The Cayley graph of a (small) group $G$ is a directed graph encoding the abstract structure of $G$.

The CayleyGraph( G ) command returns the Cayley graph of the group G, in which the elements of G have been labeled by the integers 1..n, where n is the order of G.

You can specify a particular ordering for the elements of the group by passing the optional argument elements = E, where E is an explicit list of the members of G.

By default, the set of generators used by the CayleyGraph command is the set that is returned by Generators( G ). To specify a different set of generators, use the generators=S option, where S is a set of generators of the group G.

Note that computing the Cayley graph of a group requires that all the group elements be computed explicitly, so the command should only be used for groups of modest size.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{SymmetricGroup}\left(4\right)$

$G≔{\mathbf{S}}_4$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{CayleyGraph}\left(G\right)\,\mathrm{style}=\mathrm{spring}\right)$

$G≔\mathrm{DihedralGroup}\left(7\right)$

$G≔{\mathbf{D}}_7$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{CayleyGraph}\left(G\right)\,\mathrm{style}=\mathrm{spring}\right)$

$G≔\mathrm{PGL}\left(2\,3\right)\:$

$\mathrm{Generators}\left(G\right)$

$\left[\left(3\,4\right)\,\left(1\,2\,4\right)\right]$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{CayleyGraph}\left(G\right)\,'\mathrm{style}'='\mathrm{spring}'\right)$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{CayleyGraph}\left(G\,'\mathrm{generators}'=\left[\mathrm{Perm}\left(\left[\left[1\,2\right]\,\left[3\,4\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[3\,4\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,2\,4\right]\right]\right)\right]\right)\,'\mathrm{style}'='\mathrm{spring}'\right)$

$G≔\mathrm{PSL}\left(3\,2\right)\:$

$\mathrm{Generators}\left(G\right)$

$\left[\left(4\,6\right)\left(5\,7\right)\,\left(1\,2\,4\right)\left(3\,6\,5\right)\right]$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{CayleyGraph}\left(G\right)\,'\mathrm{style}'='\mathrm{spring}'\right)$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{CayleyGraph}\left(G\,'\mathrm{generators}'=\left[\mathrm{Perm}\left(\left[\left[1\,6\,2\,7\,4\,5\,3\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[4\,6\right]\,\left[5\,7\right]\right]\right)\right]\right)\,'\mathrm{style}'='\mathrm{spring}'\right)$

"Cayley graph", Wikipedia. http://en.wikipedia.org/wiki/Cayley_graph

The GroupTheory[CayleyGraph] command was introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.