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

## Synopsis

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

## Description

This function is provided by the package InvariantRing.

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.