 Orbit-stabilizer theorem

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.