What is group action? — Mathematics

Definition 1.8 University Mathematics — Year 3 · Chapter 1 — Group Theory

An action of $G$ on a set $X$ is a morphism $\varphi \colon G \to \mathfrak{S}(X)$ into the group of bijections of $X$ ; one writes $g \cdot x$ for $\varphi(g)(x)$ . Equivalently: a map $G \times X \to X$ with $e \cdot x = x$ and $g \cdot (h \cdot x) = (gh) \cdot x$ . The orbit of $x$ is $\mathcal O_x = \{g \cdot x : g \in G\}$ , its stabilizer is the subgroup $G_x = \{g : g\cdot x = x\}$ , and $X^G = \{x : \forall g,\ g \cdot x = x\}$ is the set of fixed points. The action is transitive if there is exactly one orbit, faithful if $\varphi$ is injective, free if all stabilizers are trivial.

Examples

Example 1.9

Five actions run all of finite group theory:

$G$ on itself by left translation $g \cdot x = gx$ : free and transitive.
$G$ on itself by conjugation $g \cdot x = gxg^{-1}$ : orbits are the conjugacy classes, stabilizers the centralizers $Z_G(x) = \{g : gx = xg\}$ , fixed points the center $Z(G)$ .
$G$ on the coset space $G/H$ by $g \cdot xH = gxH$ : transitive, with stabilizer of the coset $H$ equal to $H$ . Every transitive action is of this form (Exercise 1.8).
$G$ on its set of subgroups by conjugation: the stabilizer of $H$ is the normalizer $N_G(H) = \{g : gHg^{-1} = H\}$ , the largest subgroup of $G$ in which $H$ is normal.
$S_n$ on $\intint{1}{n}$ : the mother of all examples.

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