### PCF and abelian groups

by Shelah. [Sh:898]

Forum Math, 2013

We deal with some pcf investigations mostly motivated
by abelian group theory problems and deal their applications to
test problems (we expect reasonably wide applications). We prove
almost always the existence of aleph_omega-free abelian
groups with trivial dual, i.e. no non-trivial homomorphisms
to the integers. This relies on investigation of pcf; more
specifically, for this we prove that ``almost always'' there
are F subseteq {}^kappa lambda which are quite
free and has black boxes. The ``almost always'' means that
there are strong restrictions on cardinal arithmetic if the
universe fails this, this restriction are ``everywhere''.
Those are irrating results; we replace Abelian groups by
R-modules, so in some sense our advantage over earlier
results becomes clearer.

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