= Orbit
{disambiguate=group theory}
{title2=$Gx=\operatorname{Orb}_G(x)$}
{wiki}
= Orbit of a group action
{synonym}
= Group orbit
{synonym}
= Orbits of a group action
{synonym}
For a <group action> of a <group> $G$ on a <set> $X$, the <orbit of a group action> of $x\in X$ is the <subset> of all positions reachable from $x$:
$$
Gx=G\cdot x=\{g\cdot x:g\in G\}\subseteq X.
$$
Two points belong to the same <group orbit> exactly when one can be moved to the other by a <group> element. This is an <equivalence relation>: the <identity element> gives $x\sim x$, the <inverse element> of a transformation reverses a move, and a product combines two moves. Hence the <group orbits> form a <partition of a set> of $X$ into <equivalence classes>. A <transitive group action> on a nonempty <set> has just one <group orbit>. 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, lemma 0.5.1.3, for the orbit partition].
For the <square> and the eight-element <dihedral group> $D_4$ defined under <group action>, direct enumeration gives the following separate <group orbits>:
* On <vertices of a polygon>, $D_4\cdot A=\{A,B,C,D\}$ has size four.
* On unoriented <sides of a polygon>, $D_4\cdot AB=\{AB,BC,CD,DA\}$ has size four; $AB$ and $BA$ denote the same <side of a polygon>.
* On unoriented <diagonals of a polygon>, $D_4\cdot AC=\{AC,BD\}$ has size two.
* On the <singleton set> containing the <centroid> $O=(0,0)$, $D_4\cdot O=\{O\}$ has size one, so $O$ is a <fixed point of a group action>.
These counts depend on the chosen <group action>, not just the <group>. An <orbit of a group action> describes all configurations equivalent under the permitted transformations. For <ordered pairs> under the <symmetric group>, equality of the two entries is preserved, separating the two <group orbits> in <equality-pattern orbits of ordered pairs>; the <orbit-stabilizer theorem> computes their sizes when the <symmetric group> is $S_3$.
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