# 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 $\omega = g \circ \theta$.

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