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. Then $\{\tau, \gamma \}$ is a generating set of $S_3$ and $\langle\tau, \gamma \vert \tau^2=\varepsilon, \gamma^3=\varepsilon, \tau\gamma =\gamma^2\tau\rangle$ 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 of one, then the Cayley graph is a hyperbolic metric space, as it is 2 hyperbolic. Thus, $S_3$ is a hyperbolic group.