# ApCoCoA-1:Baumslag groups

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### Computation of Non-abelian Groups

#### Baumslag groups

Baumslag-Solitar groups have the following presentation.

```BS(m,n)<a, b | ba^{m} = a^{n}b> where m, n are natural numbers
```

XML data:

```<vars>a[1],a[2],b[1],b[2]</vars>
<params>m,n</params>
<rels>
<ncpoly>a[1]*a[2]-1</ncpoly>
<ncpoly>a[2]*a[1]-1</ncpoly>
<ncpoly>b[1]*b[2]-1</ncpoly>
<ncpoly>b[2]*b[1]-1</ncpoly>
<ncpoly>b[1]*a[1]^{m}-a[1]^{n}*b[1]</ncpoly>
</rels>
```

We enumerate partial Groebner bases for the Baumslag-Solitar groups as follows.

```/*Use the ApCoCoA package ncpoly.*/
Use ZZ/(2)[a[1..2],b[1..2]];
NC.SetOrdering("LLEX");
A1:=[[a[1],a[2]],[1]];
A2:=[[a[2],a[1]],[1]];
B1:=[[b[1],b[2]],[1]];
B2:=[[b[2],b[1]],[1]];
-- Relation ba^2=a^3b. Change 2 and 3 in "()" to make another relation
R:=[[b[1],a[1]^(2)],[a[1]^(3),b[1]]];
G:=[A1,A2,B1,B2,R];
-- Enumerate a partial Groebner basis (see NC.GB for more details)
NC.GB(G,31,1,100,1000);
```
```/*Use the ApCoCoA package gbmr.*/
-- See NCo.BGB for more details on the parameters DB, LB and OFlag.
Define BS(M,N,DB,LB,OFlag)
\$apcocoa/gbmr.SetX("aAbB");
\$apcocoa/gbmr.SetOrdering("LLEX");
G:= [["aA",""],["Aa",""],["bB",""],["bB",""]];
BA:= "b";
AB:= "b";
For I:= 1 To ARGV[1] Do
BA:= BA + "a";
EndFor;
For I:= 1 To ARGV[2] Do
AB:= "a" + Ab;
EndFor;
Append(G,[BA,AB]);
Return \$apcocoa/gbmr.BGB(G,DB,LB,OFlag);
EndDefine;
```