proof of Cayley’s theorem

Let G be a group, and let SG be the permutation groupMathworldPlanetmath of the underlying set G. For each gG, define ρg:GG by ρg(h)=gh. Then ρg is invertible with inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ρg-1, and so is a permutationMathworldPlanetmath of the set G.

Define Φ:GSG by Φ(g)=ρg. Then Φ is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, since


And Φ is injectivePlanetmathPlanetmath, since if Φ(g)=Φ(h) then ρg=ρh, so gx=hx for all xX, and so g=h as required.

So Φ is an embedding of G into its own permutation group. If G is finite of order n, then simply numbering the elements of G gives an embedding from G to Sn.

Title proof of Cayley’s theorem
Canonical name ProofOfCayleysTheorem
Date of creation 2013-03-22 12:30:50
Last modified on 2013-03-22 12:30:50
Owner Evandar (27)
Last modified by Evandar (27)
Numerical id 5
Author Evandar (27)
Entry type Proof
Classification msc 20B99