How to convert one presentation into another? Please explain using a dihedral group as an example.












4














How can we convert a given presentation of a group $G$ into an another presentation?



Would anyone please explain to me by converting two different presentations of a dihedral group?



Thanks in advance.






group-theory group-presentation dihedral-groups combinatorial-group-theory







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  • 6




    See en.wikipedia.org/wiki/Tietze_transformations
    – Derek Holt
    Nov 20 '16 at 19:32
















4














How can we convert a given presentation of a group $G$ into an another presentation?



Would anyone please explain to me by converting two different presentations of a dihedral group?



Thanks in advance.






group-theory group-presentation dihedral-groups combinatorial-group-theory







share|cite|improve this question




















  • 6




    See en.wikipedia.org/wiki/Tietze_transformations
    – Derek Holt
    Nov 20 '16 at 19:32














4












4








4


3





How can we convert a given presentation of a group $G$ into an another presentation?



Would anyone please explain to me by converting two different presentations of a dihedral group?



Thanks in advance.






group-theory group-presentation dihedral-groups combinatorial-group-theory







share|cite|improve this question















How can we convert a given presentation of a group $G$ into an another presentation?



Would anyone please explain to me by converting two different presentations of a dihedral group?



Thanks in advance.






group-theory group-presentation dihedral-groups combinatorial-group-theory




group-theory group-presentation dihedral-groups combinatorial-group-theory






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share|cite|improve this question













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share|cite|improve this question








edited Nov 30 at 2:00









Shaun

8,339113578




8,339113578










asked Nov 20 '16 at 14:09









Prince Thomas

593210




593210








  • 6




    See en.wikipedia.org/wiki/Tietze_transformations
    – Derek Holt
    Nov 20 '16 at 19:32














  • 6




    See en.wikipedia.org/wiki/Tietze_transformations
    – Derek Holt
    Nov 20 '16 at 19:32








6




6




See en.wikipedia.org/wiki/Tietze_transformations
– Derek Holt
Nov 20 '16 at 19:32




See en.wikipedia.org/wiki/Tietze_transformations
– Derek Holt
Nov 20 '16 at 19:32










1 Answer
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One way to change a presentation of a group $G$ into another presentation of the same group is via Tietze transformations. They introduce or delete either generators or relations.



To use the example of the dihedral group $D_n$ of $2n$ elements, let's start with



$$P=langle a, bmid a^2, b^n, (ab)^2rangle.$$



Let $x=ab$ be an element of $D_n$. Then $astackrel{(1)}{=}xb^{-1}$, so, introducing $x$ as a generator gives



$$Pconglangle a, b, xmid a^2, b^n, (ab)^2, x=abrangle,$$



which is then isomorphic to



$$Q=langle b, xmid (xb^{-1})^2, b^n, x^2rangle$$



by eliminating $a$ (since $(1)$ tells us that it can be written as a product of the other generators, not including $a$).



Then $Q$ is a "new" presentation of $D_n$, although not entirely different from $P$.






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    One way to change a presentation of a group $G$ into another presentation of the same group is via Tietze transformations. They introduce or delete either generators or relations.



    To use the example of the dihedral group $D_n$ of $2n$ elements, let's start with



    $$P=langle a, bmid a^2, b^n, (ab)^2rangle.$$



    Let $x=ab$ be an element of $D_n$. Then $astackrel{(1)}{=}xb^{-1}$, so, introducing $x$ as a generator gives



    $$Pconglangle a, b, xmid a^2, b^n, (ab)^2, x=abrangle,$$



    which is then isomorphic to



    $$Q=langle b, xmid (xb^{-1})^2, b^n, x^2rangle$$



    by eliminating $a$ (since $(1)$ tells us that it can be written as a product of the other generators, not including $a$).



    Then $Q$ is a "new" presentation of $D_n$, although not entirely different from $P$.






    share|cite|improve this answer




























      1














      One way to change a presentation of a group $G$ into another presentation of the same group is via Tietze transformations. They introduce or delete either generators or relations.



      To use the example of the dihedral group $D_n$ of $2n$ elements, let's start with



      $$P=langle a, bmid a^2, b^n, (ab)^2rangle.$$



      Let $x=ab$ be an element of $D_n$. Then $astackrel{(1)}{=}xb^{-1}$, so, introducing $x$ as a generator gives



      $$Pconglangle a, b, xmid a^2, b^n, (ab)^2, x=abrangle,$$



      which is then isomorphic to



      $$Q=langle b, xmid (xb^{-1})^2, b^n, x^2rangle$$



      by eliminating $a$ (since $(1)$ tells us that it can be written as a product of the other generators, not including $a$).



      Then $Q$ is a "new" presentation of $D_n$, although not entirely different from $P$.






      share|cite|improve this answer


























        1












        1








        1






        One way to change a presentation of a group $G$ into another presentation of the same group is via Tietze transformations. They introduce or delete either generators or relations.



        To use the example of the dihedral group $D_n$ of $2n$ elements, let's start with



        $$P=langle a, bmid a^2, b^n, (ab)^2rangle.$$



        Let $x=ab$ be an element of $D_n$. Then $astackrel{(1)}{=}xb^{-1}$, so, introducing $x$ as a generator gives



        $$Pconglangle a, b, xmid a^2, b^n, (ab)^2, x=abrangle,$$



        which is then isomorphic to



        $$Q=langle b, xmid (xb^{-1})^2, b^n, x^2rangle$$



        by eliminating $a$ (since $(1)$ tells us that it can be written as a product of the other generators, not including $a$).



        Then $Q$ is a "new" presentation of $D_n$, although not entirely different from $P$.






        share|cite|improve this answer














        One way to change a presentation of a group $G$ into another presentation of the same group is via Tietze transformations. They introduce or delete either generators or relations.



        To use the example of the dihedral group $D_n$ of $2n$ elements, let's start with



        $$P=langle a, bmid a^2, b^n, (ab)^2rangle.$$



        Let $x=ab$ be an element of $D_n$. Then $astackrel{(1)}{=}xb^{-1}$, so, introducing $x$ as a generator gives



        $$Pconglangle a, b, xmid a^2, b^n, (ab)^2, x=abrangle,$$



        which is then isomorphic to



        $$Q=langle b, xmid (xb^{-1})^2, b^n, x^2rangle$$



        by eliminating $a$ (since $(1)$ tells us that it can be written as a product of the other generators, not including $a$).



        Then $Q$ is a "new" presentation of $D_n$, although not entirely different from $P$.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Nov 30 at 3:07

























        answered Nov 30 at 2:12









        Shaun

        8,339113578




        8,339113578






























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