Math 2.0

Group theory and generalizations Group actions and permutation groups Theorem

Orbit-stabiliser theorem

Builds on OrbitStabiliserLeft cosetAction on left cosets

Statement

Let a group GG act on a set XX, and let x∈Xx \in X. Then the map gGx↦g⋅xgG_x \mapsto g \cdot x is a well defined bijection from the set of the left cosets of the stabiliser GxG_x to the orbit G⋅xG \cdot x. In particular, ∣G⋅x∣=[G:Gx]|G \cdot x| = [G : G_x].

Remarks

The key equivalence is g⋅x=h⋅xg \cdot x = h \cdot x if and only if g−1h∈Gxg^{-1}h \in G_x, if and only if gGx=hGxgG_x = hG_x. The bijection is compatible with the actions: it sends k⋅gGxk \cdot gG_x to k⋅(g⋅x)k \cdot (g \cdot x) for each k∈Gk \in G. So each transitive action of GG is isomorphic to the action on the left cosets of a stabiliser.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptConjugate elements, definition, no evidence levelConjugate elementsdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionSymmetric group, definition, no evidence levelSymmetric groupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionGroup action, definition, no evidence levelGroup actiondefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementNormal subgroup, definition, no evidence levelNormal subgroupdefinitionOrbit, definition, no evidence levelOrbitdefinitionStabiliser, definition, no evidence levelStabiliserdefinitionIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementNormal core, definition, no evidence levelNormal coredefinitionTransitive action, definition, no evidence levelTransitive actiondefinitionAction on left cosets, example, no evidence levelAction on left cosetsexampleOrbit-stabiliser theorem, statement, no evidence levelOrbit-stabiliser theo…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. Group concept
  2. Conjugate elements definition
  3. Group homomorphism definition
  4. Subgroup definition
  5. Symmetric group definition
  6. Conjugate subgroup definition
  7. Group action definition
  8. Left coset definition
  9. Subgroup criterion statement
  10. Normal subgroup definition
  11. Orbit definition
  12. Stabiliser definition
  13. Intersection of subgroups is a subgroup statement
  14. Normal core definition
  15. Transitive action definition
  16. Action on left cosets example
  17. Orbit-stabiliser theorem statement

Where this leads

23 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Group_action, Orbit–stabilizer theorem
About this page
Tag0E2HSCKZ6C
URIm2://0E2HSCKZ6C
Aliasesorbit-stabilizer-theorem
Kindstatement, theorem
WritingW0 stub
MSC20B05
StatusThis copy of the site has no build data. Run m2 status 0E2HSCKZ6C.
UsesOrbit, Stabiliser, Left coset, Action on left cosets

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