Using basic counting we can show there are precisely symettries of a pentagon. We have five slots to place the the number . Haven chosen a slot we can place either to the right or left of that point, hence there are just two more slots. Haven done that all the other numbers are completely determined because they need to preserve the configuration of the pentagon. Thus, there are a total of symettries.
Let be a rotation by degree clockwise. Then are all distinct rotations. While , where is the identity transformation.
Draw a line segment from through the middle of the pentagon. Let be the rotation through this line (so stays fixed). Then . Now confirm that are all distinct. Thus, we have created distinct symettries. And we see that each one is made out of , a rotation, and - a reflection.