# ApCoCoA-1:Other2 groups

### Other Groups

#### Description

The first group is called Rosenberger-Monster and is the largest finite generalized triangle group. A finite representation of G is given below:

``` G = <a,b | a^2 = b^3 = (abababab^{2}ab^{2}abab^{2}ab^{2})^2 = 1>
```

The second group is already infinite and denoted by H:

``` H = <a,b | a^2 = b^3 = (abababab^{2}abab^{2}ab^{2})^2 = 1>
```

#### Reference

On the Rosenberger Monster Robert Fitzgerald Morse, Department of Electrical Engineering and Computer Science, University of Evansville IN 47722 USA

#### Computation of G

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

Use ZZ/(2)[a,b];
NC.SetOrdering("LLEX");

Define CreateRelationsOther2()
Relations:=[];

// add the relations a^2 = b^3 = 1
Append(Relations,[[a,a],[1]]);
Append(Relations,[[b,b,b],[1]]);

// add the relation (abababab^{2}ab^{2}abab^{2}ab^{2})^2 = 1
Append(Relations,[[a,b,a,b,a,b,a,b^2,a,b^2,a,b,a,b^2,a,b^2,a,b,a,b,a,b,a,b^2,a,b^2,a,b,a,b^2,a,b^2],[1]]);

Return Relations;
EndDefine;

Relations:=CreateRelationsOther2();
Gb:=NC.GB(Relations,31,1,100,1000);
```

#### G in Symbolic Data Format

``` <FREEALGEBRA createdAt="2014-02-27" createdBy="strohmeier">
<vars>a,b</vars>
<uptoDeg>39</uptoDeg>
<basis>
<ncpoly>a*a-1</ncpoly>
<ncpoly>b*b*b-1</ncpoly>
<Comment>polynomials to define inverse elements</Comment>
<ncpoly>(a*b*a*b*a*b*a*b*b*a*b*b*a*b*a*b*b*a*b*b)^2-1</ncpoly>
<Comment>(abababab^{2}ab^{2}abab^{2}ab^{2})^2 = 1 </Comment>
</basis>
<Comment> The partial LLex Gb has 3 elements</Comment>
<Comment>Other_groups2</Comment>
</FREEALGEBRA>
```

#### Computation of H

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

Use ZZ/(2)[a,b];
NC.SetOrdering("LLEX");

Define CreateRelationsOther3()
Relations:=[];

// add the relations a^2 = b^3 = 1
Append(Relations,[[a,a],[1]]);
Append(Relations,[[b,b,b],[1]]);

// add the relation (abababab^{2}abab^{2}ab^{2})^2 = 1
Append(Relations,[[a,b,a,b,a,b,a,b^2,a,b,a,b^2,a,b^2,a,b,a,b,a,b,a,b^2,a,b,a,b^2,a,b^2],[1]]);

Return Relations;
EndDefine;

Relations:=CreateRelationsOther3();
Gb:=NC.GB(Relations,31,1,100,1000);
```

#### H in Symbolic Data Format

``` <FREEALGEBRA createdAt="2014-02-27" createdBy="strohmeier">
<vars>a,b</vars>
<uptoDeg>34</uptoDeg>
<basis>
<ncpoly>a*a-1</ncpoly>
<ncpoly>b*b*b-1</ncpoly>
<ncpoly>(a*b*a*b*a*b*a*b*b*a*b*a*b*b*a*b*b)^2-1</ncpoly>
<Comment>relation: (abababab^{2}abab^{2}ab^{2})^2 = 1</Comment>
</basis>
<Comment>The partial LLex Gb has 249 elements</Comment>
<Comment>Other_groups3</Comment>
</FREEALGEBRA>
```