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The symmetries of a pentagram

  • Thread starter fk378
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  • #1
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Homework Statement


The group of symmetries of a regular pentagram is isomorphic to the dihedral group of order 10.

Show that this is true.


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?
 

Answers and Replies

  • #2
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The rotational symmetries are isomorphic to the cyclic group of order 5, but there are also reflectional symmetries that need to be considered.
 
  • #3
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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?
 
  • #4
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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.
 
  • #5
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Oh okay...so when I said this---

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?
---was I wrong?
 
  • #6
6
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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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