Michael K. Johnson and Don Kreher report that they are working on the
development of course material using GAP and ask where else work of
this kind is done.
It will be no surprise that we do use GAP in teaching in Aachen,
although we have not written a laboratory manual or systematic course
material. In order to explain the situation it should perhaps first be
explained that the contents of courses is less fixed in German
universities than it usually seems to be in the US, that is, each of
us rather goes his own way in teaching a course on group theory, say,
and may also change his course from one year to another. With this
reservation made, one can say that we have perhaps two main lines in
integrating algorithmic methods and the use of a system such as GAP
into such a course.
In one line, which I have followed two years ago, I gave a course that
was entitled "Groups, theory and algorithms" parts I and II over a
full year, in which algorithmic aspects and methods were closely knit
into the theory, e.g. the course - that assumed a course on Algebra I
which gave the basics up to Jordan-Hoelder and Sylow - started with
free groups and presentations and alongside with the theory introduced
computational methods such as Todd-Coxeter, Reidemeister-Schreier,
Low-index and IMD. These then were treated through easy
hand-calculations as well as examples using programs in the exercises
( at that time we had partially to resort to SPAS because the
algorithms were not all in GAP yet, but they will be in GAP 3.2 to be
released soon ). In a similar way then permutation groups, soluble
groups and p-groups were treated. This course was followed by a
further year on representation theory, of which I gave the first
semester on ordinary representation theory, again interlacing theory
with computational methods mainly for charactertheory, again using
GAP, which provides quite a lot of possibilities in this field.
For these courses we have files with the weekly exercises given to
the students and some percentage of these involve the use of GAP. If
somebody is interested to get these ( in German and not specially
organized for export ) we will be happy to send them.
In another setup, which we follow this year, my colleague, Prof.
Pahlings will give a more traditional one-semester course on group
theory, in which again GAP may be used occasionally, but more as a
black box, while most of the algorithmic aspects will be treated in a
separate course by me next summer, in which GAP will naturally play a
more central role. Prof Pahlings meanwhile will already go on to
representation theory next summer.
We have followed that line also some years ago, both seem to have
advantages and drawbacks and I really cannot say that I recommend one
of them as the better setup.
Generally we tend to allow or even recommend the use of GAP also in
other courses such as the introductory algebra course. We hope that
for students, who nowadays tend to come being pretty well used to
PASCAL or the like, using GAP is not so difficult, so in these courses
usually we have made no attempt with a systematic introduction to GAP
but rather have "let things happen" and this is perhaps even so with
the above-mentioned courses. But I am sure we could do better than
that and hence I would be very interested to get whatever course
material is developed. I would also very much welcome if such material
- perhaps after some test with students - could be made generally
available alongside with GAP.