I'm supposed to find the Cayley Table of the group of symmetries for a regular pentagon. But to find the Cayley table, I need to be able to figure out the symmetries of the pentagon.

I can see 6 symmetries of a pentagon. The identity, 4 rotations, and 1 reflections on the y-axis. There should be 4 more reflections, but I can't see it visually. Could anyone help me out?

  • 2
    $\begingroup$ Hint: The symmetries you've named so far generate the group. So to find the others, try composing the ones you have. (ie what happens if you rotate and then reflect on the y-axis?) $\endgroup$ – MartianInvader Sep 11 '13 at 0:48
  • $\begingroup$ Oh wow... Now I feel a little stupid. Thank you! $\endgroup$ – fernand Sep 11 '13 at 0:50
  • $\begingroup$ For someone who knows zero group theory, is there a way to tell that the "symmetries... named so far generate the group"? $\endgroup$ – Bennett Gardiner Sep 11 '13 at 1:15

The symmetries of the pentagon under flips and rotations are illustrated below:

Symmetries of the pentagon

Now it's a matter of filling in a $10 \times 10$ Cayley table.

  • $\begingroup$ +1 Specifically for using TikZ to make awesome diagrams and posting the source. $\endgroup$ – joshphysics Sep 18 '13 at 3:00

There will always be $2n$ symmetries for any convex regular n-gon. half of them are the rotations, and the other half are the reflections. For a polygon with an even number of sides, the axes of reflection will be about lines passing through opposing vertices, while the ones with an odd number of sides will pass through a vertex through the center of the opposing edge.

I have a suggestion, and I did this even with symmetries of polyhedral solids when I did undergraduate group theory projects and papers. Get yourself some card stock, draw yourself a real nice regular pentagon or whatever polygon you are working with, label the vertices on both sides as 1 through n (1 to 5 in your case). Cut it out, and start flipping and rotating to see the effect of combining rotation and reflection operations. Your table will be much easier to build, and you might actually get a deeper understanding of polygonal symmetries/dihedral groups.

This really is supposed to be the most enjoyable time in your mathematical education (symmetry groups are undeniably close to mathematical recreation), take advantage of it by having the fun! Get out your crayons and scissors and get at it! I remember when I first learned about what you are starting to get into, and I could not believe they were actually giving us college credit for it.


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