CayleyGraph - Maple Help

GroupTheory

 CayleyGraph
 construct the Cayley graph of a group

 Calling Sequence CayleyGraph( G ) CayleyGraph( G, elements = E, generators = S )

Parameters

 G - a small group E - (optional) list ; an ordering of the elements of G S - (optional) list ; a list of generators for G

Description

 • 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.

Examples

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

Draw the Cayley graph of the symmetric group of degree 4.

 > $G≔\mathrm{SymmetricGroup}\left(4\right)$
 ${G}{≔}{{\mathbf{S}}}_{{4}}$ (1)
 > $\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{CayleyGraph}\left(G\right),\mathrm{style}=\mathrm{spring}\right)$

Draw the Cayley graph of the dihedral group of degree 7.

 > $G≔\mathrm{DihedralGroup}\left(7\right)$
 ${G}{≔}{{\mathbf{D}}}_{{7}}$ (2)
 > $\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{CayleyGraph}\left(G\right),\mathrm{style}=\mathrm{spring}\right)$

The default set of generators for the group $PGL\left(2,3\right)$ is given by the following command.

 > $G≔\mathrm{PGL}\left(2,3\right):$
 > $\mathrm{Generators}\left(G\right)$
 $\left[\left({3}{,}{4}\right){,}\left({1}{,}{2}{,}{4}\right)\right]$ (3)

These are used by default for the Cayley graph.

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

To specify a different generating set, use the generators= option.

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

The simple group of order $168$ is $2,3$-generated.

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

It is also generated by the involution above and and element of order $7$, leading to a very different Cayley graph.

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

References

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

Compatibility

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