Math 2.0

Group theory and generalizations Proposition

Stabiliser sizes over an orbit add up to the order of the group

Builds on GroupGroup actionOrbitStabiliserOrbit-stabiliser theorem

Statement

Let a group GG act on a finite set XX, and let x∈Xx \in X. Write G⋅xG \cdot x for the orbit of xx, and GyG_y for the stabiliser of a point yy. Then the sizes of the stabilisers of all points of the orbit add up to the order of the group:

∑y∈G⋅x∣Gy∣=∣G∣.\sum_{y \in G \cdot x} |G_y| = |G|.

Remarks

The stabilisers of the points of one orbit are conjugate, so they all have the same size. By the orbit-stabiliser theorem, this common size is ∣G∣/∣G⋅x∣|G| / |G \cdot x|. The orbit has ∣G⋅x∣|G \cdot x| terms, so the sum is ∣G∣|G|.

For example, let G=ℤ/4ℤG = \mathbb{Z}/4\mathbb{Z} rotate the corners of a square. The orbit of a corner has 44 points, each with trivial stabiliser, so the sum is 4=∣G∣4 = |G|.

Reading path

To understand this page from the ground up, read these 28 pages in order. The foundations come first. The path takes the proofs Proof that an intersection of subgroups is a subgroup, Proof of the uniqueness of inverses, Proof of the subgroup criterion.

Show the path as a graph
Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptGroup action, concept, no evidence levelGroup actionconceptOrbit, concept, no evidence levelOrbitconceptStabiliser, concept, no evidence levelStabiliserconceptIdentity element, definition, no evidence levelIdentity elementdefinitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionSymmetric group, definition, no evidence levelSymmetric groupdefinitionInverse element, definition, no evidence levelInverse elementdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionProof that an intersection of subgroups is a subgroup, proof, no evidence levelProof that an interse…proofGroup action, definition, no evidence levelGroup actiondefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionProof of the uniqueness of inverses, proof, no evidence levelProof of the uniquene…proofNormal subgroup, definition, no evidence levelNormal subgroupdefinitionOrbit, definition, no evidence levelOrbitdefinitionStabiliser, definition, no evidence levelStabiliserdefinitionUniqueness of inverses, statement, no evidence levelUniqueness of inversesstatementTransitive action, definition, no evidence levelTransitive actiondefinitionProof of the subgroup criterion, proof, no evidence levelProof of the subgroup…proofSubgroup criterion, statement, no evidence levelSubgroup criterionstatementIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementNormal core, definition, no evidence levelNormal coredefinitionAction on left cosets, example, no evidence levelAction on left cosetsexampleOrbit-stabiliser theorem, statement, no evidence levelOrbit-stabiliser theo…statementStabiliser sizes over an orbit add up to the order of the group, statement, no evidence levelStabiliser sizes over…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Binary operation definition
  2. Group concept
  3. Group action concept
  4. Orbit concept
  5. Stabiliser concept
  6. Identity element definition
  7. Conjugate elements definition
  8. Group homomorphism definition
  9. Subgroup definition
  10. Symmetric group definition
  11. Inverse element definition
  12. Conjugate subgroup definition
  13. Proof that an intersection of subgroups is a subgroup proof
  14. Group action definition
  15. Left coset definition
  16. Proof of the uniqueness of inverses proof
  17. Normal subgroup definition
  18. Orbit definition
  19. Stabiliser definition
  20. Uniqueness of inverses statement
  21. Transitive action definition
  22. Proof of the subgroup criterion proof
  23. Subgroup criterion statement
  24. Intersection of subgroups is a subgroup statement
  25. Normal core definition
  26. Action on left cosets example
  27. Orbit-stabiliser theorem statement
  28. Stabiliser sizes over an orbit add up to the order of the group statement

Further reading

  1. Web https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_polya_burnside.ptx#L9, source/s_polya_burnside.ptx:9
About this page
TagCQTDS8B6R7
URIm2://CQTDS8B6R7
Kindstatement, proposition
WritingW1 stated
MSC20B05, 05E18
StatusThis copy of the site has no build data. Run m2 status CQTDS8B6R7.
UsesGroup, Group action, Orbit, Stabiliser, Orbit-stabiliser theorem
Changed4 November 2026

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