# Symmetries of a Pentagon.

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?

• 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?) – MartianInvader Sep 11 '13 at 0:48
• Oh wow... Now I feel a little stupid. Thank you! – fernand Sep 11 '13 at 0:50
• For someone who knows zero group theory, is there a way to tell that the "symmetries... named so far generate the group"? – Bennett Gardiner Sep 11 '13 at 1:15

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