernels, co-kernels, and module structures in finitely generated groups.

Findings

Kernel of N^{ab} to G^{ab} satisfies Hilbert 90 property

N^{ab} is a pseudo permutation module under certain conditions

Relation between transfer kernel and co-kernel determines Hilbert-Suzuki multiplier

Abstract

It is shown that for a normal subgroup $N$ of a group $G$, $G/N$ cyclic, the kernel of the map $N^{\mathrm{ab}}\to G^{\mathrm{ab}}$ satisfies the classical Hilbert 90 property (cf. Thm. A). As a consequence, if $G$ is finitely generated, $|G:N|<\infty$, and all abelian groups $H^{\mathrm{ab}}$, $N\subseteq H\subseteq G$, are torsion free, then $N^{\mathrm{ab}}$ must be a pseudo permutation module for $G/N$ (cf. Thm. B). From Theorem A one also deduces a non-trivial relation between the order of the transfer kernel and co-kernel which determines the Hilbert-Suzuki multiplier (cf. Thm. C). Translated into a number theoretic context one obtains a strong form of Hilbert's theorem 94. In case that $G$ is finitely generated and $N$ has prime index $p$ in $G$ there holds a "generalized Schreier formula" involving the torsion free ranks of $G$ and $N$ and the ratio of the order of the transfer kernel and co-kernel (cf. Thm. D).