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 for g_1 = h^{-1} g_2 h we have 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?

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 n = 2 and n = 3 on Sheet 6.

What are the conjugacy classes in D_{14}?  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?

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