Burnside Necklace Symmetry

Burnside's Lemma (or the Cauchy-Frobenius lemma) provides a powerful way to count the number of distinct configurations of an object under a group of symmetries, like a necklace under rotation or a bracelet under both rotation and reflection. It tells us that the number of orbits (distinct colorings) is exactly the average number of configurations fixed by the group elements.

Experiment

You can interactively explore the relationship between the group action, stabilizers, and orbits using the module:

1. Open the Burnside Necklace Symmetry module.
2. Set the Length to 6.
3. Set the Colors to 2.
4. Change the Symmetry to "Rotations and reflections" to see how reflections group configurations that were previously distinct into a single "bracelet".
5. Adjust the Coloring index to select a specific coloring. Observe its stabilizer size and the resulting orbit size. The product of the orbit size and stabilizer size always equals the group order (Orbit-Stabilizer Theorem).