Math 2.0

statement theorem

Orbit-stabiliser theorem

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.

Explanations

L1 human

Two group elements move xx to the same point exactly when they differ by an element that fixes xx. So the points of the orbit correspond to the cosets of the stabiliser. A big stabiliser means a small orbit, and a small stabiliser means a big orbit.

What this node needs for W1
  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 17 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • 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

What depends on this

18 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Group_action, Orbit–stabilizer theorem

What links here

Used by