# 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 $\struct {A \otimes B, \theta}$ where:

- $A \otimes B$ is an abelian group
- $\theta : A \times B \to A \otimes B$ is a biadditive mapping such that, for every ordered pair $\struct {C, \omega}$ where:
- $C$ is an abelian group
- $\omega : A \times B \to C$ is a biadditive mapping

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

### Definition 2: construction

Their **tensor product** is the pair $\struct {A \otimes B, \theta}$ where:

- $A \otimes B$ is the quotient of the free abelian group $\Z^{\paren {A \times B} }$ on the cartesian product $A \times B$ by the subgroup generated by the elements of the form:
- $\tuple {a_1 + a_2, b} - \tuple {a_1, b} - \tuple {a_2, b}$
- $\tuple {a, b_1 + b_2} - \tuple {a, b_1} - \tuple {a, b_2}$

- for $a, a_1, a_2 \in A$, $b, b_1, b_2 \in B$, where we denote $\tuple {a, b}$ for its image under the canonical mapping $A \times B \to \Z^{\paren {A \times B} }$.

- $\theta : A \times B \to A \otimes B$ is the composition of the canonical mapping $A \times B \to \Z^{\paren {A \times B} }$ with the quotient group epimorphism $\Z^{\paren {A \times B} } \to A \otimes B$.

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

### Definition 1: by universal property

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 }$

### Definition 2: construction

Their **tensor product** is the ordered pair:

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

where:

- $\ds \bigotimes_{i \mathop \in I} G_i$ 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 $j \in I$, $x, y \in G_j$, $\family {z_i}_{i \mathop \ne j} \in \ds \prod_{i \mathop \ne j} G_i$, where we denote $\tuple {x, \family {z_i}_{i \mathop \ne j} }$ for:
- the family in $G$ whose $j$th term is $x$ and whose $i$th term is $z_i$, for $i \ne j$
- its image under the canonical mapping $G \to \Z \sqbrk G$.

- for $j \in I$, $x, y \in G_j$, $\family {z_i}_{i \mathop \ne j} \in \ds \prod_{i \mathop \ne j} G_i$, where we denote $\tuple {x, \family {z_i}_{i \mathop \ne j} }$ for:
- $\theta : G \to \ds \bigotimes_{i \mathop \in I} G_i$ is the composition of the canonical mapping $G \to \Z \sqbrk G$ with the quotient group epimorphism $\Z \sqbrk G \to \ds \bigotimes_{i \mathop \in I} G_i$:
- $G \hookrightarrow \Z \sqbrk G \twoheadrightarrow \ds \bigotimes_{i \mathop \in I} G_i$