Cayley graph of $S_{3}$

In this entry, a Cayley graph of $S_{3}$, the group of permutations of $\{1,2,3\}$, will be investigated.

Let $\varepsilon\in S_{3}$ denote the identity element, $\tau\in S_{3}$ be a transposition, and $\gamma\in S_{3}$ be a three-cycle (http://planetmath.org/SymmetricGroup). Then $\{\tau,\gamma\}$ is a generating set of $S_{3}$ and $\langle\tau,\gamma|\tau^{2}=\varepsilon,\gamma^{3}=\varepsilon,\tau\gamma=% \gamma^{2}\tau\rangle$ is a presentation of $S_{3}$. The corresponding Cayley graph $\Gamma(S_{3},\{\tau,\gamma\})$ is:

If each of the edges is assigned a length (http://planetmath.org/BasicLength) of one, then the Cayley graph is a hyperbolic metric space, as it is 2 hyperbolic. Thus, $S_{3}$ is a hyperbolic group.

 Title Cayley graph of $S_{3}$ \metatable