## 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.