**GAP Lesson 4
**

*Change to the directory where you store your GAP files, and then start up GAP.
*

gap> At:=KnownAttributesOfObject;

gap> a:=FreeGroup(2);

gap> a.1*a.2;

gap> f1*f2;

gap> b:=FreeGroup(2,"b");

gap> b.1/b.2^3;

gap> b1*b2;

gap> c:=FreeGroup(2,"b");

gap> c.1*c.2;

gap> b.1*b.2;

gap> c.1*c.2=b.1*b.2;

gap> c.1 in c;

gap> c.1 in b;

gap> d:=FreeGroup("u","v");

gap> u;

gap> d.1;

gap> v:=d.2;

gap> u*v/(v*u);

gap> d;

gap> Size(d);

gap> d:=d/[u^4,v^2,v*u*v*u];

gap> At(d);

gap> Size(d);

gap> At(d);

gap> Elements(d);

gap> At(d);

gap> Center(d);

gap> isod:=IsomorphismPermGroup(d);

gap> Image(isod);

gap> Image(isod, d.1*d.2));

Some of the last lines were intended to show you that the commands the worked for permutation and matrix groups may also work for groups given by generators and relations. What is going on in the following lines?

gap> v in d;

gap> GeneratorsOfGroup(d);

gap> d.2;

gap> d.2 in d;

We had started to feel comfortable about GAP's conventions, and now we see more pitfalls! The following lines fix the problem.

gap> u:=d.1;

gap> v:=d.2;

gap> v in d;

The following computes the homomorphism to a permutation group which GAP has already done:

gap> FactorCosetOperation(d,Subgroup(d,[v]));

[ u, v ] -> [ (1,2,3,4), (2,4) ]

The following command is also possible at this stage.

gap> Action(d,RightCosets(d,Subgroup(d,[v])),OnRight);

gap> e:=FreeGroup(3,"e"); e1:=e.1; e2:=e.2; e3:=e.3;

gap> e:=e/[e1^2/e2^3,e2^3/e3^5,e3^5/(e1*e2*e3)];

gap> Size(e);

gap> At(e);

gap> ce:=Centre(e);

Exercises: 1. Determine whether or not the center is generated by the element rst. Discuss what issues came up, and whether anything happened that might lead to an incorrect conclusion.

2. Determine the isomorphism type of the factor group e/ce.

3. Show that e is not isomorphic to the direct product of ce and (e/ce).

In doing these exercises you will explore what commands seem to work for groups given by a presentation.