# ApCoCoA-1:VonDyck groups

From ApCoCoAWiki

#### 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>

(Reference: not found yet)

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

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