Consider a necklace contains $2m$ elements. We want to determine some combinatorical number (for example number of ways to color them). We may use P.E.T. But firstly it's need to determine group of representation of necklace. This group $G$ contains $2m$ rotations and the same amount of symmetries.
But there is a problem. If we consider regular polygon with $2m$ vertices and consider the same problem - then we need to consider only rotations. What's the defference between them?