# relations(FiniteGroupAction) -- relations of a finite group

## Synopsis

• Function: relations
• Usage:
relations G
• Inputs:
• Outputs:
• a list, a list of relations of the group

## Description

This function is provided by the package InvariantRing.

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