### How rigid are reduced products?

by Goebel and Shelah. [GbSh:831]

J Pure and Applied Algebra, 2005

For any cardinal mu let Z^mu be the
additive group of all
integer-valued functions f: mu-> Z . The support of f is
[f]= {i in mu : f(i)=f_i ne 0} . Also let Z_mu =
Z^mu / Z^{< mu} with Z^{< mu}=
{f in Z^mu : |[f]|< mu} .
If mu <= chi are regular cardinals we analyze the question when
Hom (Z_mu, Z_chi) = 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 mu with
Hom (Z_mu,G) ne 0 - this is an infinite, regular cardinal (or
infty). 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 mu for which there is a family {G_i: i in mu} of
groups such that R does not commute with the product
prod_{i in mu}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. J{a}nos
Bolyai 61 (1992) 77--107) and a seminal
result by {L}o{s} 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 Pess, 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 An interesting connection to earlier (unpublished)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model theory by (unpublished, circulated notes, 1973) is elaborated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.

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