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GroupTheory

  

CayleyGraph

  

construct the Cayley graph of a group

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

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

withGroupTheory:

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

GSymmetricGroup4

GS4

(1)

GraphTheory:-DrawGraphCayleyGraphG,style=spring

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

GDihedralGroup7

GD7

(2)

GraphTheory:-DrawGraphCayleyGraphG,style=spring

The default set of generators for the group PGL2,3 is given by the following command.

GPGL2,3:

GeneratorsG

3,4,1,2,4

(3)

These are used by default for the Cayley graph.

GraphTheory:-DrawGraphCayleyGraphG,style=spring

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

GraphTheory:-DrawGraphCayleyGraphG,generators=Perm1,2,3,4,Perm3,4,Perm1,2,4,style=spring

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

GPSL3,2:

GeneratorsG

4,65,7,1,2,43,6,5

(4)

GraphTheory:-DrawGraphCayleyGraphG,style=spring

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

GraphTheory:-DrawGraphCayleyGraphG,generators=Perm1,6,2,7,4,5,3,Perm4,6,5,7,style=spring

References

  

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

Compatibility

• 

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

• 

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

See Also

GraphTheory

GroupTheory

GroupTheory[CayleyTable]

GroupTheory[SymmetricGroup]