Math 2.0

Group theory and generalizations Group actions and permutation groups Theorem

Cayley's theorem

Builds on Action by left multiplicationSymmetric groupFaithful actionGroup isomorphism

Statement

Let GG be a group. Then GG is isomorphic to a subgroup of the symmetric group Sym⁡(G)\operatorname{Sym}(G). Explicitly, the map g↦ℓgg \mapsto \ell_g, where ℓg(x)=gx\ell_g(x) = gx, is an injective group homomorphism from GG to Sym⁡(G)\operatorname{Sym}(G).

Remarks

The map comes from the action by left multiplication, and it is injective because ℓg(e)=g\ell_g(e) = g. If ∣G∣=n|G| = n is finite, then GG is isomorphic to a subgroup of SnS_n. Often a smaller symmetric group is enough, for example S3S_3 is isomorphic to a subgroup of S3S_3 and not only of S6S_6.

History

Arthur Cayley introduced abstract groups in 1854. At that time, groups were usually groups of permutations, and the theorem shows that the abstract notion gives nothing more, up to isomorphism.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptGroup 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 theoremstatement
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. Group homomorphism definition
  3. Symmetric group definition
  4. Group isomorphism definition
  5. Kernel of a homomorphism definition
  6. Group action definition
  7. Faithful action definition
  8. Orbit definition
  9. Transitive action definition
  10. Action by left multiplication example
  11. Cayley's theorem statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cayley's_theorem, lead section
  2. Web https://en.wikipedia.org/wiki/Cayley's_theorem, History
About this page
Tag0151VYVX27
URIm2://0151VYVX27
Kindstatement, theorem
WritingW1 stated
MSC20B30
StatusThis copy of the site has no build data. Run m2 status 0151VYVX27.
UsesAction by left multiplication, Symmetric group, Faithful action, Group isomorphism
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