In which we meet the Orbit-Counting Formula
- Definition of Image may be NSFW.
Clik here to view.
for an element Image may be NSFW.
Clik here to view. of a group Image may be NSFW.
Clik here to view. acting on a set. - Theorem 65 (Orbit-Counting Formula): Let Image may be NSFW.
Clik here to view. be a finite group acting on a finite set Image may be NSFW.
Clik here to view.. Then Image may be NSFW.
Clik here to view.. We defined a set Image may be NSFW.
Clik here to view. and counted its elements in two ways. - Lemma 66: Let Image may be NSFW.
Clik here to view. be a group acting on a finite set Image may be NSFW.
Clik here to view.. Take Image may be NSFW.
Clik here to view., Image may be NSFW.
Clik here to view. with Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. conjugate. Then Image may be NSFW.
Clik here to view.. This was a quick check: we showed that for Image may be NSFW.
Clik here to view. we have Image may be NSFW.
Clik here to view. if and only if Image may be NSFW.
Clik here to view.. - Corollary 67: Let Image may be NSFW.
Clik here to view. be a finite group acting on a finite set Image may be NSFW.
Clik here to view.. Say Image may be NSFW.
Clik here to view. has Image may be NSFW.
Clik here to view. conjugacy classes, and pick a representative from each, say Image may be NSFW.
Clik here to view., …, Image may be NSFW.
Clik here to view.. Then Image may be NSFW.
Clik here to view.. 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?
You could 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 Image may be NSFW.
Clik here to view.
and Image may be NSFW.
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on Sheet 6.
What are the conjugacy classes in Image may be NSFW.
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? I just stated them in the lecture, you could check that this fits with your work on conjugacy classes in dihedral groups on Sheet 5.
Further reading
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 bring it along to the lecture for reference?
