# 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 $\u27e8\tau ,\gamma |{\tau}^{2}=\epsilon ,{\gamma}^{3}=\epsilon ,\tau \gamma ={\gamma}^{2}\tau \u27e9$ 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 ${S}_{3}$ |

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