# ApCoCoA:Free groups

### Free Groups

#### Description

The relations of a free group with n generators only consists of the existence of the invers elements. Any element of a free group has a unique representation.

``` F(n) = <a_{1},...,a_{n} | a_{i}a_{i}^{-1} = a_{i}^{-1}a_{i} = 1>
```

#### Reference

Kharlampovich, Olga; Myasnikov, Alexei, "Elementary theory of free non-abelian groups". J. Algebra 302 (2): 451–552, 2006.

#### Computation

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

// Number of free group
MEMORY.N:=4;

Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]];
NC.SetOrdering("LLEX");

Define CreateRelationsFree()
Relations:=[];
For Index1 := 1 To MEMORY.N Do
Append(Relations,[[x[Index1],y[Index1]],[1]]);
Append(Relations,[[y[Index1],x[Index1]],[1]]);
EndFor;
Return Relations;
EndDefine;

Relations:=CreateRelationsFree();
Relations;

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

#### Examples in Symbolic Data Format

##### Free group 4
``` <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
<vars>x1,x2,x3,x4,y1,y2,y3,y4</vars>
<basis>
<ncpoly>x1*y1-1</ncpoly>
<ncpoly>y1*x1-1</ncpoly>
<ncpoly>x2*y2-1</ncpoly>
<ncpoly>y2*x2-1</ncpoly>
<ncpoly>x3*y3-1</ncpoly>
<ncpoly>y3*x3-1</ncpoly>
<ncpoly>x4*y4-1</ncpoly>
<ncpoly>y4*x4-1</ncpoly>
</basis>
<Comment>Free_group_4</Comment>
</FREEALGEBRA>
```
##### Free group 5
``` <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
<vars>x1,x2,x3,x4,x5,y1,y2,y3,y4,y5</vars>
<basis>
<ncpoly>x1*y1-1</ncpoly>
<ncpoly>y1*x1-1</ncpoly>
<ncpoly>x2*y2-1</ncpoly>
<ncpoly>y2*x2-1</ncpoly>
<ncpoly>x3*y3-1</ncpoly>
<ncpoly>y3*x3-1</ncpoly>
<ncpoly>x4*y4-1</ncpoly>
<ncpoly>y4*x4-1</ncpoly>
<ncpoly>x5*y5-1</ncpoly>
<ncpoly>y5*x5-1</ncpoly>
</basis>
<Comment>Free_group_5</Comment>
</FREEALGEBRA>
```
##### Free group 6
``` <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
<vars>x1,x2,x3,x4,x5,x6,y1,y2,y3,y4,y5,y6</vars>
<basis>
<ncpoly>x1*y1-1</ncpoly>
<ncpoly>y1*x1-1</ncpoly>
<ncpoly>x2*y2-1</ncpoly>
<ncpoly>y2*x2-1</ncpoly>
<ncpoly>x3*y3-1</ncpoly>
<ncpoly>y3*x3-1</ncpoly>
<ncpoly>x4*y4-1</ncpoly>
<ncpoly>y4*x4-1</ncpoly>
<ncpoly>x5*y5-1</ncpoly>
<ncpoly>y5*x5-1</ncpoly>
<ncpoly>x6*y6-1</ncpoly>
<ncpoly>y6*x6-1</ncpoly>
</basis>
<Comment>Free_group_6</Comment>
</FREEALGEBRA>
```
##### Free group 7
``` <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
<vars>x1,x2,x3,x4,x5,x6,x7,y1,y2,y3,y4,y5,y6,y7</vars>
<basis>
<ncpoly>x1*y1-1</ncpoly>
<ncpoly>y1*x1-1</ncpoly>
<ncpoly>x2*y2-1</ncpoly>
<ncpoly>y2*x2-1</ncpoly>
<ncpoly>x3*y3-1</ncpoly>
<ncpoly>y3*x3-1</ncpoly>
<ncpoly>x4*y4-1</ncpoly>
<ncpoly>y4*x4-1</ncpoly>
<ncpoly>x5*y5-1</ncpoly>
<ncpoly>y5*x5-1</ncpoly>
<ncpoly>x6*y6-1</ncpoly>
<ncpoly>y6*x6-1</ncpoly>
<ncpoly>x7*y7-1</ncpoly>
<ncpoly>y7*x7-1</ncpoly>
</basis>
<Comment>Free_group_7</Comment>
</FREEALGEBRA>
```
##### Free group 8
``` <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
<vars>x1,x2,x3,x4,x5,x6,x7,x8,y1,y2,y3,y4,y5,y6,y7,y8</vars>
<basis>
<ncpoly>x1*y1-1</ncpoly>
<ncpoly>y1*x1-1</ncpoly>
<ncpoly>x2*y2-1</ncpoly>
<ncpoly>y2*x2-1</ncpoly>
<ncpoly>x3*y3-1</ncpoly>
<ncpoly>y3*x3-1</ncpoly>
<ncpoly>x4*y4-1</ncpoly>
<ncpoly>y4*x4-1</ncpoly>
<ncpoly>x5*y5-1</ncpoly>
<ncpoly>y5*x5-1</ncpoly>
<ncpoly>x6*y6-1</ncpoly>
<ncpoly>y6*x6-1</ncpoly>
<ncpoly>x7*y7-1</ncpoly>
<ncpoly>y7*x7-1</ncpoly>
<ncpoly>x8*y8-1</ncpoly>
<ncpoly>y8*x8-1</ncpoly>
</basis>
<Comment>Free_group_8</Comment>
</FREEALGEBRA>
```
##### Free group 9
``` <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
<vars>x1,x2,x3,x4,x5,x6,x7,x8,x9,y1,y2,y3,y4,y5,y6,y7,y8,y9</vars>
<basis>
<ncpoly>x1*y1-1</ncpoly>
<ncpoly>y1*x1-1</ncpoly>
<ncpoly>x2*y2-1</ncpoly>
<ncpoly>y2*x2-1</ncpoly>
<ncpoly>x3*y3-1</ncpoly>
<ncpoly>y3*x3-1</ncpoly>
<ncpoly>x4*y4-1</ncpoly>
<ncpoly>y4*x4-1</ncpoly>
<ncpoly>x5*y5-1</ncpoly>
<ncpoly>y5*x5-1</ncpoly>
<ncpoly>x6*y6-1</ncpoly>
<ncpoly>y6*x6-1</ncpoly>
<ncpoly>x7*y7-1</ncpoly>
<ncpoly>y7*x7-1</ncpoly>
<ncpoly>x8*y8-1</ncpoly>
<ncpoly>y8*x8-1</ncpoly>
<ncpoly>x9*y9-1</ncpoly>
<ncpoly>y9*x9-1</ncpoly>
</basis>
<Comment>Free_group_9</Comment>
</FREEALGEBRA>
```
##### Free group 10
``` <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
<vars>x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10</vars>
<basis>
<ncpoly>x1*y1-1</ncpoly>
<ncpoly>y1*x1-1</ncpoly>
<ncpoly>x2*y2-1</ncpoly>
<ncpoly>y2*x2-1</ncpoly>
<ncpoly>x3*y3-1</ncpoly>
<ncpoly>y3*x3-1</ncpoly>
<ncpoly>x4*y4-1</ncpoly>
<ncpoly>y4*x4-1</ncpoly>
<ncpoly>x5*y5-1</ncpoly>
<ncpoly>y5*x5-1</ncpoly>
<ncpoly>x6*y6-1</ncpoly>
<ncpoly>y6*x6-1</ncpoly>
<ncpoly>x7*y7-1</ncpoly>
<ncpoly>y7*x7-1</ncpoly>
<ncpoly>x8*y8-1</ncpoly>
<ncpoly>y8*x8-1</ncpoly>
<ncpoly>x9*y9-1</ncpoly>
<ncpoly>y9*x9-1</ncpoly>
<ncpoly>x10*y10-1</ncpoly>
<ncpoly>y10*x10-1</ncpoly>
</basis>
<Comment>Free_group_10</Comment>
</FREEALGEBRA>
```
##### Free group 11
``` <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
<vars>x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11</vars>
<basis>
<ncpoly>x1*y1-1</ncpoly>
<ncpoly>y1*x1-1</ncpoly>
<ncpoly>x2*y2-1</ncpoly>
<ncpoly>y2*x2-1</ncpoly>
<ncpoly>x3*y3-1</ncpoly>
<ncpoly>y3*x3-1</ncpoly>
<ncpoly>x4*y4-1</ncpoly>
<ncpoly>y4*x4-1</ncpoly>
<ncpoly>x5*y5-1</ncpoly>
<ncpoly>y5*x5-1</ncpoly>
<ncpoly>x6*y6-1</ncpoly>
<ncpoly>y6*x6-1</ncpoly>
<ncpoly>x7*y7-1</ncpoly>
<ncpoly>y7*x7-1</ncpoly>
<ncpoly>x8*y8-1</ncpoly>
<ncpoly>y8*x8-1</ncpoly>
<ncpoly>x9*y9-1</ncpoly>
<ncpoly>y9*x9-1</ncpoly>
<ncpoly>x10*y10-1</ncpoly>
<ncpoly>y10*x10-1</ncpoly>
<ncpoly>x11*y11-1</ncpoly>
<ncpoly>y11*x11-1</ncpoly>
</basis>
<Comment>Free_group_11</Comment>
</FREEALGEBRA>
```