  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 lambda^kappa 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, the
restrictions are "everywhere". Also we replace Abelian groups by R-modules, so
in some sense our advantage over earlier results becomes clearer.
