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

### Higman Group

#### Description

The Higman group is an infinite finitely presented group with no non-trivial finite quotients and is generated by the elements a,b,c and d with the following relations:

``` H = <a,b,c,d | a^{-1}ba = b^{2}, b^{-1}cb = c^{2}, c^{-1}dc = d^{2}, d^{-1}ad = a^{2}>
```

#### Reference

Graham Higman, "A finitely generated infinite simple group", Journal of the London Mathematical Society. Second Series 26 (1): 61–64, 1951.

#### Computation

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

// a is inverse to e, b is inverse to f, c is inverse to g and d is inverse to h
Use ZZ/(2)[a,b,c,d,e,f,g,h];
NC.SetOrdering("LLEX");

Define CreateRelationsHigman()
Relations:=[];

// add the relations of the inverse elements
Append(Relations,[[a,e],]);
Append(Relations,[[e,a],]);
Append(Relations,[[b,f],]);
Append(Relations,[[f,b],]);
Append(Relations,[[c,g],]);
Append(Relations,[[g,c],]);
Append(Relations,[[d,h],]);
Append(Relations,[[h,d],]);

// add the relation a^{-1}ba = b^2
Append(Relations,[[e,b,a],[b^2]]);

// add the relation b^{-1}cb = c^2
Append(Relations,[[f,c,b],[c^2]]);

// add the relation c^{-1}dc = d^2
Append(Relations, [[g,d,c],[d^2]]);

Append(Relations, [[h,a,d],[a^2]]);

Return Relations;
EndDefine;

Relations:=CreateRelationsHigman();
Relations;

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

#### Example in Symbolic Data Format

``` <FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier">
<vars>a,b,c,d,e,f,g,h</vars>
<uptoDeg>5</uptoDeg>
<basis>
<ncpoly>a*e-1</ncpoly>
<ncpoly>e*a-1</ncpoly>
<ncpoly>b*f-1</ncpoly>
<ncpoly>f*b-1</ncpoly>
<ncpoly>c*g-1</ncpoly>
<ncpoly>g*c-1</ncpoly>
<ncpoly>d*h-1</ncpoly>
<ncpoly>h*d-1</ncpoly>
<ncpoly>e*b*a-b*b</ncpoly>
<ncpoly>f*c*b-c*c</ncpoly>
<ncpoly>g*d*c-d*d</ncpoly>
<ncpoly>h*a*d-a*a</ncpoly>
</basis>
<Comment>The partial LLex Gb has 144 elements</Comment>
<Comment>Higman_group</Comment>
</FREEALGEBRA>
```