group -- list all elements of the group of a finite group action

Synopsis

• Usage:

  group G

• Inputs:
  • G, an instance of the type FiniteGroupAction
• Outputs:
  • a list, of all elements in the finite matrix group associated to the given group action

Description

The following example defines the permutation action of a symmetric group on three elements.

i1 : R = QQ[x_1..x_3]

o1 = R

o1 : PolynomialRing

i2 : L = apply(2, i -> permutationMatrix(3, [i + 1, i + 2] ) )

o2 = {| 0 1 0 |, | 1 0 0 |}
      | 1 0 0 |  | 0 0 1 |
      | 0 0 1 |  | 0 1 0 |

o2 : List

i3 : S3 = finiteAction(L, R)

o3 = R <- {| 0 1 0 |, | 1 0 0 |}
           | 1 0 0 |  | 0 0 1 |
           | 0 0 1 |  | 0 1 0 |

o3 : FiniteGroupAction

i4 : group S3

o4 = {| 1 0 0 |, | 0 1 0 |, | 1 0 0 |, | 0 0 1 |, | 0 1 0 |, | 0 0 1 |}
      | 0 0 1 |  | 1 0 0 |  | 0 1 0 |  | 0 1 0 |  | 0 0 1 |  | 1 0 0 |
      | 0 1 0 |  | 0 0 1 |  | 0 0 1 |  | 1 0 0 |  | 1 0 0 |  | 0 1 0 |

o4 : List

The computation of all elements in the group is actually performed by the method schreierGraph since the process of computing the Schreier graph of the group yields other useful information about the group besides just its elements.

See also

• schreierGraph -- Schreier graph of a finite group
• words -- associate a word in the generators of a group to each element