Tensor and Torsion Products
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).