Daer GAP-Forum, Dear Jamshid

I am sending my reply to the GAP Forum, since you also sent your
question there.

Habtay Ghebrewold had already written to the address 'gap@...' with an
almost identical question and I have replied a daylater by the letter
that I append.

While we try to answer all questions that are sent to us, we must ask
for a bit of patience if a reply does not come by return mail. All
this answering of questions and requests is done by (mostly young)
colleagues who are not employed for this help but have regular duties
in teaching and research.

We therefore also very much appreciate if we get help from other users
of GAP with part of that service, in this case e.g. from GAP users in
South Africa..

Kind regards    Joachim
_____________________________________________________________________
Prof. em. J. Neubueser 
Lehrstuhl D fuer Mathematik
RWTH Aachen

Your letter:

Dear Gap-Forum

I have received the following email from "Habtay Ghebrewold" a PhD
student at the University of the Western Cape.
Jamshid Moori.
-----------------

>Dear Sir:

I am currently Working on Group Theory for my PhD Project.
Thus, I want to use GAP for Algebraic computations. Such
as, computing Automorphism Group of Finite Abelian Groups
and checking If two groups are isomorphic, where the
groups are semidirect products of Finite Abelian Group with
a Free Abelian Group of finite rank ( free abelian group
of finite rank acting on a finite abelian group).
Now, I want to know, if GAP can help me to deal with
the above computations? And if possible would you please
give me an idea how to deal with the above problems using
computers?

Thank you.
                        Sincerely Yours
                        Habtay Ghebrewold
                        University of the Western Cape
                        Department of Mathematics 
                        Private  Bag  X17
                        7535  Bellville,  South Africa

My reply to Habtay Ghebrewold

Subject: Re: Need Help
In-Reply-To:  HABTAY GHEBREWOLD at "Sep 7, 2000 09:33:26 am"
To: HABTAY GHEBREWOLD <hghebrewold@uwc.ac.za>
Date: Fri, 8 Sep 2000 17:30:07 +0200 (CEST)

Dear Habtay Ghebrewold,

I am afraid, I can give you only a partial answer to your request:

I am currently Working on Group Theory for my PhD Project.
Thus, I want to use GAP for Algebraic computations. Such
as:
1. Computing Automorphism Group of Finite Abelian Groups;
2. Checking If two groups are isomorphic, where the
groups are semidirect products of Finite Abelian Group
with a Free Abelian Group of finite rank ( free abelian
group of finite rank acting on a finite abelian group).
Now, I want to know, if GAP can help me to deal with
the above computations? And if possible would you please
give me an idea how to deal with the above problems using
computers?
Note that I am currently having GAP version 3 release
4 for UNIX.

Let me first of all strongly recommend to get GAP4.2 and install it
under Unix (or have it installed by your local system andministrator)
*together with the so far 4 bugfixes that have been issued*. GAP4 has
a number of additional possibilities compared to GAP3 and in
particular some share packages, to which I am going to refer, are
written for GAP4 only.

As to your first question:

In the GAP4 manual you find a section (35.6) on groups of
automorphisms. E.G. you find a descripton of a function

AutomorphismGroup(obj)

where for object you can put in a group of which you want the
automorphism group. In the example given the group for which the
automorphism group is calculated is a dihedral group of order 8, but
you can as well use a group given by a polycyclic presentation. A
wealth of examples is accessible in the Small Groups library, you can
call groups from it e.g. by

SmallGroup(8,4).

which will give you the fourth group of order 8 in that library.

E.g.  

SmallGroup(8.1) is the cyclicgroup of order 8. SmallGroup(8.2) is the
direct product of a cyclic group of order 4 with a cyclic group of
order 2, while SmallGroup(8.5) is the elementary abelian group of
order 8.

But you can also define finite abelian groups not in that library by a
polycyclic presentation yourself.

There is also a chapter about automorphism groups of finite soluble
groups in the GAP3 manual (chapter 58), by the way, but neither the
Small Groups Library nor the share packages mentioned below are
available in GAP3.

As to your second question:

At present in the released GAP no function is available that would
test the isomorphism of semidirect products of finite abelian groups
by free abelian groups of finite rank, although in principle I think
that question should be decidable using Taunt's criterium for the
isomorphism of semidirect products. (Ask you supervisor for it, if you
do not know it).

There is a GAP share package on polycyclic groups (to which such
extensions do belong) under development by Dr. Bettina Eick from the
University of Kassel, who has also developed a package on group
constructions (for finite groups). I will forward this letter to her,
however at present she is travelling, so that you should not expect an
answer (positive or negative) from her very soon.

Sorry for not being able to help better right now.

Joachim Neubueser