How rigid are reduced products?

Abstract

For any cardinal μ let Z μ be the additive group of all integer-valued functions f : μ ⟶ Z . The support of f is [ f ] = { i ∈ μ : f ( i ) = f i ≠ 0 } . Also let Z μ = Z μ / Z < μ with Z < μ = { f ∈ Z μ : | [ f ] | < μ } . If μ ⩽ χ are regular cardinals we analyze the question when Hom ( Z μ , Z χ ) = 0 and obtain a complete answer under GCH and independence results in Section 8. These results and some extensions are applied to a problem on groups: Let the norm ∥ G ∥ of a group G be the smallest cardinal μ with Hom ( Z μ , G ) ≠ 0 —this is an infinite, regular cardinal (or ∞ ). As a consequence we characterize those cardinals which appear as norms of groups. This allows us to analyze another problem on radicals: The norm ∥ R ∥ of a radical R is the smallest cardinal μ for which there is a family { G i : i ∈ μ } of groups such that R does not commute with the product ∏ i ∈ μ G i . Again these norms are infinite, regular cardinals and we show which cardinals appear as norms of radicals. The results extend earlier work (Arch. Math. 71 (1998) 341–348; Pacific J. Math. 118 (1985) 79–104; Colloq. Math. Soc. János Bolyai 61 (1992) 77–107) and a seminal result by Łoś on slender groups. (His elegant proof appears here in new light; Proposition 4.5.), see Fuchs [Vol. 2] (Infinite Abelian Groups, vols. I and II, Academic Press, New York, 1970 and 1973). An interesting connection to earlier (unpublished) work on model theory by (unpublished, circulated notes, 1973) is elaborated in Section 3.