It is shown that for a normal subgroup N of a group G, G/N cyclic, the kernel of the map Nab→&gt; Gad satisfies the classical Hilbert 90 property (cf. Theorem A). As a consequence, if G is finitely generated, |G: N| &lt; ∞, and all abelian groups Hab, N ⊆ H ⊆ G, are torsion free, then Na 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).

### A group-theoretical version of Hilbert's theorem 90

#### Abstract

It is shown that for a normal subgroup N of a group G, G/N cyclic, the kernel of the map Nab→> Gad satisfies the classical Hilbert 90 property (cf. Theorem A). As a consequence, if G is finitely generated, |G: N| < ∞, and all abelian groups Hab, N ⊆ H ⊆ G, are torsion free, then Na 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).
##### Scheda breve Scheda completa Scheda completa (DC) 2015
http://blms.oxfordjournals.org/
Hilbert’s theorem 90; pseudo-permutation modules; transfer kernels; generalized Schreier formula; cohomological Mackey functors; Herbrand quotient
File in questo prodotto:
File
1502.01146v1.pdf

non disponibili

Tipologia: Documento in Post-print
Licenza: DRM non definito
Dimensione 198.24 kB
BLMSClaudio.pdf

non disponibili

Tipologia: Versione Editoriale (PDF)
Licenza: DRM non definito
Dimensione 201.95 kB
Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11383/2129354`
• ND
• 1
• 1