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# 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. 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 of one, then the Cayley graph is a hyperbolic metric space, as it is 2 hyperbolic. Thus, $S_{3}$ is a hyperbolic group.

Related:

SymmetricGroup, Presentationgroup

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Reference

Type of Math Object:

Example

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## Mathematics Subject Classification

20F06*no label found*20B30

*no label found*05C25

*no label found*

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