# schreierGraph -- Schreier graph of a finite group

## Synopsis

• Usage:
schreierGraph G
• Inputs:
• Outputs:
• , representing the Schreier graph of the group

## Description

This function is provided by the package InvariantRing.

For a finite group action, we form a HashTable whose keys are the generators provided by the user. The value corresponding to a generator g is a HashTable containing all pairs a => b such that a*g == b. This represents the Schreier graph of the group relative to the generating set provided by the user.

The following example defines the permutation action of a symmetric group on three elements using only two generators, a transposition and a 3-cycle.

 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}} } o2 = {| 0 1 0 |, | 0 0 1 |} | 1 0 0 | | 0 1 0 | | 0 0 1 | | 1 0 0 | o2 : List i3 : G = finiteAction(L, R) o3 = R <- {| 0 1 0 |, | 0 0 1 |} | 1 0 0 | | 0 1 0 | | 0 0 1 | | 1 0 0 | o3 : FiniteGroupAction i4 : schreierGraph G o4 = HashTable{0 => HashTable{| 0 0 1 | => | 0 0 1 |}} | 0 1 0 | | 1 0 0 | | 1 0 0 | | 0 1 0 | | 0 0 1 | => | 0 0 1 | | 1 0 0 | | 0 1 0 | | 0 1 0 | | 1 0 0 | | 0 1 0 | => | 1 0 0 | | 0 0 1 | | 0 0 1 | | 1 0 0 | | 0 1 0 | | 0 1 0 | => | 1 0 0 | | 1 0 0 | | 0 1 0 | | 0 0 1 | | 0 0 1 | | 1 0 0 | => | 0 1 0 | | 0 0 1 | | 0 0 1 | | 0 1 0 | | 1 0 0 | | 1 0 0 | => | 0 1 0 | | 0 1 0 | | 1 0 0 | | 0 0 1 | | 0 0 1 | 1 => HashTable{| 0 0 1 | => | 1 0 0 |} | 0 1 0 | | 0 1 0 | | 1 0 0 | | 0 0 1 | | 0 0 1 | => | 1 0 0 | | 1 0 0 | | 0 0 1 | | 0 1 0 | | 0 1 0 | | 0 1 0 | => | 0 1 0 | | 0 0 1 | | 1 0 0 | | 1 0 0 | | 0 0 1 | | 0 1 0 | => | 0 1 0 | | 1 0 0 | | 0 0 1 | | 0 0 1 | | 1 0 0 | | 1 0 0 | => | 0 0 1 | | 0 0 1 | | 1 0 0 | | 0 1 0 | | 0 1 0 | | 1 0 0 | => | 0 0 1 | | 0 1 0 | | 0 1 0 | | 0 0 1 | | 1 0 0 | o4 : HashTable

## Ways to use schreierGraph :

• "schreierGraph(FiniteGroupAction)"

## For the programmer

The object schreierGraph is .