Abelian Groups pp 255-298 | Cite as

Groups of Extensions and Cotorsion Groups

  • László Fuchs
Part of the Springer Monographs in Mathematics book series (SMM)


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 [1]; 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.


Exact Sequence Short Exact Sequence Inverse Limit Torsion Group Pure Subgroup 
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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • László Fuchs
    • 1
  1. 1.Mathematics DepartmentTulane UniversityNew OrleansUSA

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