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Abelian GroupsGroups of Extensions and Cotorsion Groups

Abelian Groups: 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 [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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Abelian GroupsGroups of Extensions and Cotorsion Groups

Part of the Springer Monographs in Mathematics Book Series
Fuchs, László
Springer Journals — Jun 12, 2015
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44 pages

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Publisher
Springer International Publishing
Copyright
© Springer International Publishing Switzerland 2015
ISBN
978-3-319-19421-9
Pages
255 –298
DOI
10.1007/978-3-319-19422-6_9
Publisher site
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Abstract

[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.]

Published: Jun 12, 2015

Keywords: Exact Sequence; Short Exact Sequence; Inverse Limit; Torsion Group; Pure Subgroup

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