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^ Subject:

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

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