Calculating a presentation of $mathbb{Z}_{3}$ in detail.











up vote
1
down vote

favorite













Theorem: Let $G$ groups and $Ssubset G$ such that
$langle Srangle =G$. (Here $G=left{s_1ldots, s_n:s_iin Scup S^{-1}, ninmathbb{N}right}$.) Let $varphi:Sto G$ with $varphi(s)=s$. By the universal property of free groups there exists a unique homomorphism (in fact, epimorphism) $varphi:F(S)to G$ with $$F(S)=left{win S^{ast}: w text{ reduced word} right}.$$Then $$Gsimeq F(S)/{ker(varphi)}.$$




Here $langle langle Sranglerangle=langle left{gsg^{-1}:sin Scup S^{-1}, gin Gright}rangle.$



Let $Ssubset G$ and $G=langle Srangle.$ Then $langle Smid Trangle $ presentation of $G$ if $G=langle Srangle$ and $Tsubset kervarphi$ and $langle langle Tranglerangle=kervarphi$.




I want prove in a detailed way that $mathbb{Z}_{3}=langle amid a^3rangle.$




I have this.



Here $S=left{aright}.$



First, $a$ must be different from $0$. Because if $a=0$, then $mathbb{Z}_{3}=left{0right}$.



If $a=1,$ then $0=1+1+1, 1=1, 2=1+1$.



If $a=2$, then $0=2+2+2, 1=2+2^{-1}, 2=2$.



Therefore, $mathbb{Z}_{3}=langle arangle$, with $a=1$ or $a=2$.



So . . .




How prove $ker(varphi)=langlelangle left{a^3right}ranglerangle$?




I have this:



$kervarphi=left{s_1cdots s_nin F(S): varphi(s_1cdots s_n)=0, s_iin left{aright}cupleft{a^{-1}right}, ninmathbb{N}right}$



$=left{s_1cdots s_nin F(S): s_1+cdots +s_n=0, s_iin left{aright}cupleft{a^{-1}right}, ninmathbb{N}right}$






group-theory finite-groups cyclic-groups group-presentation universal-property







share|cite|improve this question




















  • 1




    After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark $checkmark$ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?, What should I do if someone answers my question?.
    – Shaun
    2 days ago






  • 2




    There are many different ways of defining $Bbb Z_3$ and a sufficiently detailed answer would depend on which definition you are using. Please be more specific in future.
    – Shaun
    2 days ago






  • 1




    Note, too, that there is no such thing as the presentation of a group, strictly speaking.
    – Shaun
    2 days ago

















up vote
1
down vote

favorite













Theorem: Let $G$ groups and $Ssubset G$ such that
$langle Srangle =G$. (Here $G=left{s_1ldots, s_n:s_iin Scup S^{-1}, ninmathbb{N}right}$.) Let $varphi:Sto G$ with $varphi(s)=s$. By the universal property of free groups there exists a unique homomorphism (in fact, epimorphism) $varphi:F(S)to G$ with $$F(S)=left{win S^{ast}: w text{ reduced word} right}.$$Then $$Gsimeq F(S)/{ker(varphi)}.$$




Here $langle langle Sranglerangle=langle left{gsg^{-1}:sin Scup S^{-1}, gin Gright}rangle.$



Let $Ssubset G$ and $G=langle Srangle.$ Then $langle Smid Trangle $ presentation of $G$ if $G=langle Srangle$ and $Tsubset kervarphi$ and $langle langle Tranglerangle=kervarphi$.




I want prove in a detailed way that $mathbb{Z}_{3}=langle amid a^3rangle.$




I have this.



Here $S=left{aright}.$



First, $a$ must be different from $0$. Because if $a=0$, then $mathbb{Z}_{3}=left{0right}$.



If $a=1,$ then $0=1+1+1, 1=1, 2=1+1$.



If $a=2$, then $0=2+2+2, 1=2+2^{-1}, 2=2$.



Therefore, $mathbb{Z}_{3}=langle arangle$, with $a=1$ or $a=2$.



So . . .




How prove $ker(varphi)=langlelangle left{a^3right}ranglerangle$?




I have this:



$kervarphi=left{s_1cdots s_nin F(S): varphi(s_1cdots s_n)=0, s_iin left{aright}cupleft{a^{-1}right}, ninmathbb{N}right}$



$=left{s_1cdots s_nin F(S): s_1+cdots +s_n=0, s_iin left{aright}cupleft{a^{-1}right}, ninmathbb{N}right}$






group-theory finite-groups cyclic-groups group-presentation universal-property







share|cite|improve this question




















  • 1




    After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark $checkmark$ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?, What should I do if someone answers my question?.
    – Shaun
    2 days ago






  • 2




    There are many different ways of defining $Bbb Z_3$ and a sufficiently detailed answer would depend on which definition you are using. Please be more specific in future.
    – Shaun
    2 days ago






  • 1




    Note, too, that there is no such thing as the presentation of a group, strictly speaking.
    – Shaun
    2 days ago















up vote
1
down vote

favorite









up vote
1
down vote

favorite












Theorem: Let $G$ groups and $Ssubset G$ such that
$langle Srangle =G$. (Here $G=left{s_1ldots, s_n:s_iin Scup S^{-1}, ninmathbb{N}right}$.) Let $varphi:Sto G$ with $varphi(s)=s$. By the universal property of free groups there exists a unique homomorphism (in fact, epimorphism) $varphi:F(S)to G$ with $$F(S)=left{win S^{ast}: w text{ reduced word} right}.$$Then $$Gsimeq F(S)/{ker(varphi)}.$$




Here $langle langle Sranglerangle=langle left{gsg^{-1}:sin Scup S^{-1}, gin Gright}rangle.$



Let $Ssubset G$ and $G=langle Srangle.$ Then $langle Smid Trangle $ presentation of $G$ if $G=langle Srangle$ and $Tsubset kervarphi$ and $langle langle Tranglerangle=kervarphi$.




I want prove in a detailed way that $mathbb{Z}_{3}=langle amid a^3rangle.$




I have this.



Here $S=left{aright}.$



First, $a$ must be different from $0$. Because if $a=0$, then $mathbb{Z}_{3}=left{0right}$.



If $a=1,$ then $0=1+1+1, 1=1, 2=1+1$.



If $a=2$, then $0=2+2+2, 1=2+2^{-1}, 2=2$.



Therefore, $mathbb{Z}_{3}=langle arangle$, with $a=1$ or $a=2$.



So . . .




How prove $ker(varphi)=langlelangle left{a^3right}ranglerangle$?




I have this:



$kervarphi=left{s_1cdots s_nin F(S): varphi(s_1cdots s_n)=0, s_iin left{aright}cupleft{a^{-1}right}, ninmathbb{N}right}$



$=left{s_1cdots s_nin F(S): s_1+cdots +s_n=0, s_iin left{aright}cupleft{a^{-1}right}, ninmathbb{N}right}$






group-theory finite-groups cyclic-groups group-presentation universal-property







share|cite|improve this question
















Theorem: Let $G$ groups and $Ssubset G$ such that
$langle Srangle =G$. (Here $G=left{s_1ldots, s_n:s_iin Scup S^{-1}, ninmathbb{N}right}$.) Let $varphi:Sto G$ with $varphi(s)=s$. By the universal property of free groups there exists a unique homomorphism (in fact, epimorphism) $varphi:F(S)to G$ with $$F(S)=left{win S^{ast}: w text{ reduced word} right}.$$Then $$Gsimeq F(S)/{ker(varphi)}.$$




Here $langle langle Sranglerangle=langle left{gsg^{-1}:sin Scup S^{-1}, gin Gright}rangle.$



Let $Ssubset G$ and $G=langle Srangle.$ Then $langle Smid Trangle $ presentation of $G$ if $G=langle Srangle$ and $Tsubset kervarphi$ and $langle langle Tranglerangle=kervarphi$.




I want prove in a detailed way that $mathbb{Z}_{3}=langle amid a^3rangle.$




I have this.



Here $S=left{aright}.$



First, $a$ must be different from $0$. Because if $a=0$, then $mathbb{Z}_{3}=left{0right}$.



If $a=1,$ then $0=1+1+1, 1=1, 2=1+1$.



If $a=2$, then $0=2+2+2, 1=2+2^{-1}, 2=2$.



Therefore, $mathbb{Z}_{3}=langle arangle$, with $a=1$ or $a=2$.



So . . .




How prove $ker(varphi)=langlelangle left{a^3right}ranglerangle$?




I have this:



$kervarphi=left{s_1cdots s_nin F(S): varphi(s_1cdots s_n)=0, s_iin left{aright}cupleft{a^{-1}right}, ninmathbb{N}right}$



$=left{s_1cdots s_nin F(S): s_1+cdots +s_n=0, s_iin left{aright}cupleft{a^{-1}right}, ninmathbb{N}right}$






group-theory finite-groups cyclic-groups group-presentation universal-property




group-theory finite-groups cyclic-groups group-presentation universal-property






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 days ago









Shaun

8,205113578




8,205113578










asked Dec 5 at 1:34









eraldcoil

342111




342111








  • 1




    After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark $checkmark$ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?, What should I do if someone answers my question?.
    – Shaun
    2 days ago






  • 2




    There are many different ways of defining $Bbb Z_3$ and a sufficiently detailed answer would depend on which definition you are using. Please be more specific in future.
    – Shaun
    2 days ago






  • 1




    Note, too, that there is no such thing as the presentation of a group, strictly speaking.
    – Shaun
    2 days ago
















  • 1




    After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark $checkmark$ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?, What should I do if someone answers my question?.
    – Shaun
    2 days ago






  • 2




    There are many different ways of defining $Bbb Z_3$ and a sufficiently detailed answer would depend on which definition you are using. Please be more specific in future.
    – Shaun
    2 days ago






  • 1




    Note, too, that there is no such thing as the presentation of a group, strictly speaking.
    – Shaun
    2 days ago










1




1




After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark $checkmark$ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?, What should I do if someone answers my question?.
– Shaun
2 days ago




After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark $checkmark$ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?, What should I do if someone answers my question?.
– Shaun
2 days ago




2




2




There are many different ways of defining $Bbb Z_3$ and a sufficiently detailed answer would depend on which definition you are using. Please be more specific in future.
– Shaun
2 days ago




There are many different ways of defining $Bbb Z_3$ and a sufficiently detailed answer would depend on which definition you are using. Please be more specific in future.
– Shaun
2 days ago




1




1




Note, too, that there is no such thing as the presentation of a group, strictly speaking.
– Shaun
2 days ago






Note, too, that there is no such thing as the presentation of a group, strictly speaking.
– Shaun
2 days ago












1 Answer
1






active

oldest

votes

















up vote
2
down vote













In order to show $ker(varphi)=langle langle a^3ranglerangle$, one must first be clear on what $varphi$ is, and instead of writing, say, $color{red}{a=1}$, one writes $varphi(a)=1$.



You are on the right track in that you have evidence to suggest that $varphi: amapsto 1text{ or } 2$; that is, that the generator $a$ of the presentation $langle amid a^3rangle$ maps via $varphi$ to one of the elements of $Bbb Z_3$ given by a number coprime to $3$.



Ask yourself,




What reduced words, with letters in ${ a, a^{-1}}$, get mapped to the identity of $Bbb Z_3$ via $varphi$?




But can you deduce what each word maps to under $varphi$ in general?



Hover your cursor over (or, if you're on a touchscreen device, tap) the box below for some hints.




Hint: Use Bézout's Identity, assuming $varphi(a)=p$ for some $p$ coprime to $3$. Here $varphi$ is a homomorphism. Don't forget to show both $ker(varphi)subseteq langlelangle a^3ranglerangle$ and $langlelangle a^3rangleranglesubseteqker(varphi)$.




I hope this helps :)






share|cite|improve this answer























    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',
    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%2f3026474%2fcalculating-a-presentation-of-mathbbz-3-in-detail%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








    up vote
    2
    down vote













    In order to show $ker(varphi)=langle langle a^3ranglerangle$, one must first be clear on what $varphi$ is, and instead of writing, say, $color{red}{a=1}$, one writes $varphi(a)=1$.



    You are on the right track in that you have evidence to suggest that $varphi: amapsto 1text{ or } 2$; that is, that the generator $a$ of the presentation $langle amid a^3rangle$ maps via $varphi$ to one of the elements of $Bbb Z_3$ given by a number coprime to $3$.



    Ask yourself,




    What reduced words, with letters in ${ a, a^{-1}}$, get mapped to the identity of $Bbb Z_3$ via $varphi$?




    But can you deduce what each word maps to under $varphi$ in general?



    Hover your cursor over (or, if you're on a touchscreen device, tap) the box below for some hints.




    Hint: Use Bézout's Identity, assuming $varphi(a)=p$ for some $p$ coprime to $3$. Here $varphi$ is a homomorphism. Don't forget to show both $ker(varphi)subseteq langlelangle a^3ranglerangle$ and $langlelangle a^3rangleranglesubseteqker(varphi)$.




    I hope this helps :)






    share|cite|improve this answer



























      up vote
      2
      down vote













      In order to show $ker(varphi)=langle langle a^3ranglerangle$, one must first be clear on what $varphi$ is, and instead of writing, say, $color{red}{a=1}$, one writes $varphi(a)=1$.



      You are on the right track in that you have evidence to suggest that $varphi: amapsto 1text{ or } 2$; that is, that the generator $a$ of the presentation $langle amid a^3rangle$ maps via $varphi$ to one of the elements of $Bbb Z_3$ given by a number coprime to $3$.



      Ask yourself,




      What reduced words, with letters in ${ a, a^{-1}}$, get mapped to the identity of $Bbb Z_3$ via $varphi$?




      But can you deduce what each word maps to under $varphi$ in general?



      Hover your cursor over (or, if you're on a touchscreen device, tap) the box below for some hints.




      Hint: Use Bézout's Identity, assuming $varphi(a)=p$ for some $p$ coprime to $3$. Here $varphi$ is a homomorphism. Don't forget to show both $ker(varphi)subseteq langlelangle a^3ranglerangle$ and $langlelangle a^3rangleranglesubseteqker(varphi)$.




      I hope this helps :)






      share|cite|improve this answer

























        up vote
        2
        down vote










        up vote
        2
        down vote









        In order to show $ker(varphi)=langle langle a^3ranglerangle$, one must first be clear on what $varphi$ is, and instead of writing, say, $color{red}{a=1}$, one writes $varphi(a)=1$.



        You are on the right track in that you have evidence to suggest that $varphi: amapsto 1text{ or } 2$; that is, that the generator $a$ of the presentation $langle amid a^3rangle$ maps via $varphi$ to one of the elements of $Bbb Z_3$ given by a number coprime to $3$.



        Ask yourself,




        What reduced words, with letters in ${ a, a^{-1}}$, get mapped to the identity of $Bbb Z_3$ via $varphi$?




        But can you deduce what each word maps to under $varphi$ in general?



        Hover your cursor over (or, if you're on a touchscreen device, tap) the box below for some hints.




        Hint: Use Bézout's Identity, assuming $varphi(a)=p$ for some $p$ coprime to $3$. Here $varphi$ is a homomorphism. Don't forget to show both $ker(varphi)subseteq langlelangle a^3ranglerangle$ and $langlelangle a^3rangleranglesubseteqker(varphi)$.




        I hope this helps :)






        share|cite|improve this answer














        In order to show $ker(varphi)=langle langle a^3ranglerangle$, one must first be clear on what $varphi$ is, and instead of writing, say, $color{red}{a=1}$, one writes $varphi(a)=1$.



        You are on the right track in that you have evidence to suggest that $varphi: amapsto 1text{ or } 2$; that is, that the generator $a$ of the presentation $langle amid a^3rangle$ maps via $varphi$ to one of the elements of $Bbb Z_3$ given by a number coprime to $3$.



        Ask yourself,




        What reduced words, with letters in ${ a, a^{-1}}$, get mapped to the identity of $Bbb Z_3$ via $varphi$?




        But can you deduce what each word maps to under $varphi$ in general?



        Hover your cursor over (or, if you're on a touchscreen device, tap) the box below for some hints.




        Hint: Use Bézout's Identity, assuming $varphi(a)=p$ for some $p$ coprime to $3$. Here $varphi$ is a homomorphism. Don't forget to show both $ker(varphi)subseteq langlelangle a^3ranglerangle$ and $langlelangle a^3rangleranglesubseteqker(varphi)$.




        I hope this helps :)







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 2 days ago

























        answered 2 days ago









        Shaun

        8,205113578




        8,205113578






























            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.





            Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


            Please pay close attention to the following guidance:


            • 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.


            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%2f3026474%2fcalculating-a-presentation-of-mathbbz-3-in-detail%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

            Bressuire

            Image
            .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
            Read more

            Cabo Verde

            Image
              Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
            Read more

            Gyllenstierna

            Image
            .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
            Read more