Exercise 2151Algebra › GroupsM1Added 3 Nov 2026

Group actions and orbits

#group action#orbit#stabilizer

Statement

Let $G$ act on a set $X$ . Show that the orbits form a partition of $X$ , and state the orbit-stabilizer theorem.

Hint

Two orbits are either equal or disjoint. For the orbit-stabilizer theorem, use the bijection $G/\operatorname{Stab}(x)\to\operatorname{Orb}(x)$ , $g\mapsto gx$ .

Show solutionHide solution

Solution

Partition. Define $x\sim y$ if $y\in\operatorname{Orb}(x)$ . This is an equivalence relation: reflexive ( $x=ex$ ), symmetric ( $y=gx\Rightarrow x=g^{-1}y$ ), transitive ( $y=gx,z=hy\Rightarrow z=(hg)x$ ). So orbits are equivalence classes, hence partition $X$ .

Orbit-stabilizer. The map $G/\operatorname{Stab}(x)\to\operatorname{Orb}(x)$ given by $g\operatorname{Stab}(x)\mapsto gx$ is well defined and bijective.

Conclusion. $|\operatorname{Orb}(x)|=[G:\operatorname{Stab}(x)]$ . For $G$ finite: $|G|=|\operatorname{Stab}(x)|\cdot|\operatorname{Orb}(x)|$ .

Group actions and orbits

Statement

Solution

More problems in Groups