Suborbits and double-cosets

Throughout we will assume that $G$ acts transitively on $\Omega$. The idea motivating this section is that since each orbit is conjugate, what freedom remains when we pick some fixed $\alpha$ and consider $G_\alpha$ (The stabilizer of $\alpha$)?

Since $G_\alpha$ is not just a subset of $G$ but in fact a subgroup, the action of $G$ on $\Omega$ induces an action of $G_\alpha$ on $\Omega$.

Definition The orbits of $G_\alpha$ on $\Omega$ (by this induced action) are called suborbits, their sizes are called subdegrees and the rank is how many there are.


Recall the orbit/stabilizer theorem:

• $\alpha G = \{ \alpha g \in \Omega | g \in G\}$
• $G_\alpha = \{g \in G | \alpha g = \alpha \}$
• Since $\alpha g = \alpha g'$ iff $gg'^{-1} \in G_{\alpha}$ iff $G_\alpha g = G_\alpha g'$ the bijection $\alpha g \mapsto G_\alpha g$ is well defined, it also respects the group action.

By transitivity every element $\beta \in \Omega$ can be written as $\alpha g$ for some $g$, in fact $\{\beta\} = \alpha G_\alpha g$ so elements of $\Omega$ are in bijection with the right cosets $\{G_\alpha g| g \in G\}$.

Suborbits of $G_\alpha$ are the orbits $\beta G_\alpha$ which are thus in bijection with the double cosets $G_\alpha g G_\alpha$.

We call these $(G_\alpha,G_\alpha)$-double cosets and they partition the group. The size of a double coset divided by $|G_\alpha|$ gives the subdegree (similar to Lagrange's theorem).

Lemma The rank of the action of $G$ is $\frac{1}{|G|} \sum_{g \in G} |\operatorname{fix}(g)|^2$.
proof: Apply Burnside's lemma to the action of $G_\alpha$ on $\Omega$ to get $$\frac{|\Omega|}{|G|}\sum_{g \in G_\alpha} |\operatorname{fix}(g)|$$ since $|G_\alpha| = \frac{|G|}{|\Omega|}$ now sum over all $\alpha$ to get $$\frac{1}{|G|}\sum_{\alpha \in \Omega} \sum_{g \in G_\alpha} |\operatorname{fix}(g)| = \frac{1}{|G|}\sum_{g \in G} \sum_{\alpha \in \operatorname{fix}(g)} |\operatorname{fix}(g)|.$$

Definition Let $\alpha,\beta \in \Omega$. The 2-point stabilizer $G_{\alpha,\beta}$ is $G_\alpha \cap G_\beta$.
Definition The pointwise stabilizer of a set of points $\Gamma \subseteq \Omega$, $G_{(\Gamma)}$ is $bigcap_{\gamma \in \Gamma} G_\gamma$.
Definition The setwise stabilizer $G_{\Gamma} = \{ g \in G | \Gamma_g = \Gamma \}$.

Lemma Given $\beta \in \Omega$ the subdegree corresponding to $\beta$ is $|G_\alpha : G_{\alpha,\beta}|$.
proof: In the action of $G_\alpha$ on the suborbit $\beta G_\alpha$ the stab. of $\beta$ is $G_\alpha \cap G_\beta$. The result follows from orb/stab theorem.

Definition If $G_\alpha = 1$ the action is regular and has rank $|\Omega|$