Abelian Groups - László Fuchs [Springer]

Chapter 9
Groups of Extensions and Cotorsion Groups
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.
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.
1 Group Extensions
The Extension Problem Given two groups, A and C, theextension problem
consists in finding all groups B such that B contains a subgroup A 0 isomorphic to A
and B=A 0 Š C. This situation can be expressed in terms of the exact sequence
e W 0 ! A !B !C ! 0; (9.1)
where stands for the inclusion map, and is an epimorphism with kernel A. In
this case, we say that B is an extension of A by C.
It is our next aim to survey all extensions for fixed A and C. This can be done
in different ways. We first describe extensions via factor sets (in a pedestrian way),
and then we discuss them by using short exact sequences (which is a more attractive
and more powerful method).
Let a; b;::: denote elements of A, andu;v;w;::: those of C. Assuming that B
is an extension of A by C, wepickatransversal; this is a function g W C ! B
that assigns an element of B in the coset corresponding to u, i.e.g.u/ 2 1 u.For
convenience, it is always assumed that g.0/ D 0. Once the function g is selected,
every b 2 B has a unique form b D g.u/ C a with u 2 C; a 2 A.Sinceg.u/ C g.v/
and g.u C v/ belong to the same coset mod A, thereisanf .u;v/2 A such that
g.u/ C g.v/ D g.u C v/ C f .u;v/:
© Springer International Publishing Switzerland 2015
L. Fuchs, Abelian Groups, Springer Monographs in Mathematics,
DOI 10.1007/978-3-319-19422-6_9
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