[GAP Forum] Help/Question

[Alexander Hulpke]

Dear Gap Forum,

Abdulsatar Al-Juburie and Addam Brady both asked about `IsomorphismFpGroup' for infinite groups.

Indeed this operation is the one GAP uses to calculate presentations for a given group, but of course the computer can only do what has been programmed in before. Often this is limited to some kind of finiteness, as an algorithm otherwise won't terminate.

For IsomorphismFpGroup, such methods exist in GAP for the cases of
- Finite groups
- Subgroups of finitely presented groups of finite index

(I am not aware of other generic algorithms that would work for infinite groups, so very likely this is not a shortcoming of GAP but a fundamental algorithmic problem.)

Both GL_n(Z) and the automorphism group of a free group do not fit either of these two classes, so calling `IsomorphismFpGroup' for such a group will simply yield an error message.

This does not mean that such presentations are not known -- it simply means that GAP is not a universal library lookup tool that collects all human knowledge. (Looking them up in the library -- for example Wikipedia (automorphism group of a free group) points to Magnus/Karras/Solitar for such a presentation for the automorphism group of the free group, and
MR1079696 (92h:20050) 
Conder, Marston; Robertson, Edmund; Williams, Peter Presentations for $3$-dimensional special linear groups over integer rings. Proc. Amer. Math. Soc. 115 (1992), no. 1, 19–26
gives one for SL_n(Z) from which one can probably obtain one for GL_n(Z))

Best wishes,

   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