Publication Date:
2015-07-15
Description:
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. Theorem A). As a consequence, if $G$ is finitely generated, $|G:N|\lt \infty$ , 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. Theorem 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. Theorem C). Translated into a number-theoretical setting, one obtains a strong form of Hilbert's theorem 94 (Theorem 4.1). 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. Theorem D).
Print ISSN:
0024-6093
Electronic ISSN:
1469-2120
Topics:
Mathematics
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