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 $\epsilon \in S_3$ 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 $⟨\tau ,\gamma |{\tau }^2=\epsilon ,{\gamma }^3=\epsilon ,\tau \gamma ={\gamma }^2\tau ⟩$ is a presentation of $S_3$. The corresponding Cayley graph $\mathrm{\Gamma }(S_3,\{\tau ,\gamma \})$ 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 $S_3$