A group-theoretical version of Hilbert’s theorem 90

indri putri utari;

It is shown that for a normal subgroup N of a group G, G/N cyclic, the kernel of the map Nab → Gab 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 Nab 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).