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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 $x$: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 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:

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 $S_3$.

For a group action of any group $G$ on a set $X$, the cardinality of the orbit of a group action through $x$ is the index of a subgroup of its stabilizer subgroup:In particular, if $G$ is a finite group,For a fixed finite group, a larger stabilizer subgroup means a smaller group orbit: more transformations repeat the same image of $x$. The set $X$ itself need not be finite.

The underlying bijection uses left cosets, without requiring $G_x$ to be a normal subgroup:To check it briefly, replacing $g$ by $gh$ with $h\in G_x$ preserves $g\cdot x$, so $\Phi$ is well-defined. Every member of the group orbit has this form. Finally, $g\cdot x=k\cdot x$ implies $k^{-1}g\in G_x$, hence $g{G}_x=k{G}_x$, proving the bijection. This is the construction in MIT's group-action notes, lemma 23.5. For infinite groups, the bijection and the cardinality statement $|Gx|=[G:G_x]$ still hold; division of infinite cardinalities does not supply the finite formula.

For the eight-element dihedral group $D_4$ acting on the square described under group action, moving a marked vertex of a polygon gives $8=4\cdot 2$. Moving a marked side of a polygon also gives $8=4\cdot 2$; moving one unoriented diagonal of a polygon gives $8=2\cdot 4$; moving the centroid gives $8=1\cdot 8$. These are four different group actions, each with one group orbit. In the first group action, the four left cosets of $H=G_A=\{e,s\}$ areThey correspond respectively to $A,B,C,D$. Counting all eight transformations would count each position of the marked vertex of a polygon twice.

For a second computed example, let the symmetric group $S_3$ act on the Cartesian product $X=\{1,2,3{\}}^2$ by $\sigma \cdot (i,j)=(\sigma (i),\sigma (j))$. There are two orbits of a group action:andThe first stabilizer subgroup can exchange $2$ and $3$ while fixing $1$; the second must fix both $1$ and $2$, so also fixes $3$. The orbit-stabilizer theorem gives $6=3\cdot 2$ and $6=6\cdot 1$, and the two group orbit sizes account for all nine ordered pairs. The general reason for exactly these two group orbits is equality-pattern orbits of ordered pairs.

A permutation preserves whether the two coordinates are equal. Consequently the diagonal and its complement are invariant under this group action. Any $(i,i)$ can be carried to any $(j,j)$ by a permutation sending $i$ to $j$. Likewise, for $i\ne j$ and $k\ne l$, the assignment $i\mapsto k$, $j\mapsto l$ extends to a permutation of the remaining letters.

For $n\ge 2$ the equality-pattern orbits of ordered pairs are the two orbits of a group action are thereforewith sizes $n$ and $n(n-1)$. For $n=1$ only the diagonal orbit of a group action exists.