Abelian Groups - László Fuchs [Springer]

Chapter 8
Tensor and Torsion Products
Abstract The tensor product of groups is one of the most important concepts and
indispensable tools in the theory of abelian groups. They compete in importance with
homomorphism groups, but their features are totally different.
Tensor products can be introduced in various ways. We define them via generators and defining
relations, and then we show that they have the universal property for bilinear maps. Tensoring is
a bifunctor that is right exact in both arguments. The exact sequence of tensor products is a most
useful asset, both as a tool in proofs and as a device in discovering new facts. Exactness on the left
can be restored by introducing the functor Tor, the torsion product, that is of independent interest.
If one of the groups is a torsion group, then the tensor product can be completely described by
invariants. In particular, the tensor product of two torsion groups is always a direct sum of cyclic
groups. The torsion product behaves differently, it raises more challenging problems. The tensor
product of torsion-free groups is a difficult subject.
Various facts concerning groups that were proved originally in an ad hoc fashion may be
verified more clearly, and perhaps more elegantly, by using homological methods, in particular,
the long exact sequences connecting the tensor and torsion products (as well as Hom and Ext).
1 The Tensor Product
Bilinear Functions and the Tensor Product Suppose A and C are arbitrary
groups, and g is a function defined on the set A C with values in a group G,
g W A C ! G. We say that g is a bilinear function if it satisfies
and
g.a 1 C a 2 ; c/ D g.a 1 ; c/ C g.a 2 ; c/ (8.1)
g.a; c 1 C c 2 / D g.a; c 1 / C g.a; c 2 / (8.2)
for all a; a 1 ; a 2 2 A; c; c 1 ; c 2 2 C. It follows at once that a bilinear function obeys
the following simple rules: g.a;0/ D 0 D g.0; c/; g.na; c/ D ng.a; c/ D g.a; nc/
for all a 2 A; c 2 C, andn 2 Z.
We are going to define a group (that will be denoted by A ˝ C) aswellasa
bilinear function eW AC ! A˝C, such that e is the most general bilinear function
in the sense that any bilinear function g W A C ! G for any group G factors
© Springer International Publishing Switzerland 2015
L. Fuchs, Abelian Groups, Springer Monographs in Mathematics,
DOI 10.1007/978-3-319-19422-6_8
229