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Dear Gap-Forum@RWTH-Aachen.de

I had some experience of teaching group theory that I like to tell you

about it.

I used to give to my students to compute (even by hand when no computer was

possible) concrete examples (full computations): I think that this is the

right way to make them ASSIMILATE the theoretical concepts. But, of course,

the task is horrible when the groups have order about (more or less) than

30 elements, while many elementary concepts cannot be illustrated by

so small groups.

More over, I found that the students are really exited if they have to

"guess" what group they have while this group is defined by an abstract

presentation. The only thing that they discourage them to continue

investigations is of course a long, perhaps endless, hand calculus.

That is why I made a few years ago (before finding GAP or handling

any symbolic system) a "small" program for investigation "small groups"

(I call it Hijara=little stones, to make people remember they learned

the first things in arithmetics with little stones). These groups may be

permutation groups, abstract groups, "modular groups" (=units of Z/nZ)

or matrix groups with coefficients in some Z/nZ (matrices are not yet

implemented).

Now with such system, one can do elementary investigations in reasonable

time, on groups with some hundred elements and main elementary concepts

may be illustrated. For bigger groups, the program take a very long time

(some hours for groups of order = some thousands ).

The membership test is very expensive...and my implementation is

not optimized at all.

The fact that some groups to investigate are abstract make them return to

the theory for solving problem of how to handle that group. For them,

it is an "open question". There is no standard way to solve problems of

this nature, and all theoretical tools are potentially useful for

solving that problem.

The solution may be given later by the professor who is assumed to know

what group it is.

So the philosophy is that the computer as well as the book have the same

chance to make students thinking about what they are studying. We have only

to find the adequate pedagogy for using computer and any other tool

for communication. More over, the students need always at some moments...

professors...and this is not a discovery.

best regards

M. Taoufik Karkar

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