Dear Dr. Larson,

Thank you for the paper copy of the notes for your students which I
obtained a while ago. I also got hold of a copy of Gallian's
book. Both address students at an earlier mathematical age, than our
students are when entering our algebra course, but both impress me as
carefully thought over. There are several details in your note, that I
will remember for my teaching. e.g. the nice way to introduce the idea
of conjugation as well as that of even and odd permutations by the
sliding puzzles right at the beginning.

Nevertheless I would like to comment on the GAP notes and suggest
additions and/or alternatives.

1. My first observation is that you let the students use GAP only in
strictly interactive mode. You provide even absolutely straightforward
code as the one for finding the commutativity index as an appendix. I
do not know what the situation with your students is, but here I would
take it almost for granted that they all had some private encounter
with some computing language, like Pascal or even Basic and hence they
could write such a little double loop themselves. After all, the GAP
language is so simple and so similar to Pascal - at least for this
kind of level.

But of course in a more general way , I would leave it to their own
initiative to learn to do it if they do not know how to do it and I
would at best offer a couple of extra instruction hours. Very
bluntly, I would tell them that after all I also assume that they can
hold a pencil, that they can read a textbook in English (or can even
give a seminar talk in (some) English if necessary). I would object to
spoonfeed them too much.

2. In detail then with just this question: The little routine simply
runs through all pairs and tests commutativety. Of course introducing
the notion of the centralizer and noting that conjugate elements have
centralizers of the same order could be used to shorten the
computation. Could one not use just this nice idea of looking at the
commutativety index to motivate such notions? Or would this already
go beyond the time you have for the course?

3. What I would try to do is to use GAP to introduce a greater variety
of groups. I think one of the most important insights that a program
system like GAP can help to give the student is that the theory of
groups really is a theory about a vast set of different objects. There
are the libraries of groups that come with the GAP system. Just
letting the students look at some of these might be worthwhile,
looking at the differences of the groups of the same order, i.e. to
ask them how they could distinguish isomorphism types or e.g. ask for
the different commutativity index of the different groups of order 60
or... .

For this purpose it may be necessary to transform the list of AG -
groups into a list of permutation groups, since the idea of a
polycyclic presentation is too difficult at this level, but then this
could be done by adding another appendix.

I would be most interested to hear about any further teaching
experiments that you will make, I think there is a lot to do in this
line.

With kind regards Joachim Neubueser