= Group action
{title2=$G\times X\longrightarrow X$}
{wiki}
= Permutation action
{synonym}
A left <group action> of a <group> $G$ on a <set> $X$ is a <function> $(g,x)\mapsto g\cdot x$ satisfying
$$
e\cdot x=x,\qquad (gh)\cdot x=g\cdot(h\cdot x)\qquad(g,h\in G,\ x\in X),
$$
where $e$ is the <identity element> of $G$. The second rule means that $h$ moves the point first and $g$ moves it next. These rules let <group> multiplication describe consistent rearrangements of a <set>; see https://people.math.harvard.edu/~nate/teaching/UPenn/2007/fall/math_371/lectures/week_1/lecture_1/lecture_1.pdf[Ackerman's algebra notes, definition 0.4.1.1].
For each $g$, the <function> $x\mapsto g\cdot x$ is a <bijection>: its inverse is $x\mapsto g^{-1}\cdot x$, using the <inverse element> of $g$. Equivalently, a <group action> is a <group homomorphism> $\rho:G\to\operatorname{Sym}(X)$ into the <symmetric group> of all <permutations> of $X$, with $\rho(g)(x)=g\cdot x$ and $\rho(gh)=\rho(g)\circ\rho(h)$. See https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/96221ce2d5b1077b3e5d3c5e08c88afe_MIT18_703S13_pra_l_23.pdf[MIT's group-action notes, definition 23.1].
For a concrete <group action>, put the <vertices of a polygon> of a <square> at $A=(1,1)$, $B=(-1,1)$, $C=(-1,-1)$ and $D=(1,-1)$. Let $r(x,y)=(-y,x)$ be a counterclockwise quarter-turn <rotation> and let $s(x,y)=(y,x)$ be a <reflection>. Their eight <permutations> of the <vertices of a polygon> form the <dihedral group>
$$
D_4=\{e,r,r^2,r^3,s,rs,r^2s,r^3s\},\qquad r^4=s^2=e,\quad srs=r^{-1}.
$$
Here $D_4$ means the <symmetry group> of a four-sided <polygon>, with <group order> eight; the alternative convention calls this same <group> $D_8$. Products apply the rightmost operation first. For example, $s\cdot B=D$ and $(rs)\cdot B=r\cdot D=A$. The same <group> also has separate <group actions> on the <sides of a polygon>, the <diagonals of a polygon>, and the <singleton set> containing the <centroid> of the <square>. Specifying $X$ matters: changing what is moved changes the <orbits of a group action> and the <stabilizer subgroups>.
\Image[/square-symmetry-orbits.png]
{title=Eight <square> <symmetry group> elements grouped by the image of one <vertex of a polygon>}
{description=The <group action> sends the marked <vertex of a polygon> $A$ to four possible <vertices of a polygon>. Each column contains two <group> elements with the same image, giving a <group orbit> of size four and a <stabilizer subgroup> of <group order> two. The labels show where every original <vertex of a polygon> goes.}
A <group action> need not be a <faithful group action>. For example, the <cyclic group> $C_4=\mathbb Z/4\mathbb Z$ acts on $X=\{0,1\}$ by $k\cdot x=x+k\pmod2$, using <modular arithmetic>. Only the <parity> of $k$ matters: $0,2$ fix both points and $1,3$ exchange them. The associated <group homomorphism> $C_4\to\operatorname{Sym}(X)$ has <group kernel> $\{0,2\}$, of <group order> two. Either point has a <group orbit> of size two and a <stabilizer subgroup> of <group order> two, so the <orbit-stabilizer theorem> gives $4=2\cdot2$.
The useful questions are which positions can be exchanged, and which transformations survive after imposing a constraint such as keeping one marked point fixed. An <orbit of a group action> answers the first question; a <stabilizer subgroup> answers the second. This turns a list of transformations into a count of distinct configurations, and explains why several transformations can produce the same configuration.
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