relations(FiniteGroupAction) -- relations of a finite group

Synopsis

• Function: relations

• Usage:

  relations G

• Inputs:
  • G, an instance of the type FiniteGroupAction, the action of a finite group
• Outputs:
  • a list, a list of relations of the group

Description

Use this function to get the relations among elements of a group. Each element is represented by a word of minimal length in the Coxter generators. And each relation is represented by a list of two words that equates the group element represented by those two words.

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

i1 : R = QQ[x_1..x_3]

o1 = R

o1 : PolynomialRing

i2 : L = {matrix {{0,1,0},{1,0,0},{0,0,1}}, matrix {{0,0,1},{0,1,0},{1,0,0}}, matrix {{1,0,0},{0,0,1},{0,1,0}} }

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

o2 : List

i3 : G = finiteAction(L, R)

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

o3 : FiniteGroupAction

i4 : relations G

o4 = {{{}, {1, 1}}, {{}, {2, 2}}, {{1}, {0, 1, 2}}, {{1}, {0, 2, 0}}, {{}, {0, 0}}, {{0, 2}, {1, 0}}, {{0, 2}, {2, 1}}, {{0, 1},
     ----------------------------------------------------------------------------------------------------------------------------
     {1, 2}}, {{0, 1}, {2, 0}}, {{2}, {0, 1, 0}}, {{2}, {0, 2, 1}}}

o4 : List

See also

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