Abelian Groups - László Fuchs [Springer]

508 13 Torsion-Free Groups of Infinite Rank
Rychkov [4] considers families fA i j i < 2 g of -separable torsion-free groups for an
uncountable regular cardinal that is not weakly compact. Assuming V = L, he establishes the
existence of such families satisfying j Im j < for each homomorphism W A i ! A j with
i ¤ j in I. Grinshpon–Krylov [1] study the lattice of fully invariant subgroups of separable
torsion-free groups. This lattice—as expected—is determined by the extractable types of the
group. Grinshpon–Krylov [1] describe the fully transitive separable groups. For the transitivity
of completely decomposable and separable groups, see also Metelli [2].
Hill–Megibben [7] generalize the notion of separability by defining K-groups. The class
of these groups is closed under direct sums and direct summands, they display several useful
properties similar to separable groups. By making use of this concept, knice subgroups are
introduced, and torsion-free groups are studied that admit H.@ 0 /-families of knice subgroups.
There is a considerable body of literature on dual groups. Their in-depth discussion requires
sophisticated set-theoretical machinery; we refer to [EM] for many interesting results. See also
Mekler–Schlitt [1] where dual groups are discussed from the logical point of view. Göbel–Pokutta
[1] construct dual groups in Z @0 using MA+ :CH. It was for a while an open problem if the
groups in the Reid classes were all different. The problem was solved by Zimmermann-Huisgen
[1], Ivanov [3], and Eda [3] in the affirmative.
Nunke–Rotman [1] show that every integral singular cohomology group is of the form
Ext.G; Z/ ˚ Hom.H; Z/, whereG; H can be arbitrary groups. Thus they are direct sums of a
cotorsion group and a dual group.
Coseparability is more difficult to deal with, and the reader might wonder why we have not
given any explicit examples for non-free coseparable groups. The reason is simple: it is undecidable
in ZFC whether or not all coseparable groups are free. We do not wish to enter into the discussion of
these groups, just mention that Chase [Pac. J. Math. 12, 847–854 (1962)] proved that CH implies
that there exist non-free coseparable groups of cardinality @ 1 , while Mekler–Shelah [3] showed
that coseparable groups of cardinality @ 1 are free in the model of ZFC with @ 2 Cohen reals added.
It is an interesting fact that all @ 1 -coseparable groups are reflexive (Huber [2]).
Coseparability can also be defined to depend on a cardinality. Usually the definition is restricted
to -free groups. Accordingly, A is -coseparable if it is C@ 1 -free and every pure subgroup of
corank