In which we meet the Orbit-Counting Formula
- Definition of for an element of a group acting on a set.
- Theorem 65 (Orbit-Counting Formula): Let be a finite group acting on a finite set . Then . We defined a set and counted its elements in two ways.
- Lemma 66: Let be a group acting on a finite set . Take , with and conjugate. Then . This was a quick check: we showed that if then if and only if .
- Corollary 67: Let be a finite group acting on a finite set . Say has conjugacy classes, and pick a representative from each, say , …, . Then . This was immediate from Theorem 65 and Lemma 66.
Understanding today’s lecture
Are you happy about why the Orbit-Counting Formula in Corollary 67 follows from the version of the Orbit-Counting Formula in Theorem 65?
Check that the answer we obtained in the first example (about colourings of the edges of an equilateral triangle) matches what you obtained directly for and on Sheet 6.
What are the conjugacy classes in ? I just stated them in the lecture, check that this fits with your work on conjugacy classes in dihedral groups on Sheet 5.
The Orbit-Counting Formula has many names. It is sometimes known as Burnside‘s Lemma, although was not first proved by Burnside. It is relevant in Representation Theory. Here’s a page with some more applications of the result.
Preparation for Lecture 16
Sheet 7 Q5 is excellent preparation for Lecture 16.
What is the group of rotational symmetries of a cube? Or of a tetrahedron?
Why not build a cube and a tetrahedron and bring them along to the lecture for reference?