symmetry group

Some things to remember about the Cayley graph:

• You can think of the edges as the group elements. So for the symmetry group of a triangle, the edges either represent rotations of 120 degrees or flips.
• The binary operation is composition. How do you compose two rotations? Do one rotation, then the other.
• You can also think of each node in the Cayley graph as either:
  • Representing a state of the triangle, after applying the arrows to get from the original triangle to the transformed triangle
  • Representing a function, that moves the triangle from their original labeling to the transformed labeling.
  • In this way, the nodes can also be thought of as the group elements. There are as many group elements as there are nodes, because each node is a distinct end result from performing a series of symmetry transformations.
  • I had a little bit of trouble of thinking of the nodes as group elements at first, because I wondered "how can you compose two nodes?"