Example of non-abelian groups with these properties











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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







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    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













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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






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






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    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.






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      2 Answers
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      active

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      2 Answers
      2






      active

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      active

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      active

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      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

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              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






























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