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Math Help - torsion abelian groups category

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    torsion abelian groups category

    Let Tors be the category whose objects are torsion abelian groups; if A and B are torsion abelian groups, we define Mor_{\text{Tors}}(A, B) to be the set of all (group) homomorphisms \phi : A \rightarrow B. Prove that direct products exist in Tors; that is, show that given any indexed family \{A_i\}_{i \in I} where 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 A and B are torsion abelian groups, we define Mor_{\text{Tors}}(A, B) to be the set of all (group) homomorphisms \phi : A \rightarrow B. Prove that direct products exist in Tors; that is, show that given any indexed family \{A_i\}_{i \in I} where 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 B=\prod_{i \in I}A_i in the category of abelian groups, and let T be the torsion subgroup of B. now let \pi_i: B \longrightarrow A_i be the projection map and \tilde{\pi_i} be the restriction of \pi_i to T.

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