## Abstract

It is shown that for a normal subgroup N of a group G, G/N cyclic, the kernel of the map N^{ab}→> G^{ad} satisfies the classical Hilbert 90 property (cf. Theorem A). As a consequence, if G is finitely generated, |G: N| < ∞, and all abelian groups H^{ab}, 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).

Original language | English |
---|---|

Pages (from-to) | 704-714 |

Number of pages | 11 |

Journal | Bulletin of the London Mathematical Society |

Volume | 47 |

Issue number | 4 |

DOIs | |

State | Published - 1 Jan 2015 |

Externally published | Yes |

## ASJC Scopus subject areas

- General Mathematics