Abstract algebra? dihedral group-pentagon?

Well you could rotate the pentagon 5 ways each bringing it into coincidence with itself. That gives you 5 elements of the group, which looks like the cyclic group of order 5. Or, you could flip it upside down and then proceed to rotated it. That gives you 5 more elements. Since flipping it upside down again would make it right-side up, the flipping part is essentially the cyclic group of order 2. When you put it all together, you basically have a kind of product of the cyclic group of order 5 and the cyclic group of order 2, and you have all 10 elements.