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The symmetries of a pentagram

  1. Mar 26, 2009 #1
    1. The problem statement, all variables and given/known data
    The group of symmetries of a regular pentagram is isomorphic to the dihedral group of order 10.

    Show that this is true.


    3. The attempt at a solution
    It seems to me that the group shown by the "star" has order 5, since, by following the lines from one point, it takes 5 total paths to get back to the original point.

    So I thought it would isomorphic to the cyclic group of order 5.....how is it isomorphic to the dihedral group?
     
  2. jcsd
  3. Mar 26, 2009 #2
    The rotational symmetries are isomorphic to the cyclic group of order 5, but there are also reflectional symmetries that need to be considered.
     
  4. Mar 26, 2009 #3
    By reflectional symmetries do you mean the inverse? As in, if I choose one path from point A to point B to point C, then the reflectional symmetry is from point C to point B to point A?
     
  5. Mar 26, 2009 #4
    If you center your pentagram about the origin then the reflection across the x-axis would be a reflectional symmetry. You should be able to generate your group from that reflection and the rotation of 72 degrees about the origin.
     
  6. Mar 26, 2009 #5
    Oh okay...so when I said this---

    ---was I wrong?
     
  7. Mar 26, 2009 #6
    Yes the group will have order 10. If you label a vertex A and label a vertex adjacent to A B. Then there are 5 positions that A can be moved to and each of those allows exactly 2 positions for B.
     
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