Given an א1-free abelian group G we characterize the class CG of all torsion abelian groups T satisfying Ext(G, T ) = 0 assuming the special continuum hypothesis CH. Moreover, in Gödel’s constructable universe we prove that this characterizes CG for arbitrary torsion-free abelian G. It follows that there exist some ugly א1-free abelian groups.

where (f + g)(u) = V_,,+,,,f(v) A f(w). Since E is countable, ZcB) can be defined for any countably complete Boolean algebra (ccBa) B where Z is the group of the integers. This kind of group was first (1962) studied by Balcerzyk [l]. However, it seems that not much attention was paid to such groups for a rather long period. Under the point of view in [l, Theorem 51 and [13, Proposition 11, it c...