Orbit of Group Action

up:: Group Action

Given a Group Action of a Group $G$ upon a set $X$, we have that the orbit of a point $x\in X$ is the set of all points in the set which are reached from $x$ after being acted upon by $G$. That is,

$$Or{b}_x=\{y\in X\mid \exists g\in G:y=g\cdot x\}\subseteq X$$

Note that Group actions induce an equivalence relation over a set’s orbits, since distinct orbits are disjoint and cover the entire set $X$.

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References

• abstract algebra - Intuitive definitions of the Orbit and the Stabilizer - Mathematics Stack Exchange