Cayley graph of S3

In this entry, a Cayley graphMathworldPlanetmath of S3, the group of permutationsMathworldPlanetmath of {1,2,3}, will be investigated.

Let εS3 denote the identity elementMathworldPlanetmath, τS3 be a transpositionMathworldPlanetmath, and γS3 be a three-cycle ( Then {τ,γ} is a generating set of S3 and τ,γ|τ2=ε,γ3=ε,τγ=γ2τ is a presentationMathworldPlanetmathPlanetmath of S3. The corresponding Cayley graph Γ(S3,{τ,γ}) is:


If each of the edges is assigned a length ( of one, then the Cayley graph is a hyperbolic metric space, as it is 2 hyperbolic. Thus, S3 is a hyperbolic group.

Title Cayley graph of S3