How rigid are reduced products?

For any cardinal μ let image be the additive group of all integer-valued functions image. The support of f is [f]={iset membership, variantμ:f(i)=fi≠0}. Also let image with image. If μless-than-or-equals, slantχ are regular cardinals we analyze the question when image 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 short parallelGshort parallel of a group G be the smallest cardinal μ with image—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 short parallelRshort parallel of a radical R is the smallest cardinal μ for which there is a family {Gi:iset membership, variantμ} of groups such that R does not commute with the product image. 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 An interesting connection to earlier (unpublished) work on model theory by (unpublished, circulated notes, 1973) is elaborated in Section 3.