Abstract. We deal with some pcf (possible cofinality theory) investigations mostly motivated by questions in abelian group theory. We concentrate on applications to test problems but we expect the combinatorics will have reasonably wide applications. The main test problem is the “trivial dual conjecture” which says that there is a quite free abelian group with trivial dual. The “quite free” stands for “-free” for a suitable cardinal , the first open case is . We almost always answer it positively, that is, prove the existence of -free abelian groups with trivial dual, i.e., with no non-trivial homomorphisms to the integers. Combinatorially, we prove that “almost always” there are which are quite free and have a relevant black box. The qualification “almost always” means except when we have strong restrictions on cardinal arithmetic, in fact restrictions which hold “everywhere”. The nicest combinatorial result is probably the so-called “Black Box Trichotomy Theorem” proved in ZFC. Also we may replace abelian groups by R-modules. Part of our motivation (in dealing with modules) is that in some sense the improvement over earlier results becomes clearer in this context.