Abstract Algebra: Theory and Applications · Problem 14.5.8
The Sylow Theorems — Thomas Judson
Mathematics — Abstract Algebra · burnside-coloring-square
Problem
If a square remains fixed in the plane, how many different ways can the corners of the square be colored if three colors are used? Hint. .Intuitive explanation
This is a Burnside's lemma problem: you're counting colorings up to symmetry, not all possible colorings. The square has 8 symmetries (4 rotations and 4 reflections), and each one fixes a different subset of the 3⁴ = 81 total colorings. Rather than subtract out "equivalent" colorings by hand, Burnside's lemma says the answer is the average number of colorings fixed by each symmetry—exactly what the hint computes. Your job is to identify which symmetries fix which colorings, then verify the formula.
Intuitive explanation
This is a Burnside's lemma problem: you're counting colorings up to symmetry, not all possible colorings. The square has 8 symmetries (4 rotations and 4 reflections), and each one fixes a different subset of the 3⁴ = 81 total colorings. Rather than subtract out "equivalent" colorings by hand, Burnside's lemma says the answer is the average number of colorings fixed by each symmetry—exactly what the hint computes. Your job is to identify which symmetries fix which colorings, then verify the formula.