Subgroups of $D_4$












10












$begingroup$


I need to determine the subgroups of the dihedral group of order 4, $D_4$.



I know that the elements of $D_4$ are ${1,r,r^2,r^3, s,rs,r^2s,r^3s}$



But I don't understand how to get the subgroups..






dihedral-groups







share|cite|improve this question









$endgroup$












  • $begingroup$
    The elements don't do you any good unless you know generators/relations, or something else about D4 to get you the group structure. What do you know about D4 apart from the names of the elements?
    $endgroup$
    – Tyler
    Oct 19 '13 at 16:19












  • $begingroup$
    @Tyler we know that $D_4$ is generated by the rotation $r$ and the reflection $s$
    $endgroup$
    – user43418
    Oct 19 '13 at 16:20






  • 3




    $begingroup$
    It is very disrespectful to delete a question once you get an answer. You shouldn't do that again.
    $endgroup$
    – Pedro Tamaroff♦
    Oct 19 '13 at 17:02






  • 3




    $begingroup$
    @Pedro: in the OP's defense, the question was deleted only 39 seconds after the answer was posted. It is possible the the OP hadn't seen the answer appear yet.
    $endgroup$
    – Henning Makholm
    Oct 19 '13 at 17:05






  • 1




    $begingroup$
    @PedroTamaroff It is true I didn't see it..
    $endgroup$
    – user43418
    Oct 19 '13 at 17:23
















10












$begingroup$


I need to determine the subgroups of the dihedral group of order 4, $D_4$.



I know that the elements of $D_4$ are ${1,r,r^2,r^3, s,rs,r^2s,r^3s}$



But I don't understand how to get the subgroups..






dihedral-groups







share|cite|improve this question









$endgroup$












  • $begingroup$
    The elements don't do you any good unless you know generators/relations, or something else about D4 to get you the group structure. What do you know about D4 apart from the names of the elements?
    $endgroup$
    – Tyler
    Oct 19 '13 at 16:19












  • $begingroup$
    @Tyler we know that $D_4$ is generated by the rotation $r$ and the reflection $s$
    $endgroup$
    – user43418
    Oct 19 '13 at 16:20






  • 3




    $begingroup$
    It is very disrespectful to delete a question once you get an answer. You shouldn't do that again.
    $endgroup$
    – Pedro Tamaroff♦
    Oct 19 '13 at 17:02






  • 3




    $begingroup$
    @Pedro: in the OP's defense, the question was deleted only 39 seconds after the answer was posted. It is possible the the OP hadn't seen the answer appear yet.
    $endgroup$
    – Henning Makholm
    Oct 19 '13 at 17:05






  • 1




    $begingroup$
    @PedroTamaroff It is true I didn't see it..
    $endgroup$
    – user43418
    Oct 19 '13 at 17:23














10












10








10


4



$begingroup$


I need to determine the subgroups of the dihedral group of order 4, $D_4$.



I know that the elements of $D_4$ are ${1,r,r^2,r^3, s,rs,r^2s,r^3s}$



But I don't understand how to get the subgroups..






dihedral-groups







share|cite|improve this question









$endgroup$




I need to determine the subgroups of the dihedral group of order 4, $D_4$.



I know that the elements of $D_4$ are ${1,r,r^2,r^3, s,rs,r^2s,r^3s}$



But I don't understand how to get the subgroups..






dihedral-groups




dihedral-groups






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Oct 19 '13 at 16:18









user43418user43418

4582828




4582828












  • $begingroup$
    The elements don't do you any good unless you know generators/relations, or something else about D4 to get you the group structure. What do you know about D4 apart from the names of the elements?
    $endgroup$
    – Tyler
    Oct 19 '13 at 16:19












  • $begingroup$
    @Tyler we know that $D_4$ is generated by the rotation $r$ and the reflection $s$
    $endgroup$
    – user43418
    Oct 19 '13 at 16:20






  • 3




    $begingroup$
    It is very disrespectful to delete a question once you get an answer. You shouldn't do that again.
    $endgroup$
    – Pedro Tamaroff♦
    Oct 19 '13 at 17:02






  • 3




    $begingroup$
    @Pedro: in the OP's defense, the question was deleted only 39 seconds after the answer was posted. It is possible the the OP hadn't seen the answer appear yet.
    $endgroup$
    – Henning Makholm
    Oct 19 '13 at 17:05






  • 1




    $begingroup$
    @PedroTamaroff It is true I didn't see it..
    $endgroup$
    – user43418
    Oct 19 '13 at 17:23


















  • $begingroup$
    The elements don't do you any good unless you know generators/relations, or something else about D4 to get you the group structure. What do you know about D4 apart from the names of the elements?
    $endgroup$
    – Tyler
    Oct 19 '13 at 16:19












  • $begingroup$
    @Tyler we know that $D_4$ is generated by the rotation $r$ and the reflection $s$
    $endgroup$
    – user43418
    Oct 19 '13 at 16:20






  • 3




    $begingroup$
    It is very disrespectful to delete a question once you get an answer. You shouldn't do that again.
    $endgroup$
    – Pedro Tamaroff♦
    Oct 19 '13 at 17:02






  • 3




    $begingroup$
    @Pedro: in the OP's defense, the question was deleted only 39 seconds after the answer was posted. It is possible the the OP hadn't seen the answer appear yet.
    $endgroup$
    – Henning Makholm
    Oct 19 '13 at 17:05






  • 1




    $begingroup$
    @PedroTamaroff It is true I didn't see it..
    $endgroup$
    – user43418
    Oct 19 '13 at 17:23
















$begingroup$
The elements don't do you any good unless you know generators/relations, or something else about D4 to get you the group structure. What do you know about D4 apart from the names of the elements?
$endgroup$
– Tyler
Oct 19 '13 at 16:19






$begingroup$
The elements don't do you any good unless you know generators/relations, or something else about D4 to get you the group structure. What do you know about D4 apart from the names of the elements?
$endgroup$
– Tyler
Oct 19 '13 at 16:19














$begingroup$
@Tyler we know that $D_4$ is generated by the rotation $r$ and the reflection $s$
$endgroup$
– user43418
Oct 19 '13 at 16:20




$begingroup$
@Tyler we know that $D_4$ is generated by the rotation $r$ and the reflection $s$
$endgroup$
– user43418
Oct 19 '13 at 16:20




3




3




$begingroup$
It is very disrespectful to delete a question once you get an answer. You shouldn't do that again.
$endgroup$
– Pedro Tamaroff♦
Oct 19 '13 at 17:02




$begingroup$
It is very disrespectful to delete a question once you get an answer. You shouldn't do that again.
$endgroup$
– Pedro Tamaroff♦
Oct 19 '13 at 17:02




3




3




$begingroup$
@Pedro: in the OP's defense, the question was deleted only 39 seconds after the answer was posted. It is possible the the OP hadn't seen the answer appear yet.
$endgroup$
– Henning Makholm
Oct 19 '13 at 17:05




$begingroup$
@Pedro: in the OP's defense, the question was deleted only 39 seconds after the answer was posted. It is possible the the OP hadn't seen the answer appear yet.
$endgroup$
– Henning Makholm
Oct 19 '13 at 17:05




1




1




$begingroup$
@PedroTamaroff It is true I didn't see it..
$endgroup$
– user43418
Oct 19 '13 at 17:23




$begingroup$
@PedroTamaroff It is true I didn't see it..
$endgroup$
– user43418
Oct 19 '13 at 17:23










2 Answers
2






active

oldest

votes


















10












$begingroup$

By Lagrange's Theorem, the possible orders are $1, 2, 4,$ and $8$.



The only subgroup of order $1$ is ${1}$ and the only subgroup of order $8$ is $D_4$.



If $D_4$ has an order $2$ subgroup, it must be isomorphic to $mathbb{Z}_2$ (this is the only group of order $2$ up to isomorphism). Such a group is cyclic, it is generated by an element of order $2$. Are there any such elements in $D_4$?



If $D_4$ has an order $4$ subgroup, it must be isomorphic to either $mathbb{Z}_4$ or $mathbb{Z}_2timesmathbb{Z}_2$ (these are the only groups of order $4$ up to isomorphism). In the former case, the group is cyclic, it is generated by an element of order $4$. Are there any such elements in $D_4$? In the latter case, the group is generated by two commuting elements of order $2$. Are there any such pairs of elements in $D_4$?



In summary, first find all the elements of order $2$ and all the elements of order $4$; each of them generates a cyclic subgroup. Then consider pairs of elements of order $2$ to find which of them generate subgroups isomorphic to $mathbb{Z}_2timesmathbb{Z}_2$.






share|cite|improve this answer











$endgroup$





















    9












    $begingroup$

    The notes by K. Conrad have a nice answer: the dihedral group $D_n$ is generated by a rotation $r$ and a reflection $s$ subject to the relations $r^n=s^2=1$ and $(rs)^2=1$.



    Proposition: Every subgroup of $D_n$ is cyclic or dihedral. A complete listing of the subgroups (including $1$ and $D_n$) is as follows:

    $(1)$ $langle r^d rangle$ for all divisors $dmid n$.

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



    Very nice pictures of the subgroup diagram of $D_4$ can be found here.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      Thank you very much, I was searching the subgroups of the dihedral group. I would love if your answer was to the question: What are the subgroups of ANY dihedral group? If it was like that, I would have found it in google easier!
      $endgroup$
      – Santropedro
      Jan 16 '17 at 0:59










    • $begingroup$
      Oh, you interpreted my comment not like I intended! Indeed your answer is for all the dihedral groups. I was trying to say that it would be better if the title of this question was for any dihedral group, making the question appear in my searches! I was avoiding questions that only worked for some specific dihedral group. Then I tried with this question and your answer helped me.
      $endgroup$
      – Santropedro
      Jan 16 '17 at 14:17










    • $begingroup$
      @Santropedro Oh, I apologize! It sounded liked you would love if my answer would have been for ANY dihedral group - reading it again it is different of course.
      $endgroup$
      – Dietrich Burde
      Jan 16 '17 at 15:10












    • $begingroup$
      Yes, communicating is hard sometimes! Note: Your answer helped me a lot, thanks!
      $endgroup$
      – Santropedro
      Jan 16 '17 at 17:27











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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    10












    $begingroup$

    By Lagrange's Theorem, the possible orders are $1, 2, 4,$ and $8$.



    The only subgroup of order $1$ is ${1}$ and the only subgroup of order $8$ is $D_4$.



    If $D_4$ has an order $2$ subgroup, it must be isomorphic to $mathbb{Z}_2$ (this is the only group of order $2$ up to isomorphism). Such a group is cyclic, it is generated by an element of order $2$. Are there any such elements in $D_4$?



    If $D_4$ has an order $4$ subgroup, it must be isomorphic to either $mathbb{Z}_4$ or $mathbb{Z}_2timesmathbb{Z}_2$ (these are the only groups of order $4$ up to isomorphism). In the former case, the group is cyclic, it is generated by an element of order $4$. Are there any such elements in $D_4$? In the latter case, the group is generated by two commuting elements of order $2$. Are there any such pairs of elements in $D_4$?



    In summary, first find all the elements of order $2$ and all the elements of order $4$; each of them generates a cyclic subgroup. Then consider pairs of elements of order $2$ to find which of them generate subgroups isomorphic to $mathbb{Z}_2timesmathbb{Z}_2$.






    share|cite|improve this answer











    $endgroup$


















      10












      $begingroup$

      By Lagrange's Theorem, the possible orders are $1, 2, 4,$ and $8$.



      The only subgroup of order $1$ is ${1}$ and the only subgroup of order $8$ is $D_4$.



      If $D_4$ has an order $2$ subgroup, it must be isomorphic to $mathbb{Z}_2$ (this is the only group of order $2$ up to isomorphism). Such a group is cyclic, it is generated by an element of order $2$. Are there any such elements in $D_4$?



      If $D_4$ has an order $4$ subgroup, it must be isomorphic to either $mathbb{Z}_4$ or $mathbb{Z}_2timesmathbb{Z}_2$ (these are the only groups of order $4$ up to isomorphism). In the former case, the group is cyclic, it is generated by an element of order $4$. Are there any such elements in $D_4$? In the latter case, the group is generated by two commuting elements of order $2$. Are there any such pairs of elements in $D_4$?



      In summary, first find all the elements of order $2$ and all the elements of order $4$; each of them generates a cyclic subgroup. Then consider pairs of elements of order $2$ to find which of them generate subgroups isomorphic to $mathbb{Z}_2timesmathbb{Z}_2$.






      share|cite|improve this answer











      $endgroup$
















        10












        10








        10





        $begingroup$

        By Lagrange's Theorem, the possible orders are $1, 2, 4,$ and $8$.



        The only subgroup of order $1$ is ${1}$ and the only subgroup of order $8$ is $D_4$.



        If $D_4$ has an order $2$ subgroup, it must be isomorphic to $mathbb{Z}_2$ (this is the only group of order $2$ up to isomorphism). Such a group is cyclic, it is generated by an element of order $2$. Are there any such elements in $D_4$?



        If $D_4$ has an order $4$ subgroup, it must be isomorphic to either $mathbb{Z}_4$ or $mathbb{Z}_2timesmathbb{Z}_2$ (these are the only groups of order $4$ up to isomorphism). In the former case, the group is cyclic, it is generated by an element of order $4$. Are there any such elements in $D_4$? In the latter case, the group is generated by two commuting elements of order $2$. Are there any such pairs of elements in $D_4$?



        In summary, first find all the elements of order $2$ and all the elements of order $4$; each of them generates a cyclic subgroup. Then consider pairs of elements of order $2$ to find which of them generate subgroups isomorphic to $mathbb{Z}_2timesmathbb{Z}_2$.






        share|cite|improve this answer











        $endgroup$



        By Lagrange's Theorem, the possible orders are $1, 2, 4,$ and $8$.



        The only subgroup of order $1$ is ${1}$ and the only subgroup of order $8$ is $D_4$.



        If $D_4$ has an order $2$ subgroup, it must be isomorphic to $mathbb{Z}_2$ (this is the only group of order $2$ up to isomorphism). Such a group is cyclic, it is generated by an element of order $2$. Are there any such elements in $D_4$?



        If $D_4$ has an order $4$ subgroup, it must be isomorphic to either $mathbb{Z}_4$ or $mathbb{Z}_2timesmathbb{Z}_2$ (these are the only groups of order $4$ up to isomorphism). In the former case, the group is cyclic, it is generated by an element of order $4$. Are there any such elements in $D_4$? In the latter case, the group is generated by two commuting elements of order $2$. Are there any such pairs of elements in $D_4$?



        In summary, first find all the elements of order $2$ and all the elements of order $4$; each of them generates a cyclic subgroup. Then consider pairs of elements of order $2$ to find which of them generate subgroups isomorphic to $mathbb{Z}_2timesmathbb{Z}_2$.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Nov 27 '18 at 12:24

























        answered Oct 19 '13 at 16:25









        Michael AlbaneseMichael Albanese

        63.2k1598303




        63.2k1598303























            9












            $begingroup$

            The notes by K. Conrad have a nice answer: the dihedral group $D_n$ is generated by a rotation $r$ and a reflection $s$ subject to the relations $r^n=s^2=1$ and $(rs)^2=1$.



            Proposition: Every subgroup of $D_n$ is cyclic or dihedral. A complete listing of the subgroups (including $1$ and $D_n$) is as follows:

            $(1)$ $langle r^d rangle$ for all divisors $dmid n$.

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



            Very nice pictures of the subgroup diagram of $D_4$ can be found here.






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              Thank you very much, I was searching the subgroups of the dihedral group. I would love if your answer was to the question: What are the subgroups of ANY dihedral group? If it was like that, I would have found it in google easier!
              $endgroup$
              – Santropedro
              Jan 16 '17 at 0:59










            • $begingroup$
              Oh, you interpreted my comment not like I intended! Indeed your answer is for all the dihedral groups. I was trying to say that it would be better if the title of this question was for any dihedral group, making the question appear in my searches! I was avoiding questions that only worked for some specific dihedral group. Then I tried with this question and your answer helped me.
              $endgroup$
              – Santropedro
              Jan 16 '17 at 14:17










            • $begingroup$
              @Santropedro Oh, I apologize! It sounded liked you would love if my answer would have been for ANY dihedral group - reading it again it is different of course.
              $endgroup$
              – Dietrich Burde
              Jan 16 '17 at 15:10












            • $begingroup$
              Yes, communicating is hard sometimes! Note: Your answer helped me a lot, thanks!
              $endgroup$
              – Santropedro
              Jan 16 '17 at 17:27
















            9












            $begingroup$

            The notes by K. Conrad have a nice answer: the dihedral group $D_n$ is generated by a rotation $r$ and a reflection $s$ subject to the relations $r^n=s^2=1$ and $(rs)^2=1$.



            Proposition: Every subgroup of $D_n$ is cyclic or dihedral. A complete listing of the subgroups (including $1$ and $D_n$) is as follows:

            $(1)$ $langle r^d rangle$ for all divisors $dmid n$.

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



            Very nice pictures of the subgroup diagram of $D_4$ can be found here.






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              Thank you very much, I was searching the subgroups of the dihedral group. I would love if your answer was to the question: What are the subgroups of ANY dihedral group? If it was like that, I would have found it in google easier!
              $endgroup$
              – Santropedro
              Jan 16 '17 at 0:59










            • $begingroup$
              Oh, you interpreted my comment not like I intended! Indeed your answer is for all the dihedral groups. I was trying to say that it would be better if the title of this question was for any dihedral group, making the question appear in my searches! I was avoiding questions that only worked for some specific dihedral group. Then I tried with this question and your answer helped me.
              $endgroup$
              – Santropedro
              Jan 16 '17 at 14:17










            • $begingroup$
              @Santropedro Oh, I apologize! It sounded liked you would love if my answer would have been for ANY dihedral group - reading it again it is different of course.
              $endgroup$
              – Dietrich Burde
              Jan 16 '17 at 15:10












            • $begingroup$
              Yes, communicating is hard sometimes! Note: Your answer helped me a lot, thanks!
              $endgroup$
              – Santropedro
              Jan 16 '17 at 17:27














            9












            9








            9





            $begingroup$

            The notes by K. Conrad have a nice answer: the dihedral group $D_n$ is generated by a rotation $r$ and a reflection $s$ subject to the relations $r^n=s^2=1$ and $(rs)^2=1$.



            Proposition: Every subgroup of $D_n$ is cyclic or dihedral. A complete listing of the subgroups (including $1$ and $D_n$) is as follows:

            $(1)$ $langle r^d rangle$ for all divisors $dmid n$.

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



            Very nice pictures of the subgroup diagram of $D_4$ can be found here.






            share|cite|improve this answer











            $endgroup$



            The notes by K. Conrad have a nice answer: the dihedral group $D_n$ is generated by a rotation $r$ and a reflection $s$ subject to the relations $r^n=s^2=1$ and $(rs)^2=1$.



            Proposition: Every subgroup of $D_n$ is cyclic or dihedral. A complete listing of the subgroups (including $1$ and $D_n$) is as follows:

            $(1)$ $langle r^d rangle$ for all divisors $dmid n$.

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



            Very nice pictures of the subgroup diagram of $D_4$ can be found here.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Apr 13 '17 at 12:21









            Community♦

            1




            1










            answered Sep 16 '16 at 15:31









            Dietrich BurdeDietrich Burde

            78.7k64387




            78.7k64387












            • $begingroup$
              Thank you very much, I was searching the subgroups of the dihedral group. I would love if your answer was to the question: What are the subgroups of ANY dihedral group? If it was like that, I would have found it in google easier!
              $endgroup$
              – Santropedro
              Jan 16 '17 at 0:59










            • $begingroup$
              Oh, you interpreted my comment not like I intended! Indeed your answer is for all the dihedral groups. I was trying to say that it would be better if the title of this question was for any dihedral group, making the question appear in my searches! I was avoiding questions that only worked for some specific dihedral group. Then I tried with this question and your answer helped me.
              $endgroup$
              – Santropedro
              Jan 16 '17 at 14:17










            • $begingroup$
              @Santropedro Oh, I apologize! It sounded liked you would love if my answer would have been for ANY dihedral group - reading it again it is different of course.
              $endgroup$
              – Dietrich Burde
              Jan 16 '17 at 15:10












            • $begingroup$
              Yes, communicating is hard sometimes! Note: Your answer helped me a lot, thanks!
              $endgroup$
              – Santropedro
              Jan 16 '17 at 17:27


















            • $begingroup$
              Thank you very much, I was searching the subgroups of the dihedral group. I would love if your answer was to the question: What are the subgroups of ANY dihedral group? If it was like that, I would have found it in google easier!
              $endgroup$
              – Santropedro
              Jan 16 '17 at 0:59










            • $begingroup$
              Oh, you interpreted my comment not like I intended! Indeed your answer is for all the dihedral groups. I was trying to say that it would be better if the title of this question was for any dihedral group, making the question appear in my searches! I was avoiding questions that only worked for some specific dihedral group. Then I tried with this question and your answer helped me.
              $endgroup$
              – Santropedro
              Jan 16 '17 at 14:17










            • $begingroup$
              @Santropedro Oh, I apologize! It sounded liked you would love if my answer would have been for ANY dihedral group - reading it again it is different of course.
              $endgroup$
              – Dietrich Burde
              Jan 16 '17 at 15:10












            • $begingroup$
              Yes, communicating is hard sometimes! Note: Your answer helped me a lot, thanks!
              $endgroup$
              – Santropedro
              Jan 16 '17 at 17:27
















            $begingroup$
            Thank you very much, I was searching the subgroups of the dihedral group. I would love if your answer was to the question: What are the subgroups of ANY dihedral group? If it was like that, I would have found it in google easier!
            $endgroup$
            – Santropedro
            Jan 16 '17 at 0:59




            $begingroup$
            Thank you very much, I was searching the subgroups of the dihedral group. I would love if your answer was to the question: What are the subgroups of ANY dihedral group? If it was like that, I would have found it in google easier!
            $endgroup$
            – Santropedro
            Jan 16 '17 at 0:59












            $begingroup$
            Oh, you interpreted my comment not like I intended! Indeed your answer is for all the dihedral groups. I was trying to say that it would be better if the title of this question was for any dihedral group, making the question appear in my searches! I was avoiding questions that only worked for some specific dihedral group. Then I tried with this question and your answer helped me.
            $endgroup$
            – Santropedro
            Jan 16 '17 at 14:17




            $begingroup$
            Oh, you interpreted my comment not like I intended! Indeed your answer is for all the dihedral groups. I was trying to say that it would be better if the title of this question was for any dihedral group, making the question appear in my searches! I was avoiding questions that only worked for some specific dihedral group. Then I tried with this question and your answer helped me.
            $endgroup$
            – Santropedro
            Jan 16 '17 at 14:17












            $begingroup$
            @Santropedro Oh, I apologize! It sounded liked you would love if my answer would have been for ANY dihedral group - reading it again it is different of course.
            $endgroup$
            – Dietrich Burde
            Jan 16 '17 at 15:10






            $begingroup$
            @Santropedro Oh, I apologize! It sounded liked you would love if my answer would have been for ANY dihedral group - reading it again it is different of course.
            $endgroup$
            – Dietrich Burde
            Jan 16 '17 at 15:10














            $begingroup$
            Yes, communicating is hard sometimes! Note: Your answer helped me a lot, thanks!
            $endgroup$
            – Santropedro
            Jan 16 '17 at 17:27




            $begingroup$
            Yes, communicating is hard sometimes! Note: Your answer helped me a lot, thanks!
            $endgroup$
            – Santropedro
            Jan 16 '17 at 17:27


















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