finite abelian groups tensor product.Tensor-commutative abelian groupson finite abelian groupsProduct and...

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finite abelian groups tensor product.


Tensor-commutative abelian groupson finite abelian groupsProduct and quotient in Abelian groupsIsomorphic finite abelian groupsExplicitly computing the isomorphism class of the tensor product of two finite abelian groupsFinite rank Abelian groupIsomorphism between tensor products of abelian groupsProduct in finite abelian groupsClassification of finite rank Abelian groupsTorsion-free abelian groups, tensor product and $p$-adic integers













3












$begingroup$


Is the following question obvious ?



Let $G$ be an abelian group, such that for any finite abelian group $A$, we have
$Gotimes_{mathbf{Z}}A=0$, does it mean that $G$ is a $mathbf{Q}$-vector space ?










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    3












    $begingroup$


    Is the following question obvious ?



    Let $G$ be an abelian group, such that for any finite abelian group $A$, we have
    $Gotimes_{mathbf{Z}}A=0$, does it mean that $G$ is a $mathbf{Q}$-vector space ?










    share|cite|improve this question







    New contributor




    lab is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.







    $endgroup$















      3












      3








      3


      1



      $begingroup$


      Is the following question obvious ?



      Let $G$ be an abelian group, such that for any finite abelian group $A$, we have
      $Gotimes_{mathbf{Z}}A=0$, does it mean that $G$ is a $mathbf{Q}$-vector space ?










      share|cite|improve this question







      New contributor




      lab is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.







      $endgroup$




      Is the following question obvious ?



      Let $G$ be an abelian group, such that for any finite abelian group $A$, we have
      $Gotimes_{mathbf{Z}}A=0$, does it mean that $G$ is a $mathbf{Q}$-vector space ?







      abstract-algebra group-theory finite-groups tensor-products abelian-groups






      share|cite|improve this question







      New contributor




      lab is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question







      New contributor




      lab is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      lab is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      asked 7 hours ago









      lablab

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      183




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          $begingroup$

          You have $Gotimes(Bbb Z/nBbb Z)=0$ for all
          $nin Bbb N$. That means $G/nG=0$, so $G=nG$, that is all elements of $G$ are divisible by $n$. Then $G$ is a divisible Abelian group. Conversely if $G$ is a divisible Abelian group, then
          $Gotimes(Bbb Z/nBbb Z)=0$ and so $Gotimes A=0$ for all finitely generated
          Abelian groups.



          But not all divisible Abelian groups are $Bbb Q$-modules: they may have torsion.
          As an example, let $G=Bbb Q/Bbb Z$.






          share|cite|improve this answer









          $endgroup$













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            6












            $begingroup$

            You have $Gotimes(Bbb Z/nBbb Z)=0$ for all
            $nin Bbb N$. That means $G/nG=0$, so $G=nG$, that is all elements of $G$ are divisible by $n$. Then $G$ is a divisible Abelian group. Conversely if $G$ is a divisible Abelian group, then
            $Gotimes(Bbb Z/nBbb Z)=0$ and so $Gotimes A=0$ for all finitely generated
            Abelian groups.



            But not all divisible Abelian groups are $Bbb Q$-modules: they may have torsion.
            As an example, let $G=Bbb Q/Bbb Z$.






            share|cite|improve this answer









            $endgroup$


















              6












              $begingroup$

              You have $Gotimes(Bbb Z/nBbb Z)=0$ for all
              $nin Bbb N$. That means $G/nG=0$, so $G=nG$, that is all elements of $G$ are divisible by $n$. Then $G$ is a divisible Abelian group. Conversely if $G$ is a divisible Abelian group, then
              $Gotimes(Bbb Z/nBbb Z)=0$ and so $Gotimes A=0$ for all finitely generated
              Abelian groups.



              But not all divisible Abelian groups are $Bbb Q$-modules: they may have torsion.
              As an example, let $G=Bbb Q/Bbb Z$.






              share|cite|improve this answer









              $endgroup$
















                6












                6








                6





                $begingroup$

                You have $Gotimes(Bbb Z/nBbb Z)=0$ for all
                $nin Bbb N$. That means $G/nG=0$, so $G=nG$, that is all elements of $G$ are divisible by $n$. Then $G$ is a divisible Abelian group. Conversely if $G$ is a divisible Abelian group, then
                $Gotimes(Bbb Z/nBbb Z)=0$ and so $Gotimes A=0$ for all finitely generated
                Abelian groups.



                But not all divisible Abelian groups are $Bbb Q$-modules: they may have torsion.
                As an example, let $G=Bbb Q/Bbb Z$.






                share|cite|improve this answer









                $endgroup$



                You have $Gotimes(Bbb Z/nBbb Z)=0$ for all
                $nin Bbb N$. That means $G/nG=0$, so $G=nG$, that is all elements of $G$ are divisible by $n$. Then $G$ is a divisible Abelian group. Conversely if $G$ is a divisible Abelian group, then
                $Gotimes(Bbb Z/nBbb Z)=0$ and so $Gotimes A=0$ for all finitely generated
                Abelian groups.



                But not all divisible Abelian groups are $Bbb Q$-modules: they may have torsion.
                As an example, let $G=Bbb Q/Bbb Z$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 7 hours ago









                Lord Shark the UnknownLord Shark the Unknown

                106k1161133




                106k1161133






















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