[GAP Forum] finding automorphisms of finitely presented groups

[Alexander Hulpke]

Dear Robert Heffernan,

> I am dealing with groups constructed in this manner:
> F:=FreeGroup("x","y","z");G:=F/rels;
> where rels is some list of relations in terms of x,y and z.
> If the group is large GAP seems to have trouble constructing the  
> groupof automorphisms of G.
> I understand that doing this:G:=Image(IsomorphismPermGroup 
> (G));would give me a representation of the group that GAP can deal  
> witheasily (and easily compute Aut G, etc.), but I want to look at  
> theautomorphisms in terms of the generators x,y and z above.

That is not a contradiction. Compute the automorphism group for the  
permutation representation and then pull the generators back to the  
finitely presented group.
Concretely, if
phi:=IsomorphismPermGroup(G);
P:=Image(phi);
A:=AutomorphismGroup(P);

you can do
List(GeneratorsOfGroup(A),a->GroupHomomorphismByImagesNC 
(G,G,GeneratorsOfGroup(G),
   List(GeneratorsOfGroup(G),x->PreImagesRepresentative(phi,Image 
(a,Image(phi,x)))));

to get generators of the automorphism group in terms of x,y and z.

Best,

    Alexander Hulpke

-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke