Notes - Groups TT23, Counting orbits
Flashcards
Can you define both $\text{Stab}(x)$ and $\text{fix}(g)$ (assuming $G$ acts on $X$)?
\[\begin{aligned}
\text{Stab}(x) &= \\{g \in G : g \cdot x = x \\} \\\\
\text{fix}(g) &= \\{x \in X : g \cdot x = x \\}
\end{aligned}\]
Can you state the orbit counting formula (i.e. Burnsides lemma) and what it says informally?
\[\text{number of orbits} = \frac{1}{|G|} \sum_{g \in G} |\text{fix}(g)|\]
the number of orbits of $G$ is the average size of the $\text{fix}$.
What’s the general “table” that you make when using the orbit counting formula?
- $g$
- Size of its conjugacy class
- Size of its fix
Quickly prove the orbit-counting formula by first giving a set and then giving two different ways to count up the elements in that set.
Consider $A = \{(g, s) : g \cdot s = s\} \subseteq G \times S$. Then
\[\begin{aligned} |A| &= \sum_{g \in G} | \text{fix}(g)| \\\\ |A| &= \sum_{s \in S} | \text{Stab}(s) | \\\\ &= \sum^N_{i=1} \sum_{s \in O_i} \text{Stab}(s) \\\\ &= \sum^N_{i=1} \sum_{s \in O_i} \frac{|G|}{|O_i|} \\\\ &= \sum^N_{i=1} |G| \\\\ &= N |G| \end{aligned}\]which overall implies
\[N = \frac{1}{|G|} \sum_{g \in G} |\text{fix}(g)|\]Proofs
Prove that if $G$ is a finite group acting on a set $S$, then
\[\# \text{orbits} = \frac{1}{|G|} \sum_{g \in G} |\text{fix}(g)|\]