# Difference between revisions of "ApCoCoA-1:Other11 groups"

### Other Groups

#### Description

This group has the following finite representation:

```G = <x,t | xt^{r} = tx^{r},t^{n} = 1>
```

for r >= 1 and n >= 2

#### Reference

No reference available

#### Computation of G

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

// Note that r >= 1 and n >= 2
MEMORY.R := 3;
MEMORY.N := 4;

// x is invers to z, t has an implicit invers (Relation: t^{n} = 1)
Use ZZ/(2)[x,t,z];
NC.SetOrdering("LLEX");

Define CreateRelationsOther11()
Relations:=[];

// add the invers relations xz = zx = 1
Append(Relations,[[x,z],[1]]);
Append(Relations,[[z,x],[1]]);

// add the relation t^{n} = 1
RelationBuffer0:=[];
For Index0:=1 To MEMORY.N Do
Append(RelationBuffer0,t);
EndFor;
Append(Relations,[RelationBuffer0,[1]]);

// add the relation xt^{r} = tx^{r}
RelationBuffer1:=[];
RelationBuffer2:=[];
Append(RelationBuffer1,x);
Append(RelationBuffer2,t);
For Index1:= 1 To MEMORY.R Do
Append(RelationBuffer1,t);
Append(RelationBuffer2,x);
EndFor;
Append(Relations,[RelationBuffer1,RelationBuffer2]);

Return Relations;
EndDefine;

Relations:=CreateRelationsOther11();
Gb:=NC.GB(Relations,31,1,100,1000);
```

#### Examples in Symbolic Data Format

##### Other group 11 r=3 n=4
``` <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
<vars>t,x,z</vars>
<uptoDeg>13</uptoDeg>
<basis>
<ncpoly>x*z-1</ncpoly>
<ncpoly>z*x-1</ncpoly>
<ncpoly>(t^4)-1</ncpoly>
<ncpoly>(x*(t^3))-t*(x^3)</ncpoly>
</basis>
<Comment>The partial LLex Gb has 248 elements</Comment>
<Comment>Other_groups_11_r3n4</Comment>
</FREEALGEBRA>
```
##### Other group 11 r=5 n=5
``` <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
<vars>t,x,z</vars>
<uptoDeg>100</uptoDeg>
<basis>
<ncpoly>x*z-1</ncpoly>
<ncpoly>z*x-1</ncpoly>
<ncpoly>(t^5)-1</ncpoly>
<ncpoly>(x*(t^5))-t*(x^5)</ncpoly>
</basis>
<Comment>Other_groups_11_r5n5</Comment>
</FREEALGEBRA>
```
##### Other group 11 r=6 n=7
``` <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
<vars>t,x,z</vars>
<uptoDeg>12</uptoDeg>
<basis>
<ncpoly>x*z-1</ncpoly>
<ncpoly>z*x-1</ncpoly>
<ncpoly>(t^7)-1</ncpoly>
<ncpoly>(x*(t^6))-t*(x^6)</ncpoly>
</basis>
<Comment> The partial LLex Gb has 217 elements </Comment>
<Comment>Other_groups_11_r6n7</Comment>
</FREEALGEBRA>
```
##### Other group 11 r=7 n=11
``` <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
<vars>t,x,z</vars>
<uptoDeg>19</uptoDeg>
<basis>
<ncpoly>x*z-1</ncpoly>
<ncpoly>z*x-1</ncpoly>
<ncpoly>(t^11)-1</ncpoly>
<ncpoly>(x*(t^7))-t*(x^7)</ncpoly>
</basis>
<Comment> The partial LLex Gb has 228 elements </Comment>
<Comment>Other_groups_11_r7n11</Comment>
</FREEALGEBRA>
```