A group‐theoretical version of Hilbert's theorem 90. (8th October 2015)

• Record Type: Journal Article
• Authors: Quadrelli, C.; Weigel, Th.
• Is Part Of: Bulletin of the London Mathematical Society. Volume 47:Part 4(2015:Aug.)
• Page Start: 704
• Page End: 714
• Publication Date: 2015-10-08
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1112/blms/bdv043
• Languages: English
• ISSNs: 0024-6093

Abstract

Abstract : It is shown that for a normal subgroupN of a groupG, G / N cyclic, the kernel of the mapN ab → G ab satisfies the classical Hilbert 90 property (cf. Theorem A). As a consequence, ifG is finitely generated, | G : N | < ∞, and all abelian groupsH ab, N ⊆ H ⊆ G, are torsion free, thenN ab must be a pseudo‐permutation module forG / 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 thatG is finitely generated andN has prime indexp inG there holds a 'generalized Schreier formula' involving the torsion‐free ranks ofG andN and the ratio of the order of the transfer kernel and co‐kernel (cf. Theorem D).