Burnside's lemma

Burnside's Lemma is a mathematical tool used in group theory to count distinct objects under symmetry. It states that to find the number of unique arrangements (or orbits) of a set considering symmetrical transformations (like rotations or reflections), you can calculate the average number of arrangements fixed by each transformation. In simpler terms, it helps determine how many different patterns can be created from a shape or object when you take into account its symmetrical properties, ensuring that identical arrangements aren’t counted multiple times. This lemma is particularly useful in combinatorics and geometry.