Groups and Group Actions: Lecture 15

In which we meet the Orbit-Counting Formula

  • Definition of \mathrm{fix}(g) for an element g of a group G acting on a set.
  • Theorem 65 (Orbit-Counting Formula): Let G be a finite group acting on a finite set X.  Then \displaystyle\# \mathrm{orbits} = \frac{1}{|G|} \sum_{g \in G} |\mathrm{fix}(g)|.  We defined a set S = \{ (g,x) \in G \times X : g \cdot x = x\} and counted its elements in two ways.
  • Lemma 66: Let G be a group acting on a finite set X.  Take g_1, g_2 \in G with g_1 and g_2 conjugate.  Then | \mathrm{fix}(g_1)| = |\mathrm{fix}(g_2)|.  This was a quick check: we showed that if g_1 = h^{-1} g_2 h then x \in \mathrm{fix}(g_1) if and only if h \cdot x \in \mathrm{fix}(g_2).
  • Corollary 67: Let G be a finite group acting on a finite set X.  Say G has k conjugacy classes, and pick a representative from each, say g_1, …, g_k.  Then \displaystyle \# \mathrm{orbits} = \frac{1}{|G|} \sum_{i=1}^k |\mathrm{fix}(g_i)| |\mathrm{ccl}_G(g_i)|.  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 n = 2 and n = 3 on Sheet 6.

What are the conjugacy classes in D_{14}?  I just stated them in the lecture, 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 a tetrahedron and bring them along to the lecture for reference?

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