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

### Baumslag groups

#### Description

Baumslag-Solitar groups have the following presentation.

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

#### Reference

G. Baumslag and D. Solitar, Some two generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc. , 689 (1962) pp. 199–201.

#### Computation

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");
Define CreateRelations()
A1:=[[a,a],];
A2:=[[a,a],];
B1:=[[b,b],];
B2:=[[b,b],];
R:=[[b,a^(2)],[a^(3),b]];
Return [A1,A2,B1,B2,R];
EndDefine;
-- Relation ba^2=a^3b. Change 2 and 3 in "()" to make another relation
G:=CreateRelations();
-- 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 Do
BA:= BA + "a";
EndFor;
For I:= 1 To ARGV Do
AB:= "a" + Ab;
EndFor;
Append(G,[BA,AB]);
Return \$apcocoa/gbmr.BGB(G,DB,LB,OFlag);
EndDefine;
```