Math 2.0

Group theory and generalizations Definition

Permutation group

Builds on PermutationSymmetric groupSubgroupGroup actionFaithful actionCayley's theoremDihedral group

Definition

Let Ω\Omega be a set, and let Sym⁡(Ω)\operatorname{Sym}(\Omega) be the symmetric group of Ω\Omega: the group of all bijections Ω→Ω\Omega \to \Omega under composition. A permutation group on Ω\Omega is a subgroup GG of Sym⁡(Ω)\operatorname{Sym}(\Omega). The degree of GG is the cardinality ∣Ω∣\abs{\Omega}.

Remarks

Example: label the corners of a square 1,2,3,41, 2, 3, 4 in cyclic order. The eight symmetries of the square give the set G={e,(1 2 3 4),(1 3)(2 4),(1 4 3 2),(1 3),(2 4),(1 2)(3 4),(1 4)(2 3)}G = \set{e, (1\,2\,3\,4), (1\,3)(2\,4), (1\,4\,3\,2), (1\,3), (2\,4), (1\,2)(3\,4), (1\,4)(2\,3)}. It is closed under composition and inverses, because the composite of two symmetries of the square is again one. So GG is a permutation group of degree 44, isomorphic to the dihedral group of order 88.

Non-example: the set {e,(1 2),(1 3),(2 3)}\set{e, (1\,2), (1\,3), (2\,3)} in S3S_3. It contains (1 2)(1\,2) and (1 3)(1\,3) but not their product, which is a 33-cycle, so it is not a subgroup.

A group action of a group HH on Ω\Omega is the same as a homomorphism H→Sym⁡(Ω)H \to \operatorname{Sym}(\Omega), and its image is a permutation group on Ω\Omega. The image is isomorphic to HH exactly when the action is faithful. By Cayley's theorem, the action of a group HH on itself by left multiplication is faithful, so HH is isomorphic to a permutation group on HH.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Dihedral group, concept, no evidence levelDihedral groupconceptFaithful action, concept, no evidence levelFaithful actionconceptGroup, concept, no evidence levelGroupconceptGroup action, concept, no evidence levelGroup actionconceptPermutation, concept, no evidence levelPermutationconceptSubgroup, concept, no evidence levelSubgroupconceptSymmetric group, concept, no evidence levelSymmetric groupconceptGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSymmetric group, definition, no evidence levelSymmetric groupdefinitionGroup isomorphism, definition, no evidence levelGroup isomorphismdefinitionKernel of a homomorphism, definition, no evidence levelKernel of a homomorph…definitionGroup action, definition, no evidence levelGroup actiondefinitionFaithful action, definition, no evidence levelFaithful actiondefinitionOrbit, definition, no evidence levelOrbitdefinitionTransitive action, definition, no evidence levelTransitive actiondefinitionAction by left multiplication, example, no evidence levelAction by left multip…exampleCayley's theorem, statement, no evidence levelCayley's theoremstatementPermutation group, definition, no evidence levelPermutation groupdefinition
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. Dihedral group concept
  2. Faithful action concept
  3. Group concept
  4. Group action concept
  5. Permutation concept
  6. Subgroup concept
  7. Symmetric group concept
  8. Group homomorphism definition
  9. Symmetric group definition
  10. Group isomorphism definition
  11. Kernel of a homomorphism definition
  12. Group action definition
  13. Faithful action definition
  14. Orbit definition
  15. Transitive action definition
  16. Action by left multiplication example
  17. Cayley's theorem statement
  18. Permutation group definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Permutation_group
About this page
Tag9F0CPB14FQ
URIm2://9F0CPB14FQ
Kinddefinition
WritingW0 stub
MSC20B05, 20B07
DefinesPermutation group
UsesPermutation, Symmetric group, Subgroup, Group action, Faithful action, Cayley's theorem, Dihedral group
Changed2 November 2026

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