Math 2.0

Group theory and generalizations Definition

Orbit

Builds on Group action

Definition

Let a group GG act on a set XX, and let x∈Xx \in X. The orbit of xx is the set G⋅x={g⋅x:g∈G}G \cdot x = \{g \cdot x : g \in G\}. The set of all orbits is written X/GX/G.

Remarks

The point xx lies in its own orbit, because e⋅x=xe \cdot x = x. The orbits partition the set XX. For the action of the rotation group of a square on its corners, there is one orbit, which contains all four corners.

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Group, concept, no evidence levelGroupconceptGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSymmetric group, definition, no evidence levelSymmetric groupdefinitionGroup action, definition, no evidence levelGroup actiondefinitionOrbit, definition, no evidence levelOrbitdefinition
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  1. Group concept
  2. Group homomorphism definition
  3. Symmetric group definition
  4. Group action definition
  5. Orbit definition

Where this leads

34 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Group_action, Orbits and stabilizers
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