# torsion abelian groups category

• Nov 18th 2008, 08:25 PM
xianghu21
torsion abelian groups category
Let Tors be the category whose objects are torsion abelian groups; if $\displaystyle A$ and $\displaystyle B$ are torsion abelian groups, we define $\displaystyle Mor_{\text{Tors}}(A, B)$ to be the set of all (group) homomorphisms $\displaystyle \phi : A \rightarrow B$. Prove that direct products exist in Tors; that is, show that given any indexed family $\displaystyle \{A_i\}_{i \in I}$ where $\displaystyle A_i$ is a torsion abelian group, there exists a torsion abelian group which serves as a direct product for this family in Tors.
• Nov 18th 2008, 09:14 PM
NonCommAlg
Quote:

Originally Posted by xianghu21
Let Tors be the category whose objects are torsion abelian groups; if $\displaystyle A$ and $\displaystyle B$ are torsion abelian groups, we define $\displaystyle Mor_{\text{Tors}}(A, B)$ to be the set of all (group) homomorphisms $\displaystyle \phi : A \rightarrow B$. Prove that direct products exist in Tors; that is, show that given any indexed family $\displaystyle \{A_i\}_{i \in I}$ where $\displaystyle A_i$ is a torsion abelian group, there exists a torsion abelian group which serves as a direct product for this family in Tors.

let $\displaystyle B=\prod_{i \in I}A_i$ in the category of abelian groups, and let $\displaystyle T$ be the torsion subgroup of $\displaystyle B.$ now let $\displaystyle \pi_i: B \longrightarrow A_i$ be the projection map and $\displaystyle \tilde{\pi_i}$ be the restriction of $\displaystyle \pi_i$ to $\displaystyle T.$

then $\displaystyle \{T, \tilde{\pi}_i \}_{i \in I}$ is the product of $\displaystyle \{A_i \}_{i \in I}$ in the category $\displaystyle \text{Tors}.$