Groups of Extensions and Cotorsion Groups
The extension problem for abelian groups (as a special case of the general group-theoretical question formulated by O. Schreier) consists in constructing a group from a normal subgroup and the corresponding factor group. The classical way of discussing extensions is via factor sets which we follow in our presentation (simplified for the abelian case). Then we introduce Baer’s group Ext, an extremely important device, and discuss its fundamental properties. The intimate relationship between Hom and Ext has been pointed out by Eilenberg–MacLane ; this led to the interpretation of Ext as a derived functor of Hom and has been exploited extensively in Homological Algebra. Another important functor is Pext, the group of pure extensions, which appears unexpectedly as the first Ulm subgroup of Ext.
The investigation of the group structure of Ext leads to the concept of cotorsion group, a generalization of algebraic compactness. We give special prominence to cotorsion groups that occur not only as Ext, but also in several other forms.