Example of non-abelian groups with these properties











up vote
0
down vote

favorite
1












I am looking for examples of non-abelian groups of arbitrarily large size with the following properties




  1. Have order $p^a$, where $a$ is a positive integer and $p$ is prime.

  2. Contain an abelian subgroup of order $p^{a-2}$.


I know one example which is the quaternion group. I am looking for more examples of groups of arbitrarily large size.






group-theory finite-groups p-groups







share|cite|improve this question




















  • 1




    There are too many examples. You need to impose some more restrictions.
    – Derek Holt
    Nov 26 at 13:05















up vote
0
down vote

favorite
1












I am looking for examples of non-abelian groups of arbitrarily large size with the following properties




  1. Have order $p^a$, where $a$ is a positive integer and $p$ is prime.

  2. Contain an abelian subgroup of order $p^{a-2}$.


I know one example which is the quaternion group. I am looking for more examples of groups of arbitrarily large size.






group-theory finite-groups p-groups







share|cite|improve this question




















  • 1




    There are too many examples. You need to impose some more restrictions.
    – Derek Holt
    Nov 26 at 13:05













up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





I am looking for examples of non-abelian groups of arbitrarily large size with the following properties




  1. Have order $p^a$, where $a$ is a positive integer and $p$ is prime.

  2. Contain an abelian subgroup of order $p^{a-2}$.


I know one example which is the quaternion group. I am looking for more examples of groups of arbitrarily large size.






group-theory finite-groups p-groups







share|cite|improve this question















I am looking for examples of non-abelian groups of arbitrarily large size with the following properties




  1. Have order $p^a$, where $a$ is a positive integer and $p$ is prime.

  2. Contain an abelian subgroup of order $p^{a-2}$.


I know one example which is the quaternion group. I am looking for more examples of groups of arbitrarily large size.






group-theory finite-groups p-groups




group-theory finite-groups p-groups






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 26 at 12:58









the_fox

2,3191430




2,3191430










asked Nov 26 at 12:25









I_wil_break_wall

203




203








  • 1




    There are too many examples. You need to impose some more restrictions.
    – Derek Holt
    Nov 26 at 13:05














  • 1




    There are too many examples. You need to impose some more restrictions.
    – Derek Holt
    Nov 26 at 13:05








1




1




There are too many examples. You need to impose some more restrictions.
– Derek Holt
Nov 26 at 13:05




There are too many examples. You need to impose some more restrictions.
– Derek Holt
Nov 26 at 13:05










2 Answers
2






active

oldest

votes

















up vote
1
down vote



accepted










Take, for example, the direct product of a nonabelian group of order $p^3$ with an abelian group of order $p^{a-3}$.






share|cite|improve this answer




























    up vote
    0
    down vote













    Dihedral groups of order $2^a$ have both properties (the subgroup being the cyclic one generated by the square of a highest-order element), but they also have a larger abelian subgroup of order $2^{a-1}$, so might not be what you're after.






    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%2f3014253%2fexample-of-non-abelian-groups-with-these-properties%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








      up vote
      1
      down vote



      accepted










      Take, for example, the direct product of a nonabelian group of order $p^3$ with an abelian group of order $p^{a-3}$.






      share|cite|improve this answer

























        up vote
        1
        down vote



        accepted










        Take, for example, the direct product of a nonabelian group of order $p^3$ with an abelian group of order $p^{a-3}$.






        share|cite|improve this answer























          up vote
          1
          down vote



          accepted







          up vote
          1
          down vote



          accepted






          Take, for example, the direct product of a nonabelian group of order $p^3$ with an abelian group of order $p^{a-3}$.






          share|cite|improve this answer












          Take, for example, the direct product of a nonabelian group of order $p^3$ with an abelian group of order $p^{a-3}$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 26 at 12:53









          user10354138

          6,9551624




          6,9551624






















              up vote
              0
              down vote













              Dihedral groups of order $2^a$ have both properties (the subgroup being the cyclic one generated by the square of a highest-order element), but they also have a larger abelian subgroup of order $2^{a-1}$, so might not be what you're after.






              share|cite|improve this answer

























                up vote
                0
                down vote













                Dihedral groups of order $2^a$ have both properties (the subgroup being the cyclic one generated by the square of a highest-order element), but they also have a larger abelian subgroup of order $2^{a-1}$, so might not be what you're after.






                share|cite|improve this answer























                  up vote
                  0
                  down vote










                  up vote
                  0
                  down vote









                  Dihedral groups of order $2^a$ have both properties (the subgroup being the cyclic one generated by the square of a highest-order element), but they also have a larger abelian subgroup of order $2^{a-1}$, so might not be what you're after.






                  share|cite|improve this answer












                  Dihedral groups of order $2^a$ have both properties (the subgroup being the cyclic one generated by the square of a highest-order element), but they also have a larger abelian subgroup of order $2^{a-1}$, so might not be what you're after.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Nov 26 at 12:52









                  user3482749

                  2,096414




                  2,096414






























                      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%2f3014253%2fexample-of-non-abelian-groups-with-these-properties%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

                      To store a contact into the json file from server.js file using a class in NodeJS

                      Image
                      0 I have created three files -contactPresentation.js, contactService.js, contacts.json. I will run the contactService.js file. And it will navigate to contactPresentation.js file that will only choose an option to either add or display a file. If add(), then add the data into add of service file.And the data will be added to the contacts.json file . And if display(), then display the display() of service file. And the display will take the data from contacts.json file. However, i know my code is flawed. Please correct it . I am also getting an error - this.contacts.push(new ContactService(userName, contactNumber, emailId)); ^ TypeError: Cannot read property 'push' of undefined at ContactService.add (e:NodeJSProjectcontactManager2contactService.js:57:23) at new C
                      Read more

                      Redirect URL with Chrome Remote Debugging Android Devices

                      Image
                      1 How can i redirect an URL with Chrome Remote Debugging Android Devices? like Requestly Chrome extension on desktop. google-chrome debugging url redirect testing share | improve this question edited Nov 24 '18 at 8:16 sachinjain024 14.7k 23 73 129 asked Nov 23 '18 at 16:09 Momen Momen 91 10
                      Read more

                      Dieringhausen

                      Image
                      Dieringhausen Stadt Gummersbach 50.982777777778 7.5316666666667 165 Koordinaten: 50° 58′ 58″  N , 7° 31′ 54″  O Höhe: 165  (160–245)  m Einwohner: 5055  (30. Jun. 2016) Postleitzahl: 51645 Vorwahl: 02261 Lage von Dieringhausen in Gummersbach Im Hohl ca. 1927–1930 Ansicht von Südwesten bei Bünghausen Dieringhausen (im örtlichen Dialekt Dierkesen ) ist ein Ortsteil von Gummersbach im Oberbergischen Kreis, Regierungsbezirk Köln, Nordrhein-Westfalen (Deutschland). Inhaltsverzeichnis 1 Geographie 2 Geschichte 3 Kultur 3.1 Sehenswürdigkeiten 3.2 Regelmäßige Veranstaltungen 4 Infrastruktur und Wirtschaft 4.1 Verkehr 4.1.1 Schienen- und Busverkehr 4.1.2 Straßen 4.2 Öffentliche Einrichtungen 4.2.1 Schulen und Bildungseinrichtungen 4.2.2 Kirchliche Einrichtungen 5 Persönlichkeiten 5.1 In Dieringhausen geboren 5.2 In Dieringhausen gelebt 6 Einzelnachweise 7
                      Read more