## 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.