# Definition:Tensor Product of Abelian Groups

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## Definition

Let $A$ and $B$ be abelian groups.

### Definition 1: by universal property

Their tensor product is a pair $(A \otimes B, \theta)$ where:

such that, for every pair $(C, \omega)$ where:

there exists a unique group homomorphism $g : A \otimes B \to C$ with $\omega = g \circ \theta$.

### Definition 2: construction

Their tensor product is the pair $(A \otimes B, \theta)$ where:

## Tensor product of family of abelian groups

Let $I$ be an indexing set.

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

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

### Definition 1: by universal property

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 }$

### Definition 2: construction

Their tensor product is the pair:

$\tuple {\displaystyle \bigotimes_{i \mathop \in I} G_i, \theta}$

where:

• $\displaystyle \bigotimes_{i \in I} G_i$ is the quotient of the free abelian group $\Z \sqbrk G$ on $G$, by the subgroup generated by the elements of the form $\tuple {x + y, \family {z_i}_{i \mathop \ne j} } - \tuple {x, \family {z_i}_{i \mathop \ne j} } - \tuple {y, \family {z_i}_{i \mathop \ne j} }$
for $j \in I$, $x, y \in G_j$, $\family {z_i}_{i \mathop \ne j} \in \displaystyle \prod_{i \mathop \ne j} G_i$, where we denote $\tuple {x, \family {z_i}_{i \mathop \ne j} }$ for:
• $\theta : G \to \displaystyle \bigotimes_{i \mathop \in I} G_i$ is the composition of the canonical mapping $G \to \Z \sqbrk G$ with the quotient group epimorphism $\Z \sqbrk G \to \displaystyle \bigotimes_{i \mathop \in I} G_i$:
$G \hookrightarrow \Z \sqbrk G \twoheadrightarrow \displaystyle \bigotimes_{i \mathop \in I} G_i$