Definition:Tensor Product of Abelian Groups/Family/Definition 1

Definition
Let $I$ be an indexing set.

Let $(G_i)_{i \in I}$ be a family of abelian groups.

Let $G = \displaystyle \prod_{i \in I} G_i$ be their direct product.

Their tensor product is a pair $\left( \displaystyle \bigotimes_{i \in I} G_i, \theta \right)$ where: such that, for every pair $(C, \omega)$ where: there exists a unique group homomorphism $g : \displaystyle \bigotimes_{i \in I} G_i \to C$ with $\omega = g \circ \theta$.
 * $\displaystyle \bigotimes_{i \in I} G_i$ is an abelian group
 * $\theta : G \to \displaystyle\bigotimes_{i \in I} G_i$ is a multiadditive mapping
 * $C$ is an abelian group
 * $\omega : G \to C$ is a multiadditive mapping
 * $\xymatrix{

G \ar[d]_\theta \ar[r]^\omega & C\\ \displaystyle \bigotimes_{i \in I} G_i \ar@{.>}[ru]_g }$

Also see

 * Equivalence of Definitions of Tensor Product of Family of Abelian Groups