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

## Definition

Let $I$ be an indexing set.

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

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

Their tensor product is an ordered pair:

$\struct {\ds \bigotimes_{i \mathop \in I} G_i, \theta}$

where:

$\ds \bigotimes_{i \mathop \in I} G_i$ is an abelian group
$\theta: G \to \ds \bigotimes_{i \mathop \in I} G_i$ is a multiadditive mapping such that, for every pair $\tuple {C, \omega}$ where:
$C$ is an abelian group
$\omega : G \to C$ is a multiadditive mapping
there exists a unique group homomorphism $g : \ds \bigotimes_{i \mathop \in I} G_i \to C$ such that $\omega = g \circ \theta$.
$\xymatrix{ G \ar[d]_\theta \ar[r]^\omega & C\\ \ds \bigotimes_{i \mathop \in I} G_i \[email protected]{.>}[ru]_g }$