Proof that the size of an orbit times the size of the stabiliser is the order of the group

Builds on Orbit-stabiliser theorem Order of a group is the index times the order of the subgroup Stabiliser Orbit

Proof

We combine the orbit-stabiliser theorem with the index formula.

Let $G$ be finite, acting on $X$ , and let $x \in X$ . The stabiliser $G_x$ is a subgroup of $G$ . Indeed, $e \cdot x = x$ gives $e \in G_x$ , and $a, b \in G_x$ gives $(a b^{-1}) \cdot x = a \cdot (b^{-1} \cdot (b \cdot x)) = x$ .
By the orbit-stabiliser theorem, $\abs{G \cdot x} = [G : G_x]$ .
Since $G$ is finite and $G_x$ is a subgroup, [[ $\abs{G} = [G : G_x] \cdot \abs{G_x}$ |04DC0SRN2K]].
Substituting step 2 into step 3 gives $\abs{G \cdot x} \cdot \abs{G_x} = \abs{G}$ .
Since $\abs{G_x}$ is a positive integer, step 4 shows that $\abs{G \cdot x}$ divides $\abs{G}$ . So the size of each orbit divides $\abs{G}$ .

Idea

The orbit is in bijection with the cosets of the stabiliser. The cosets all have the size of the stabiliser and fill up the group, so the two sizes multiply to $\abs{G}$ .

Reading path

To understand this page from the ground up, read these 52 pages in order. The foundations come first. The path takes the proofs Proof that an intersection of subgroups is a subgroup, Proof of the orbit-stabiliser theorem, Proof that the order of a finite group is the index times the order of a subgroup, Proof of the uniqueness of inverses, Proof that cosets have the same cardinality as the subgroup, Proof of the subgroup criterion.

Binary operation definition
Element definition
Group concept
Proposition concept
Identity element definition
Set definition
Conjugate elements definition
Group homomorphism definition
Order of a group definition
Subgroup definition
Symmetric group definition
Implication definition
Predicate definition
Inverse element definition
Ordered pair definition
Conjugate subgroup definition
Proof that an intersection of subgroups is a subgroup proof
Group action definition
Left coset definition
Right coset definition
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Proof of the uniqueness of inverses proof
Normal subgroup definition
Orbit definition
Subset definition
Stabiliser definition
Index of a subgroup definition
Uniqueness quantifier definition
Uniqueness of inverses statement
Binary relation definition
Transitive action definition
Proof of the subgroup criterion proof
Function definition
Codomain definition
Injective function definition
Subgroup criterion statement
Surjective function definition
Intersection of subgroups is a subgroup statement
Proof of the orbit-stabiliser theorem proof
Bijective function definition
Normal core definition
Equinumerous sets definition
Action on left cosets example
Cardinality definition
Orbit-stabiliser theorem statement
Proof that cosets have the same cardinality as the subgroup proof
Cosets have the same cardinality as the subgroup statement
Proof that the order of a finite group is the index times the order of a subgroup proof
Order of a group is the index times the order of the subgroup statement
Proof that the size of an orbit times the size of the stabiliser is the order of the group proof

Where this leads

44 pages build on this one.

Tag	`F78G4C4M6Z`
URI	`m2://F78G4C4M6Z`
Kind	proof
Writing	W0 stub
MSC	20B05
Proves	Size of an orbit times the size of the stabiliser
Uses	Orbit-stabiliser theorem, Order of a group is the index times the order of the subgroup, Stabiliser, Orbit
Changed	4 November 2026

Proof that the size of an orbit times the size of the stabiliser is the order of the group

Proof

Idea

Reading path

Where this leads

Further reading

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