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Thread: torsion abelian groups category

  1. #1
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    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.
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    Quote Originally Posted by xianghu21 View Post
    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 $\displaystyle \text{Tors}.$
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