# Definition:Tensor Product of Abelian Groups

## Contents

## Definition

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

### Definition 1: by universal property

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

- $A \otimes B$ is an abelian group
- $\theta : A \times B \to A \otimes B$ is a biadditive mapping

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

- $C$ is an abelian group
- $\omega : A \times B \to C$ is a biadditive mapping

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

### Definition 2: construction

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

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

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

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

## 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 $G = \displaystyle \prod_{i \mathop \in I} G_i$ 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:

- $\displaystyle \bigotimes_{i \in I} G_i$ is an abelian group
- $\theta : G \to \displaystyle\bigotimes_{i \in I} G_i$ is a multiadditive mapping

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

- $C$ is an abelian group
- $\omega : G \to C$ is a multiadditive mapping

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:
- 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 $j \in I$, $x, y \in G_j$, $\family {z_i}_{i \mathop \ne j} \in \displaystyle \prod_{i \mathop \ne j} G_i$, 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 $\Z \sqbrk G \to \displaystyle \bigotimes_{i \mathop \in I} G_i$:
- $G \hookrightarrow \Z \sqbrk G \twoheadrightarrow \displaystyle \bigotimes_{i \mathop \in I} G_i$