How rigid are reduced products?

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 An interesting connection to earlier (unpublished) work on model theory by (unpublished, circulated notes, 1973) is elaborated in Section 3.