Math 2.0

Group theory and generalizations Proposition

Stabilisers of points in the same orbit are conjugate

Builds on StabiliserOrbitConjugate subgroup

Statement

Let a group GG act on a set XX, let x∈Xx \in X, and let g∈Gg \in G. Then the stabiliser of g⋅xg \cdot x is Gg⋅x=gGxg−1G_{g \cdot x} = g G_x g^{-1}.

Remarks

An element hh fixes g⋅xg \cdot x if and only if g−1hgg^{-1}hg fixes xx. So the stabilisers of the points of one orbit are conjugate subgroups. In particular, they all have the same order.

Reading path

To understand this page from the ground up, read these 10 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 actiondefinitionOrbit, definition, no evidence levelOrbitdefinitionStabiliser, definition, no evidence levelStabiliserdefinitionStabilisers of points in the same orbit are conjugate, statement, no evidence levelStabilisers of points…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. Orbit definition
  9. Stabiliser definition
  10. Stabilisers of points in the same orbit are conjugate statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Group_action, Fixed points and stabilizer subgroups
About this page
Tag05BM304NRX
URIm2://05BM304NRX
Kindstatement, proposition
WritingW0 stub
MSC20B05
StatusThis copy of the site has no build data. Run m2 status 05BM304NRX.
UsesStabiliser, Orbit, Conjugate subgroup

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