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












1












$begingroup$



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











$endgroup$








  • 2




    $begingroup$
    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.
    $endgroup$
    – Shaun
    Dec 14 '18 at 3:32






  • 1




    $begingroup$
    Note, too, that there is no such thing as the presentation of a group, strictly speaking.
    $endgroup$
    – Shaun
    Dec 14 '18 at 4:36


















1












$begingroup$



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











$endgroup$








  • 2




    $begingroup$
    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.
    $endgroup$
    – Shaun
    Dec 14 '18 at 3:32






  • 1




    $begingroup$
    Note, too, that there is no such thing as the presentation of a group, strictly speaking.
    $endgroup$
    – Shaun
    Dec 14 '18 at 4:36
















1












1








1





$begingroup$



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











$endgroup$





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 Dec 14 '18 at 4:35









Shaun

9,083113683




9,083113683










asked Dec 5 '18 at 1:34









eraldcoileraldcoil

395211




395211








  • 2




    $begingroup$
    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.
    $endgroup$
    – Shaun
    Dec 14 '18 at 3:32






  • 1




    $begingroup$
    Note, too, that there is no such thing as the presentation of a group, strictly speaking.
    $endgroup$
    – Shaun
    Dec 14 '18 at 4:36
















  • 2




    $begingroup$
    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.
    $endgroup$
    – Shaun
    Dec 14 '18 at 3:32






  • 1




    $begingroup$
    Note, too, that there is no such thing as the presentation of a group, strictly speaking.
    $endgroup$
    – Shaun
    Dec 14 '18 at 4:36










2




2




$begingroup$
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.
$endgroup$
– Shaun
Dec 14 '18 at 3:32




$begingroup$
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.
$endgroup$
– Shaun
Dec 14 '18 at 3:32




1




1




$begingroup$
Note, too, that there is no such thing as the presentation of a group, strictly speaking.
$endgroup$
– Shaun
Dec 14 '18 at 4:36






$begingroup$
Note, too, that there is no such thing as the presentation of a group, strictly speaking.
$endgroup$
– Shaun
Dec 14 '18 at 4:36












1 Answer
1






active

oldest

votes


















2












$begingroup$

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











$endgroup$













    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%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









    2












    $begingroup$

    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











    $endgroup$


















      2












      $begingroup$

      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











      $endgroup$
















        2












        2








        2





        $begingroup$

        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











        $endgroup$



        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 Dec 14 '18 at 3:34

























        answered Dec 14 '18 at 3:15









        ShaunShaun

        9,083113683




        9,083113683






























            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%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

            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

            build a pushdown automaton that recognizes the reverse language of a given pushdown automaton?

            Image
            0 I think it's similiar to NFAs. I replace the finite states of the given automaton for start-states for my new automaton. I do it with $epsilon$ -transition from the start state to the actual finite states of the given automaton. The start state of the given automaton will be my accepting(finite) state of my new one. All transitions will be reversed of course to recognize the reversed word. But how do I have to change the stack of the pushdown automaton? That's tricky. formal-languages automata context-free-grammar share | cite | improve this question edited Nov 29 '18 at 14:20 joee
            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