ABSTRACT
15.1 The class of sets allows the construction of a product, its Cartesian product introduced in Chapter3. Now for groups, the Cartesian product as sets may be endowed with the structure of a group so that the canonical projection maps are group homomorphisms. This construction enjoys a universal property (Theorem 15.7) which makes it the “right” product to consider, this means that it would be the categorical product if we had defined the notion of category. The direct sum (for abelian groups) is then defined as a subgroup of the product (Definition 15.22) and it satisfies a corresponding universal property phrased in Theorem 15.25. Again the latter should be seen in the light of category theory, the interested reader could look up some basic ideas inS. Mac Lane's book “Categories for the Working Mathematician”, GTM 5, Springer Verlag, Berlin 1971.
