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[parent] Cayley graph of $S_3$ (Example)

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:


\begin{pspicture}(-7,-7)(7,7) \pscircle(0,0){6} \rput[a](0,6){.} \rput[l](-6.05,... ...5) \psline(0.866,0.5)(1.007,0.641) \psline(0.866,0.5)(0.666,0.5) \end{pspicture}

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.



"Cayley graph of $S_3$" is owned by Wkbj79.
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See Also: symmetric group, presentation of a group


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Cross-references: hyperbolic group, hyperbolic metric space, edges, presentation, generating set, transposition, identity element, permutations, group, Cayley graph
There is 1 reference to this entry.

This is version 19 of Cayley graph of $S_3$, born on 2007-06-03, modified 2008-02-12.
Object id is 9513, canonical name is CayleyGraphOfS_3.
Accessed 722 times total.

Classification:
AMS MSC: 05C25 (Combinatorics :: Graph theory :: Graphs and groups)
 20B30 (Group theory and generalizations :: Permutation groups :: Symmetric groups)
 20F06 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Cancellation theory; application of van Kampen diagrams)
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