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Group actions of D5


Examples of the dihedral group $D_4$ acting on setsGeometric Interpretation of S3Consider group G acting on a set XSmallest graph with automorphism group the quaternion $8$-group, $Q_8$What kind of graph/group theoretical structure is that?$D_2n$ Acting on Opposite Pairs of VerticesThe order of the Symmetry Group of Platonic SolidsHow many triangles are there in the 'picture' of $K_5$? Different ways to count itSymmetry group of cubeHow should we think of the commutator of two permutations?













0












$begingroup$


I have to give 5 examples of D5 acting on a set. So far, I have D5 acting on the set of vertices of a pentagon and “rotating” each vertex one to the right, sending the vertices to a reflection in the x-axis, rotating the edges of a pentagon one to the right and reflecting the edges in the x-axis but I’m struggling to come up with a fifth example. Could someone help me out with a final example and/or tell me if I’ve made any mistakes with the examples I’ve given :)










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New contributor




m0729 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







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  • $begingroup$
    Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
    $endgroup$
    – Shaun
    15 hours ago






  • 1




    $begingroup$
    @Shaun I am afraid I don't completely agree with you in this case. It should be $D_5$ rather than D5, but apart from that, this seems to me to be a reasonably good title. I find it more helpful if the title indicates the general area of the question, but the question itself is easier to read if it is in the body of the post. A seriously bad title would be soemthing like "a problem in abstract algebra".
    $endgroup$
    – Derek Holt
    15 hours ago
















0












$begingroup$


I have to give 5 examples of D5 acting on a set. So far, I have D5 acting on the set of vertices of a pentagon and “rotating” each vertex one to the right, sending the vertices to a reflection in the x-axis, rotating the edges of a pentagon one to the right and reflecting the edges in the x-axis but I’m struggling to come up with a fifth example. Could someone help me out with a final example and/or tell me if I’ve made any mistakes with the examples I’ve given :)










share|cite|improve this question









New contributor




m0729 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$











  • $begingroup$
    Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
    $endgroup$
    – Shaun
    15 hours ago






  • 1




    $begingroup$
    @Shaun I am afraid I don't completely agree with you in this case. It should be $D_5$ rather than D5, but apart from that, this seems to me to be a reasonably good title. I find it more helpful if the title indicates the general area of the question, but the question itself is easier to read if it is in the body of the post. A seriously bad title would be soemthing like "a problem in abstract algebra".
    $endgroup$
    – Derek Holt
    15 hours ago














0












0








0


1



$begingroup$


I have to give 5 examples of D5 acting on a set. So far, I have D5 acting on the set of vertices of a pentagon and “rotating” each vertex one to the right, sending the vertices to a reflection in the x-axis, rotating the edges of a pentagon one to the right and reflecting the edges in the x-axis but I’m struggling to come up with a fifth example. Could someone help me out with a final example and/or tell me if I’ve made any mistakes with the examples I’ve given :)










share|cite|improve this question









New contributor




m0729 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




I have to give 5 examples of D5 acting on a set. So far, I have D5 acting on the set of vertices of a pentagon and “rotating” each vertex one to the right, sending the vertices to a reflection in the x-axis, rotating the edges of a pentagon one to the right and reflecting the edges in the x-axis but I’m struggling to come up with a fifth example. Could someone help me out with a final example and/or tell me if I’ve made any mistakes with the examples I’ve given :)







group-theory group-actions






share|cite|improve this question









New contributor




m0729 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









New contributor




m0729 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question








edited 15 hours ago









J. W. Tanner

2,9271217




2,9271217






New contributor




m0729 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 17 hours ago









m0729m0729

11




11




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New contributor





m0729 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






m0729 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











  • $begingroup$
    Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
    $endgroup$
    – Shaun
    15 hours ago






  • 1




    $begingroup$
    @Shaun I am afraid I don't completely agree with you in this case. It should be $D_5$ rather than D5, but apart from that, this seems to me to be a reasonably good title. I find it more helpful if the title indicates the general area of the question, but the question itself is easier to read if it is in the body of the post. A seriously bad title would be soemthing like "a problem in abstract algebra".
    $endgroup$
    – Derek Holt
    15 hours ago

















  • $begingroup$
    Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
    $endgroup$
    – Shaun
    15 hours ago






  • 1




    $begingroup$
    @Shaun I am afraid I don't completely agree with you in this case. It should be $D_5$ rather than D5, but apart from that, this seems to me to be a reasonably good title. I find it more helpful if the title indicates the general area of the question, but the question itself is easier to read if it is in the body of the post. A seriously bad title would be soemthing like "a problem in abstract algebra".
    $endgroup$
    – Derek Holt
    15 hours ago
















$begingroup$
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
$endgroup$
– Shaun
15 hours ago




$begingroup$
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here.
$endgroup$
– Shaun
15 hours ago




1




1




$begingroup$
@Shaun I am afraid I don't completely agree with you in this case. It should be $D_5$ rather than D5, but apart from that, this seems to me to be a reasonably good title. I find it more helpful if the title indicates the general area of the question, but the question itself is easier to read if it is in the body of the post. A seriously bad title would be soemthing like "a problem in abstract algebra".
$endgroup$
– Derek Holt
15 hours ago





$begingroup$
@Shaun I am afraid I don't completely agree with you in this case. It should be $D_5$ rather than D5, but apart from that, this seems to me to be a reasonably good title. I find it more helpful if the title indicates the general area of the question, but the question itself is easier to read if it is in the body of the post. A seriously bad title would be soemthing like "a problem in abstract algebra".
$endgroup$
– Derek Holt
15 hours ago











2 Answers
2






active

oldest

votes


















0












$begingroup$

The most important action that is usually forgotten is the identity operation. You keep all the vertices in place. It's an important one, because element “one” is literally in the definition of a group.



I think you might be good with this.






share|cite|improve this answer









$endgroup$








  • 1




    $begingroup$
    Also you could let $D_5$ act trivially on sets of sizes $1,2,3,4,5$, giving five distinct actions, but that might not the sort of answer that is expected.
    $endgroup$
    – Derek Holt
    16 hours ago



















0












$begingroup$

If I understood your question correctly, you listed 3 examples of $D_5$ acting on vertices of a pentagon. Here's how to get more.



Let $rho$ be an element of order $5$ in $D_5$ and let $sigma$ be an element of order $2$ in $D_5$. One action of $D_5$ on vertices of a pentagon would correspond to $rho$ rotating the pentagon by $72^o$ and $sigma$ reflecting it.



There are also actions corresponding to where $rho$ rotates the pentagon by any other multiple of $72^o$ (including $0^o$).



There are also actions where $rho$ rotates the pentagon but $sigma$ does nothing to the pentagon.



You could also consider actions of $D_5$ on itself (e.g., multiplication or conjugation).






share|cite|improve this answer











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    Your Answer





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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    0












    $begingroup$

    The most important action that is usually forgotten is the identity operation. You keep all the vertices in place. It's an important one, because element “one” is literally in the definition of a group.



    I think you might be good with this.






    share|cite|improve this answer









    $endgroup$








    • 1




      $begingroup$
      Also you could let $D_5$ act trivially on sets of sizes $1,2,3,4,5$, giving five distinct actions, but that might not the sort of answer that is expected.
      $endgroup$
      – Derek Holt
      16 hours ago
















    0












    $begingroup$

    The most important action that is usually forgotten is the identity operation. You keep all the vertices in place. It's an important one, because element “one” is literally in the definition of a group.



    I think you might be good with this.






    share|cite|improve this answer









    $endgroup$








    • 1




      $begingroup$
      Also you could let $D_5$ act trivially on sets of sizes $1,2,3,4,5$, giving five distinct actions, but that might not the sort of answer that is expected.
      $endgroup$
      – Derek Holt
      16 hours ago














    0












    0








    0





    $begingroup$

    The most important action that is usually forgotten is the identity operation. You keep all the vertices in place. It's an important one, because element “one” is literally in the definition of a group.



    I think you might be good with this.






    share|cite|improve this answer









    $endgroup$



    The most important action that is usually forgotten is the identity operation. You keep all the vertices in place. It's an important one, because element “one” is literally in the definition of a group.



    I think you might be good with this.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered 16 hours ago









    Vasily MitchVasily Mitch

    2,3841311




    2,3841311







    • 1




      $begingroup$
      Also you could let $D_5$ act trivially on sets of sizes $1,2,3,4,5$, giving five distinct actions, but that might not the sort of answer that is expected.
      $endgroup$
      – Derek Holt
      16 hours ago













    • 1




      $begingroup$
      Also you could let $D_5$ act trivially on sets of sizes $1,2,3,4,5$, giving five distinct actions, but that might not the sort of answer that is expected.
      $endgroup$
      – Derek Holt
      16 hours ago








    1




    1




    $begingroup$
    Also you could let $D_5$ act trivially on sets of sizes $1,2,3,4,5$, giving five distinct actions, but that might not the sort of answer that is expected.
    $endgroup$
    – Derek Holt
    16 hours ago





    $begingroup$
    Also you could let $D_5$ act trivially on sets of sizes $1,2,3,4,5$, giving five distinct actions, but that might not the sort of answer that is expected.
    $endgroup$
    – Derek Holt
    16 hours ago












    0












    $begingroup$

    If I understood your question correctly, you listed 3 examples of $D_5$ acting on vertices of a pentagon. Here's how to get more.



    Let $rho$ be an element of order $5$ in $D_5$ and let $sigma$ be an element of order $2$ in $D_5$. One action of $D_5$ on vertices of a pentagon would correspond to $rho$ rotating the pentagon by $72^o$ and $sigma$ reflecting it.



    There are also actions corresponding to where $rho$ rotates the pentagon by any other multiple of $72^o$ (including $0^o$).



    There are also actions where $rho$ rotates the pentagon but $sigma$ does nothing to the pentagon.



    You could also consider actions of $D_5$ on itself (e.g., multiplication or conjugation).






    share|cite|improve this answer











    $endgroup$

















      0












      $begingroup$

      If I understood your question correctly, you listed 3 examples of $D_5$ acting on vertices of a pentagon. Here's how to get more.



      Let $rho$ be an element of order $5$ in $D_5$ and let $sigma$ be an element of order $2$ in $D_5$. One action of $D_5$ on vertices of a pentagon would correspond to $rho$ rotating the pentagon by $72^o$ and $sigma$ reflecting it.



      There are also actions corresponding to where $rho$ rotates the pentagon by any other multiple of $72^o$ (including $0^o$).



      There are also actions where $rho$ rotates the pentagon but $sigma$ does nothing to the pentagon.



      You could also consider actions of $D_5$ on itself (e.g., multiplication or conjugation).






      share|cite|improve this answer











      $endgroup$















        0












        0








        0





        $begingroup$

        If I understood your question correctly, you listed 3 examples of $D_5$ acting on vertices of a pentagon. Here's how to get more.



        Let $rho$ be an element of order $5$ in $D_5$ and let $sigma$ be an element of order $2$ in $D_5$. One action of $D_5$ on vertices of a pentagon would correspond to $rho$ rotating the pentagon by $72^o$ and $sigma$ reflecting it.



        There are also actions corresponding to where $rho$ rotates the pentagon by any other multiple of $72^o$ (including $0^o$).



        There are also actions where $rho$ rotates the pentagon but $sigma$ does nothing to the pentagon.



        You could also consider actions of $D_5$ on itself (e.g., multiplication or conjugation).






        share|cite|improve this answer











        $endgroup$



        If I understood your question correctly, you listed 3 examples of $D_5$ acting on vertices of a pentagon. Here's how to get more.



        Let $rho$ be an element of order $5$ in $D_5$ and let $sigma$ be an element of order $2$ in $D_5$. One action of $D_5$ on vertices of a pentagon would correspond to $rho$ rotating the pentagon by $72^o$ and $sigma$ reflecting it.



        There are also actions corresponding to where $rho$ rotates the pentagon by any other multiple of $72^o$ (including $0^o$).



        There are also actions where $rho$ rotates the pentagon but $sigma$ does nothing to the pentagon.



        You could also consider actions of $D_5$ on itself (e.g., multiplication or conjugation).







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 15 hours ago

























        answered 15 hours ago









        J. W. TannerJ. W. Tanner

        2,9271217




        2,9271217




















            m0729 is a new contributor. Be nice, and check out our Code of Conduct.









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