Epimorphism between free Abelian groups












0












$begingroup$


I have found a statement (without proof) that there is no epimorphism in Group category $mathbb{Z}times mathbb{Z} to mathbb{Z}times mathbb{Z}times mathbb{Z} $ existing.
Could you please help, why it's true?






abstract-algebra group-theory abelian-groups







share|cite|improve this question











$endgroup$












  • $begingroup$
    You need to use capital Z in the mathbb font. Lower case z doesn't work. And no epimorphism between those as what? Sets? Groups? Modules? Rings?
    $endgroup$
    – Arthur
    Dec 11 '18 at 14:29










  • $begingroup$
    In Groups category
    $endgroup$
    – Igor Kozyrev
    Dec 11 '18 at 14:31










  • $begingroup$
    Tensor with $Bbb{Q}$.
    $endgroup$
    – jgon
    Dec 11 '18 at 14:33










  • $begingroup$
    Look at the matrix of your morphism.
    $endgroup$
    – Pierre-Yves Gaillard
    Dec 11 '18 at 15:08
















0












$begingroup$


I have found a statement (without proof) that there is no epimorphism in Group category $mathbb{Z}times mathbb{Z} to mathbb{Z}times mathbb{Z}times mathbb{Z} $ existing.
Could you please help, why it's true?






abstract-algebra group-theory abelian-groups







share|cite|improve this question











$endgroup$












  • $begingroup$
    You need to use capital Z in the mathbb font. Lower case z doesn't work. And no epimorphism between those as what? Sets? Groups? Modules? Rings?
    $endgroup$
    – Arthur
    Dec 11 '18 at 14:29










  • $begingroup$
    In Groups category
    $endgroup$
    – Igor Kozyrev
    Dec 11 '18 at 14:31










  • $begingroup$
    Tensor with $Bbb{Q}$.
    $endgroup$
    – jgon
    Dec 11 '18 at 14:33










  • $begingroup$
    Look at the matrix of your morphism.
    $endgroup$
    – Pierre-Yves Gaillard
    Dec 11 '18 at 15:08














0












0








0





$begingroup$


I have found a statement (without proof) that there is no epimorphism in Group category $mathbb{Z}times mathbb{Z} to mathbb{Z}times mathbb{Z}times mathbb{Z} $ existing.
Could you please help, why it's true?






abstract-algebra group-theory abelian-groups







share|cite|improve this question











$endgroup$




I have found a statement (without proof) that there is no epimorphism in Group category $mathbb{Z}times mathbb{Z} to mathbb{Z}times mathbb{Z}times mathbb{Z} $ existing.
Could you please help, why it's true?






abstract-algebra group-theory abelian-groups




abstract-algebra group-theory abelian-groups






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 11 '18 at 21:18









user26857

39.4k124183




39.4k124183










asked Dec 11 '18 at 14:26









Igor KozyrevIgor Kozyrev

464




464












  • $begingroup$
    You need to use capital Z in the mathbb font. Lower case z doesn't work. And no epimorphism between those as what? Sets? Groups? Modules? Rings?
    $endgroup$
    – Arthur
    Dec 11 '18 at 14:29










  • $begingroup$
    In Groups category
    $endgroup$
    – Igor Kozyrev
    Dec 11 '18 at 14:31










  • $begingroup$
    Tensor with $Bbb{Q}$.
    $endgroup$
    – jgon
    Dec 11 '18 at 14:33










  • $begingroup$
    Look at the matrix of your morphism.
    $endgroup$
    – Pierre-Yves Gaillard
    Dec 11 '18 at 15:08


















  • $begingroup$
    You need to use capital Z in the mathbb font. Lower case z doesn't work. And no epimorphism between those as what? Sets? Groups? Modules? Rings?
    $endgroup$
    – Arthur
    Dec 11 '18 at 14:29










  • $begingroup$
    In Groups category
    $endgroup$
    – Igor Kozyrev
    Dec 11 '18 at 14:31










  • $begingroup$
    Tensor with $Bbb{Q}$.
    $endgroup$
    – jgon
    Dec 11 '18 at 14:33










  • $begingroup$
    Look at the matrix of your morphism.
    $endgroup$
    – Pierre-Yves Gaillard
    Dec 11 '18 at 15:08
















$begingroup$
You need to use capital Z in the mathbb font. Lower case z doesn't work. And no epimorphism between those as what? Sets? Groups? Modules? Rings?
$endgroup$
– Arthur
Dec 11 '18 at 14:29




$begingroup$
You need to use capital Z in the mathbb font. Lower case z doesn't work. And no epimorphism between those as what? Sets? Groups? Modules? Rings?
$endgroup$
– Arthur
Dec 11 '18 at 14:29












$begingroup$
In Groups category
$endgroup$
– Igor Kozyrev
Dec 11 '18 at 14:31




$begingroup$
In Groups category
$endgroup$
– Igor Kozyrev
Dec 11 '18 at 14:31












$begingroup$
Tensor with $Bbb{Q}$.
$endgroup$
– jgon
Dec 11 '18 at 14:33




$begingroup$
Tensor with $Bbb{Q}$.
$endgroup$
– jgon
Dec 11 '18 at 14:33












$begingroup$
Look at the matrix of your morphism.
$endgroup$
– Pierre-Yves Gaillard
Dec 11 '18 at 15:08




$begingroup$
Look at the matrix of your morphism.
$endgroup$
– Pierre-Yves Gaillard
Dec 11 '18 at 15:08










1 Answer
1






active

oldest

votes


















4












$begingroup$

Assume that there is an epimorphism $f:mathbb{Z}timesmathbb{Z}tomathbb{Z}timesmathbb{Z}timesmathbb{Z}$, then by quotient $mathbb{Ztimesmathbb{Z}}timesmathbb{Z}$ with $2mathbb{Z}times2mathbb{Z}times2mathbb{Z}$, we also get an epimorphism $hat{f}:mathbb{Z}timesmathbb{Z}to(mathbb{Z}/2mathbb{Z})^{oplus3}$. Since $(mathbb{Z}/2mathbb{Z})^{oplus3}$ is 2-torsion, $hat{f}$ factors as $hat{f}:mathbb{Z}timesmathbb{Z}to(mathbb{Z}/2mathbb{Z})^{oplus2}to(mathbb{Z}/2mathbb{Z})^{oplus3}$, in which each morphism is an epimorphism, this is impossible if you compute the order of the two finite groups (4 is less than 8).






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
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3035334%2fepimorphism-between-free-abelian-groups%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









    4












    $begingroup$

    Assume that there is an epimorphism $f:mathbb{Z}timesmathbb{Z}tomathbb{Z}timesmathbb{Z}timesmathbb{Z}$, then by quotient $mathbb{Ztimesmathbb{Z}}timesmathbb{Z}$ with $2mathbb{Z}times2mathbb{Z}times2mathbb{Z}$, we also get an epimorphism $hat{f}:mathbb{Z}timesmathbb{Z}to(mathbb{Z}/2mathbb{Z})^{oplus3}$. Since $(mathbb{Z}/2mathbb{Z})^{oplus3}$ is 2-torsion, $hat{f}$ factors as $hat{f}:mathbb{Z}timesmathbb{Z}to(mathbb{Z}/2mathbb{Z})^{oplus2}to(mathbb{Z}/2mathbb{Z})^{oplus3}$, in which each morphism is an epimorphism, this is impossible if you compute the order of the two finite groups (4 is less than 8).






    share|cite|improve this answer









    $endgroup$


















      4












      $begingroup$

      Assume that there is an epimorphism $f:mathbb{Z}timesmathbb{Z}tomathbb{Z}timesmathbb{Z}timesmathbb{Z}$, then by quotient $mathbb{Ztimesmathbb{Z}}timesmathbb{Z}$ with $2mathbb{Z}times2mathbb{Z}times2mathbb{Z}$, we also get an epimorphism $hat{f}:mathbb{Z}timesmathbb{Z}to(mathbb{Z}/2mathbb{Z})^{oplus3}$. Since $(mathbb{Z}/2mathbb{Z})^{oplus3}$ is 2-torsion, $hat{f}$ factors as $hat{f}:mathbb{Z}timesmathbb{Z}to(mathbb{Z}/2mathbb{Z})^{oplus2}to(mathbb{Z}/2mathbb{Z})^{oplus3}$, in which each morphism is an epimorphism, this is impossible if you compute the order of the two finite groups (4 is less than 8).






      share|cite|improve this answer









      $endgroup$
















        4












        4








        4





        $begingroup$

        Assume that there is an epimorphism $f:mathbb{Z}timesmathbb{Z}tomathbb{Z}timesmathbb{Z}timesmathbb{Z}$, then by quotient $mathbb{Ztimesmathbb{Z}}timesmathbb{Z}$ with $2mathbb{Z}times2mathbb{Z}times2mathbb{Z}$, we also get an epimorphism $hat{f}:mathbb{Z}timesmathbb{Z}to(mathbb{Z}/2mathbb{Z})^{oplus3}$. Since $(mathbb{Z}/2mathbb{Z})^{oplus3}$ is 2-torsion, $hat{f}$ factors as $hat{f}:mathbb{Z}timesmathbb{Z}to(mathbb{Z}/2mathbb{Z})^{oplus2}to(mathbb{Z}/2mathbb{Z})^{oplus3}$, in which each morphism is an epimorphism, this is impossible if you compute the order of the two finite groups (4 is less than 8).






        share|cite|improve this answer









        $endgroup$



        Assume that there is an epimorphism $f:mathbb{Z}timesmathbb{Z}tomathbb{Z}timesmathbb{Z}timesmathbb{Z}$, then by quotient $mathbb{Ztimesmathbb{Z}}timesmathbb{Z}$ with $2mathbb{Z}times2mathbb{Z}times2mathbb{Z}$, we also get an epimorphism $hat{f}:mathbb{Z}timesmathbb{Z}to(mathbb{Z}/2mathbb{Z})^{oplus3}$. Since $(mathbb{Z}/2mathbb{Z})^{oplus3}$ is 2-torsion, $hat{f}$ factors as $hat{f}:mathbb{Z}timesmathbb{Z}to(mathbb{Z}/2mathbb{Z})^{oplus2}to(mathbb{Z}/2mathbb{Z})^{oplus3}$, in which each morphism is an epimorphism, this is impossible if you compute the order of the two finite groups (4 is less than 8).







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 11 '18 at 14:45









        BonbonBonbon

        32118




        32118






























            draft saved

            draft discarded




















































            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%2f3035334%2fepimorphism-between-free-abelian-groups%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

            Ellipse (mathématiques)

            Image
            Pour les articles homonymes, voir Ellipse. L'ellipse est le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante. Section du cône ou projection du cercle. En géométrie, une ellipse est une courbe plane fermée obtenue par l’intersection d’un cône de révolution avec un plan, à condition que celui-ci coupe l'axe de rotation du cône ou du cylindre : c'est une conique d'excentricité strictement comprise entre 0 et 1. On peut également la définir comme le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante (sa construction par la méthode du jardinier est très simple). Dans la vie courante, l’ellipse est la forme qu'on perçoit en regardant un cercle en perspective, ou la figure formée par l’ombre d'un disque sur une surface plane. On retrouve aussi, en première approximation [ 1 ] , des ellipses dans les trajectoires des corps célestes (planètes, comètes ou satellites arti...
            Read more

            Quarter-circle Tiles

            Image
            up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2 ...
            Read more

            Mont Emei

            Image
            Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog...
            Read more