Math 2.0

statement corollary

Size of an orbit times the size of the stabiliser

Statement

Let a finite group GG act on a set XX, and let x∈Xx \in X. Then ∣G⋅x∣⋅∣Gx∣=∣G∣|G \cdot x| \cdot |G_x| = |G|. In particular, the size of each orbit divides ∣G∣|G|.

Remarks

This follows from the orbit-stabiliser theorem and from [G:Gx]⋅∣Gx∣=∣G∣[G : G_x] \cdot |G_x| = |G|, which is Lagrange's theorem. For example, the rotation group of a cube acts transitively on the 66 faces, and the stabiliser of a face has 44 elements. So the group has order 2424.

Explanations

L1 human

Each point of the orbit is reached by exactly ∣Gx∣|G_x| group elements. So the group elements are counted by the orbit size times the stabiliser size. This is a good way to compute the order of a symmetry group.

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 24 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 homomorphismdefinitionLagrange's theorem, statement, no evidence levelLagrange's theoremstatementOrder of a group, definition, no evidence levelOrder of a groupdefinitionSubgroup, 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 cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementOrbit, definition, no evidence levelOrbitdefinitionIndex of a subgroup, definition, no evidence levelIndex of a subgroupdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionCosets have the same cardinality as the subgroup, statement, no evidence levelCosets have the same …statementStabiliser, definition, no evidence levelStabiliserdefinitionIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementTransitive action, definition, no evidence levelTransitive actiondefinitionNormal core, definition, no evidence levelNormal coredefinitionOrder of a group is the index times the order of the subgroup, statement, no evidence levelOrder of a group is t…statementAction on left cosets, example, no evidence levelAction on left cosetsexampleOrbit-stabiliser theorem, statement, no evidence levelOrbit-stabiliser theo…statementSize of an orbit times the size of the stabiliser, statement, no evidence levelSize of an orbit time…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. Lagrange's theorem statement
  5. Order of a group definition
  6. Subgroup definition
  7. Symmetric group definition
  8. Conjugate subgroup definition
  9. Group action definition
  10. Left coset definition
  11. Right coset definition
  12. Subgroup criterion statement
  13. Orbit definition
  14. Index of a subgroup definition
  15. Normal subgroup definition
  16. Cosets have the same cardinality as the subgroup statement
  17. Stabiliser definition
  18. Intersection of subgroups is a subgroup statement
  19. Transitive action definition
  20. Normal core definition
  21. Order of a group is the index times the order of the subgroup statement
  22. Action on left cosets example
  23. Orbit-stabiliser theorem statement
  24. Size of an orbit times the size of the stabiliser statement

What depends on this

12 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