Permutation groups: actions, orbit and stabilizer

Notation backwards notation: $(1\,2\,3)(1\,3\,2\,4) = (1\,4)$

Notation $(2\,4)^{(1\,2\,4\,5\,3)}=(2\,5)$, in general $\sigma^{\pi}$ sends $i \pi \mapsto i \sigma \pi$ so you can compute these by hand.

 Definition A permutation group is a finite set $\Omega$ and a group of permutations (that is, bijections $\Omega \to \Omega$). We'll write $S_{\Omega}$ for the group of all permutations on a set. The degree of a permutation group is the cardinality $|\Omega|$.

Notation Let $\alpha \in \Omega$ then $\alpha g$ for the image of $\alpha$ through $g$. Group homomorphisms are written after elements too.

Definition If $G$ acts on two sets $\Omega$ and $\Omega'$ then the actions are equivalent if there is a bijection $\theta$ between them such that $$\forall g \in G, \forall \alpha \in \Omega,\,(\alpha \theta) g = (\alpha g) \theta.$$

Definition We write $\alpha G = \{\alpha g | g \in G \}$ for the orbit of $G$ containing $\alpha$.

Definition We write $G_{\alpha} = \{g \in G | \alpha g = \alpha \}$ for the stabilizer of $\alpha$.

Definition A group action is transitive if $\Omega$ is a single orbit.

Definition The kernel of the action is $G_{(\Omega)} = \{g \in G| \forall \alpha \in \Omega, \alpha g = \alpha \}$. Clearly $G \to S_{(\Omega)}$ defined by $g \mapsto (\alpha \mapsto \alpha g)$ is a homomorphism with kernel $G_{(\Omega)}$, this $G/G_{(\Omega)}$ can be identified with its image in $S_{\Omega}$ giving a permutation group $(G/G_{(\Omega)},\Omega)$.

Definition An action is said to be faithful if the kernel is trivial.

Definition The core of $H$ in $G$ is $H_G = \bigcap_{x \in G}H^x$, this is the largest normal subgroup of $G$ contained in $H$.

Examples
(i) If $(G,\Omega)$ is a permutation gp and $G$ acts on $\Omega$ faithfully this is called the natural action.
(ii) If $H \le G$ we have an action of $G$ on the set $(G:H)$ of right cosets of $H$ in $G$ by $(Hx)g = H(xg)$. This is called a coset action and if $H=1$ it is the regular action. The kernel of the regular action is the core: We have $g \in G_{(\Omega)}$ iff $\forall x, Hxg = Hx$ iff $\forall x, xgx^{-1} \in H$ iff $\forall x, g \in H^x$.
(iii) The action of $G$ on $(G:H)$ given by $Hx \in \Omega$ is clearly transitive. The stabilizer of $H$ is $H$ while that of $Hx$ is $\{g \in G | Hxg = Hx \} = H^x$.

Lemma If $G$ acts on $\Omega$, given $g \in G$ and $\alpha \in \Omega$ we have $G_{\alpha g} = G_{\alpha}^g$.
proof: $x \in G_{\alpha g}$ iff $\alpha g x = \alpha g$ iff $\alpha g x g^{-1} = \alpha$ iff  $x \in G_{\alpha}^g$.

Theorem (orbit stabilizer) If $G$ acts on $\Omega$ and $\alpha \in \Omega$ then the actions of $G$ on $\alpha G$ and $(G:G_\alpha)$ are equivalent.
proof: given $g,h \in G$, $\alpha g = \alpha h$ iff $gh^{-1} \in G_{\alpha}$ iff $G_{\alpha} g = G_{\alpha} h$. Thus $\theta = \alpha g \mapsto G_{\alpha} g$ is a well defined bijection, and it respects the action since $((\alpha g) \theta)x = G_\alpha g x = ((\alpha g ) x) \theta$.

Corollary Any transitive action is equivalent to a coset action.

Corollary $$|G| = |G_\alpha| |\alpha_G|.$$ (because $|\alpha G| = |(G:G_\alpha)|$ by the theorem and $|G| = |G_\alpha||(G:G_\alpha)|$ is a triviality just write it down).