How rigid are reduced products?

For any cardinal μ let Zμ be the additive group of all integer-valued functions f:μ⟶Z. The support of f is [f]={i∈μ:f(i)=fi≠0}. Also let Zμ=Zμ/Z<μ with Z<μ={f∈Zμ:|[f]|<μ}. If μ⩽χ are regular cardinals we analyze the question when Hom(Zμ,Zχ)=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(Zμ,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 {Gi:i∈μ} of groups such that R does not commute with the product ∏i∈μGi. 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 An interesting connection to earlier (unpublished) work on model theory by (unpublished, circulated notes, 1973) is elaborated in Section 3.