Group theory and generalizations Corollary
Size of an orbit times the size of the stabiliser
Builds on Orbit-stabiliser theoremLagrange's theoremOrder of a group is the index times the order of the subgroup
Statement
Let a finite group act on a set , and let . Then . In particular, the size of each orbit divides .
Remarks
This follows from the orbit-stabiliser theorem and from , which is Lagrange's theorem. For example, the rotation group of a cube acts transitively on the faces, and the stabiliser of a face has elements. So the group has order .
Reading path
To understand this page from the ground up, read these 53 pages in order. The foundations come first. The path takes the proof Proof of Lagrange's theorem.
- Element definition
- Group concept
- Left coset concept
- Proposition concept
- Set definition
- Conjugate elements definition
- Group homomorphism definition
- Order of a group definition
- Subgroup definition
- Symmetric group definition
- Conjunction definition
- Disjunction definition
- Implication definition
- Predicate definition
- Empty set object
- Family of sets definition
- Ordered pair definition
- Intersection definition
- Union definition
- Conjugate subgroup definition
- Group action definition
- Left coset definition
- Right coset definition
- Subgroup criterion statement
- Set-builder notation concept
- Existential quantifier definition
- Universal quantifier definition
- Disjoint sets definition
- Cartesian product definition
- Normal subgroup definition
- Stabiliser definition
- Subset definition
- Union of a family of sets definition
- Orbit definition
- Intersection of subgroups is a subgroup statement
- Index of a subgroup definition
- Cosets have the same cardinality as the subgroup statement
- Binary relation definition
- Partition of a set definition
- Normal core definition
- Transitive action definition
- Order of a group is the index times the order of the subgroup statement
- Reflexive relation definition
- Symmetric relation definition
- Transitive relation definition
- Action on left cosets example
- Equivalence relation definition
- Orbit-stabiliser theorem statement
- Equivalence class definition
- Equivalence classes form a partition statement
- Proof of Lagrange's theorem proof
- Lagrange's theorem statement
- Size of an orbit times the size of the stabiliser statement
Where this leads
17 pages build on this one.
- A finite group is nilpotent iff it is the product of its Sylow subgroups
- Fixed points of a p-group action
- Second Sylow theorem
- The number of Sylow subgroups is the index of the normaliser
- Abelian simple groups have prime order
- Cauchy's theorem
- The number of Sylow subgroups divides the index
- Third Sylow theorem
- Frattini argument
- First Sylow theorem
- Groups of order pq
- Burnside's lemma
- Galois group of a prime degree polynomial with two nonreal roots
- Hall's theorem on solvable groups
- Hall subgroup
- Sylow theorems
- Frattini subgroup
Further reading
- Web https://en.wikipedia.org/wiki/Group_action, Orbit–stabilizer theorem
About this page
| Tag | 0S19CWFZM2 |
|---|---|
| URI | m2://0S19CWFZM2 |
| Kind | statement, corollary |
| Writing | W0 stub |
| MSC | 20B05 |
| Status | This copy of the site has no build data. Run m2 status 0S19CWFZM2. |
| Uses | Orbit-stabiliser theorem, Lagrange's theorem, Order of a group is the index times the order of the subgroup |
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