Find a subgroup of $D_{42}$ that is isomorphic to $S_3$.












0












$begingroup$


Note that $D_{42}$ is generated by $rho, r$ where $ord(rho) = 21$, $ord(r)= 2$. To locate a copy of $S_3$, there must be a subgroup of order $3$, which can be $e, rho^{7}, rho^{14}$. What about order $2$ elements? Inevitably, I have to introduce some reflections which will enlarge the group.






abstract-algebra







share|cite|improve this question









$endgroup$












  • $begingroup$
    Doesn't $S_3$ have 6 elements ?
    $endgroup$
    – Digitalis
    Dec 14 '18 at 17:49










  • $begingroup$
    Yes, and?@Digitalis
    $endgroup$
    – koch
    Dec 14 '18 at 17:55
















0












$begingroup$


Note that $D_{42}$ is generated by $rho, r$ where $ord(rho) = 21$, $ord(r)= 2$. To locate a copy of $S_3$, there must be a subgroup of order $3$, which can be $e, rho^{7}, rho^{14}$. What about order $2$ elements? Inevitably, I have to introduce some reflections which will enlarge the group.






abstract-algebra







share|cite|improve this question









$endgroup$












  • $begingroup$
    Doesn't $S_3$ have 6 elements ?
    $endgroup$
    – Digitalis
    Dec 14 '18 at 17:49










  • $begingroup$
    Yes, and?@Digitalis
    $endgroup$
    – koch
    Dec 14 '18 at 17:55














0












0








0





$begingroup$


Note that $D_{42}$ is generated by $rho, r$ where $ord(rho) = 21$, $ord(r)= 2$. To locate a copy of $S_3$, there must be a subgroup of order $3$, which can be $e, rho^{7}, rho^{14}$. What about order $2$ elements? Inevitably, I have to introduce some reflections which will enlarge the group.






abstract-algebra







share|cite|improve this question









$endgroup$




Note that $D_{42}$ is generated by $rho, r$ where $ord(rho) = 21$, $ord(r)= 2$. To locate a copy of $S_3$, there must be a subgroup of order $3$, which can be $e, rho^{7}, rho^{14}$. What about order $2$ elements? Inevitably, I have to introduce some reflections which will enlarge the group.






abstract-algebra




abstract-algebra






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 14 '18 at 17:47









kochkoch

18418




18418












  • $begingroup$
    Doesn't $S_3$ have 6 elements ?
    $endgroup$
    – Digitalis
    Dec 14 '18 at 17:49










  • $begingroup$
    Yes, and?@Digitalis
    $endgroup$
    – koch
    Dec 14 '18 at 17:55


















  • $begingroup$
    Doesn't $S_3$ have 6 elements ?
    $endgroup$
    – Digitalis
    Dec 14 '18 at 17:49










  • $begingroup$
    Yes, and?@Digitalis
    $endgroup$
    – koch
    Dec 14 '18 at 17:55
















$begingroup$
Doesn't $S_3$ have 6 elements ?
$endgroup$
– Digitalis
Dec 14 '18 at 17:49




$begingroup$
Doesn't $S_3$ have 6 elements ?
$endgroup$
– Digitalis
Dec 14 '18 at 17:49












$begingroup$
Yes, and?@Digitalis
$endgroup$
– koch
Dec 14 '18 at 17:55




$begingroup$
Yes, and?@Digitalis
$endgroup$
– koch
Dec 14 '18 at 17:55










2 Answers
2






active

oldest

votes


















1












$begingroup$

$S_3$ has $3!=6$ elements and you need three more.
$1,rho^7,rho^{14},r,rrho^7,rrho^{14}$ does the trick nicely.






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    We have a complete classification of all possible subgroups of $D_n$:



    Theorem: Every subgroup of $D_n=langle r,s rangle$ is is either cyclic or dihedral, and a complete listing of the subgroups is as follows:



    (1) $langle r^drangle$, where $dmid n$, with index $2d$,



    (2) $langle r^d, r^is rangle$, where $dmid n$, $0le ile d-1$, with index $d$.



    Every subgroup of $D_n$ occurs exactly once in this listing.
    For a proof see Theorem 3.1 of Keith Conrad's notes.



    For the appropriate $d$ we obtain the subgroup $D_3=S_3$ with $6$ elements in $(2)$.






    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%2f3039698%2ffind-a-subgroup-of-d-42-that-is-isomorphic-to-s-3%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      1












      $begingroup$

      $S_3$ has $3!=6$ elements and you need three more.
      $1,rho^7,rho^{14},r,rrho^7,rrho^{14}$ does the trick nicely.






      share|cite|improve this answer









      $endgroup$


















        1












        $begingroup$

        $S_3$ has $3!=6$ elements and you need three more.
        $1,rho^7,rho^{14},r,rrho^7,rrho^{14}$ does the trick nicely.






        share|cite|improve this answer









        $endgroup$
















          1












          1








          1





          $begingroup$

          $S_3$ has $3!=6$ elements and you need three more.
          $1,rho^7,rho^{14},r,rrho^7,rrho^{14}$ does the trick nicely.






          share|cite|improve this answer









          $endgroup$



          $S_3$ has $3!=6$ elements and you need three more.
          $1,rho^7,rho^{14},r,rrho^7,rrho^{14}$ does the trick nicely.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 14 '18 at 17:55









          Ross MillikanRoss Millikan

          295k23198371




          295k23198371























              0












              $begingroup$

              We have a complete classification of all possible subgroups of $D_n$:



              Theorem: Every subgroup of $D_n=langle r,s rangle$ is is either cyclic or dihedral, and a complete listing of the subgroups is as follows:



              (1) $langle r^drangle$, where $dmid n$, with index $2d$,



              (2) $langle r^d, r^is rangle$, where $dmid n$, $0le ile d-1$, with index $d$.



              Every subgroup of $D_n$ occurs exactly once in this listing.
              For a proof see Theorem 3.1 of Keith Conrad's notes.



              For the appropriate $d$ we obtain the subgroup $D_3=S_3$ with $6$ elements in $(2)$.






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                We have a complete classification of all possible subgroups of $D_n$:



                Theorem: Every subgroup of $D_n=langle r,s rangle$ is is either cyclic or dihedral, and a complete listing of the subgroups is as follows:



                (1) $langle r^drangle$, where $dmid n$, with index $2d$,



                (2) $langle r^d, r^is rangle$, where $dmid n$, $0le ile d-1$, with index $d$.



                Every subgroup of $D_n$ occurs exactly once in this listing.
                For a proof see Theorem 3.1 of Keith Conrad's notes.



                For the appropriate $d$ we obtain the subgroup $D_3=S_3$ with $6$ elements in $(2)$.






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  We have a complete classification of all possible subgroups of $D_n$:



                  Theorem: Every subgroup of $D_n=langle r,s rangle$ is is either cyclic or dihedral, and a complete listing of the subgroups is as follows:



                  (1) $langle r^drangle$, where $dmid n$, with index $2d$,



                  (2) $langle r^d, r^is rangle$, where $dmid n$, $0le ile d-1$, with index $d$.



                  Every subgroup of $D_n$ occurs exactly once in this listing.
                  For a proof see Theorem 3.1 of Keith Conrad's notes.



                  For the appropriate $d$ we obtain the subgroup $D_3=S_3$ with $6$ elements in $(2)$.






                  share|cite|improve this answer









                  $endgroup$



                  We have a complete classification of all possible subgroups of $D_n$:



                  Theorem: Every subgroup of $D_n=langle r,s rangle$ is is either cyclic or dihedral, and a complete listing of the subgroups is as follows:



                  (1) $langle r^drangle$, where $dmid n$, with index $2d$,



                  (2) $langle r^d, r^is rangle$, where $dmid n$, $0le ile d-1$, with index $d$.



                  Every subgroup of $D_n$ occurs exactly once in this listing.
                  For a proof see Theorem 3.1 of Keith Conrad's notes.



                  For the appropriate $d$ we obtain the subgroup $D_3=S_3$ with $6$ elements in $(2)$.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 14 '18 at 17:56









                  Dietrich BurdeDietrich Burde

                  79.1k647103




                  79.1k647103






























                      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%2f3039698%2ffind-a-subgroup-of-d-42-that-is-isomorphic-to-s-3%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