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

Are there historical instances of the capital of a colonising country being temporarily or permanently shifted to one of its colonies?

An alternative proof of an application of Hahn-Banach

How to detect if C code (which needs 'extern C') is compiled in C++

Why was Goose renamed from Chewie for the Captain Marvel film?

Accepted offer letter, position changed

How does one describe somebody who is bi-racial?

Recommendation letter by significant other if you worked with them professionally?

How strictly should I take "Candidates must be local"?

If I receive an SOS signal, what is the proper response?

What are some noteworthy "mic-drop" moments in math?

Difference on montgomery curve equation between EFD and RFC7748

When traveling to Europe from North America, do I need to purchase a different power strip?

Does a warlock using the Darkness/Devil's Sight combo still have advantage on ranged attacks against a target outside the Darkness?

What wound would be of little consequence to a biped but terrible for a quadruped?

What's the "normal" opposite of flautando?

How can I ensure my trip to the UK will not have to be cancelled because of Brexit?

Is "history" a male-biased word ("his+story")?

Does this video of collapsing warehouse shelves show a real incident?

Intuition behind counterexample of Euler's sum of powers conjecture

How did Alan Turing break the enigma code using the hint given by the lady in the bar?

List elements digit difference sort

Can I pump my MTB tire to max (55 psi / 380 kPa) without the tube inside bursting?

NASA's RS-25 Engines shut down time

Conservation of Mass and Energy



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 ?










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












    $begingroup$


    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.
      Check out our Code of Conduct.









      share|cite|improve this question




      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.









      asked 7 hours ago









      lablab

      183




      183




      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.





      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.






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






















          1 Answer
          1






          active

          oldest

          votes


















          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









          $endgroup$













            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "69"
            };
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function() {
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled) {
            StackExchange.using("snippets", function() {
            createEditor();
            });
            }
            else {
            createEditor();
            }
            });

            function createEditor() {
            StackExchange.prepareEditor({
            heartbeatType: 'answer',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });






            lab is a new contributor. Be nice, and check out our Code of Conduct.










            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3144008%2ffinite-abelian-groups-tensor-product%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            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






















                    lab is a new contributor. Be nice, and check out our Code of Conduct.










                    draft saved

                    draft discarded


















                    lab is a new contributor. Be nice, and check out our Code of Conduct.













                    lab is a new contributor. Be nice, and check out our Code of Conduct.












                    lab is a new contributor. Be nice, and check out our Code of Conduct.
















                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3144008%2ffinite-abelian-groups-tensor-product%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    Popular posts from this blog

                    El tren de la libertad Índice Antecedentes "Porque yo decido" Desarrollo de la...

                    Castillo d'Acher Características Menú de navegación

                    Puerta de Hutt Referencias Enlaces externos Menú de navegación15°58′00″S 5°42′00″O /...