# ApCoCoA-1:VonDyck groups

### Von Dyck Groups

#### Description

The von Dyck groups are sometimes referred to as ordinary triangle groups and are subgroups of index 2 in Triangle(l, m, n) generated by words of even length in the generators a, b, c. A specific representation is given for x = ab, y = ca, yx = cb:

``` D(l,m,n) = <x,y | x^{l} = y^{m} = (xy)^{n} = 1>
```

#### Computation

``` /*Use the ApCoCoA package ncpoly.*/

// Parameters of von Dyck group
MEMORY.L:=3;
MEMORY.M:=5;
MEMORY.N:=2;

Use ZZ/(2)[x,y];
NC.SetOrdering("LLEX");

Define CreateRelationsVonDyck()
Relations:=[];

// add the relation x^l = 1
Append(Relations,[[x^MEMORY.L],]);

// add the relation y^m = 1
Append(Relations,[[y^MEMORY.M],]);

// add the relation (xy)^n = 1
BufferXY:=[];
For Index1 := 1 To MEMORY.N Do
Append(BufferXY,x);
Append(BufferXY,y);
EndFor;
Append(Relations,[BufferXY,]);

Return Relations;
EndDefine;

Relations:=CreateRelationsVonDyck();
Relations;

Gb:=NC.GB(Relations);
Gb;
```

#### Example in Symbolic Data Format

``` <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
<vars>x,y</vars>
<basis>
<ncpoly>x^3-1</ncpoly>
<ncpoly>y^5-1</ncpoly>
<ncpoly>(x*y)^2-1</ncpoly>
</basis>
<Comment>Von_Dyck_group_l3m5n2</Comment>
</FREEALGEBRA>
```