Orbit under group action

Suppose $G$ is a group with a group action on a set $S$. Then, for any point $s\in S$, the orbit of $s$ under the action of $G$, denoted $G.s$, is defined as:

$\{t\in S\mid \mathrm{\exists }g\in G,g.t=s\}$

In other words, the orbit of a point is the set of all points that can be reached from that point under the action of the group.

Because of the reversibility of the action of elements of the group, it turns out that if $t$ is in the orbit of $s$, $s$ is also in the orbit of $t$. Specifically, if $g.s=t$, then $g^{-1}.t=s$. Hence we can talk of the relation of being in the same orbit. This relation is reflexive (because of the identity element), symmetric (because of invertibility) and transitive (because of the homomorphism nature of the group action), and hence gives an equivalence relation. The equivalence relation thus partitions $S$ into a disjoint union of orbits.

Examples

Consider the conjugation action of a group $G$, defined by acting on itself via $\alpha :G\times G\to G$; $(g,h)\mapsto gh{g}^{-1}$. The orbit of an element $h\in G$ is precisely the conjugacy class of $h$.

Relation to stabilizer

Further information: fundamental theorem of group actions

The fundamental theorem of group actions, also known as the orbit-stabilizer theorem, provides a result relating the orbits and stabilizers of a group action.