Thanks to Joachim Neubueser, Frank Celler, and Derek Holt for their
responses regarding my question on infinite matrix groups. I'd like
to make these comments in return:
(1) I think the first method suggested by Derek Holt is very close to what I
had in mind for getting the 'first' 1000 elements of an infinite group;
I will try generating more and more elements of the free group while
watching for repetitions (until I get 1000). As noted there doesn't
seem a GAP function that does that, but I agree that it should be easy
to implement one; an advantage of this is that I know how to add
'selection rules' for which elements to accept. I was hoping to find
something more sophisticated and it seems that such an algorithm exists.
Two catches though! : it is not part of GAP (yet), and it assumes that
you know the defining relations of the group (for this particular problem,
I don't, I only have matrices). This algorithm would be a nice addition
to GAP someday; in the mean time I would be interseted to know more about
the existing software and I think that others might be too. I'll leave it
to Derek's discretion to tell more about his software by e-mail or through
the forum -- when he's not too busy of course :-)
(2) Frank Celler's heuristic method would probably involve the least amount of
work on my part and so I will try it first. I suspect that this would not
really give me the elements that I am looking for but hopefully I'm wrong!
(3) I am glad to get a response from Dr. Neubueser on this since I am well
aware of his work (with others) on the classification of the the four
dimensional space groups; this also makes it easier for me to describe
my "wish-list" as far as crytallographic groups are concerned: the
short (and and I hope not too blunt :-) ) answer is : all the tables
in the book 'Crystallographic Groups in Four Dimensional Space';
It would be so nice to have the generators for all the point groups
as well as the full space groups in dimensions 1 to 4 complemented by
routines to give such standard data as equivalent positions (for
different cernterings) of these groups -- that would make a good part
of the International Tables available at your fingertips.
I realize that the classification was carried out a number of yeares
before GAP was introduced; I am curious to know how much of it can be
recreated within GAP today.
Once again, thanks to all for their help.