From AbstractAlgebra for Mathematica to GAP.

A few notes about the Mathematica video's I created on YouTube.

AbstractAlgebra is a Mathematica add-on package. It is open source, has been written entirely in the Mathematica programming language by two mathematics professors Hibbard and Levasseur. The download url is http://www.central.edu/EAAM/Downloads/AAPackage.asp.

The functions I used like FormGroupoid, GenerateGroupoidByRelations and FormMorphoid are defined in a Mathematica package called Master in the AbstractAlgebra directory. The group I created as G1=FormGroupoid[Range[0,11],Mod[#1+#2,12]&] is by default available as Z[12] ( when SwitchStructureTo[Group] has been used ), in fact all groups I used are available as defaults. I wanted to demonstrate that you can define any group you want, in any case much more than the familiar 'textbook example groups'.

The AbstractAlgebra package is meant as a tool to visualize the often abstract concepts in Group Theory and other topics in Abstract Algebra. There is a book available from which you can learn Abstract Algebra with Labs and Exercises in Mathematica with the Abstract Algebra package. Other algebraic structures which can be created are Rings, Polynomials, Polynomials over Rings, Galois Fields, Permutations, Permutation Cycles and more.

The package AbstractAlgebra is NOT a tool for Computational Group Theory. Group Theory is alive as it is being actively researched. A state-of-the-art tool for Computational Group Theory is GAP. ( I might give some GAP demo's soon. )

POLL-3 -> POLL-3.1

I kindly ask those who voted on POLL-3 to vote again. I had to delete and then add the poll again because a running POLL can't be modified in Blogger. Why the change? I forgot two rather important, open source, programs: SAGE and GAP. SAGE uses GAP as its Group Theory kernel. I have experimented a lot with GAP in the past. Currently I am still in a Mathematica phase. There are many other excellent programs around but I think that the list is fairly complete for what one might use when preparing a TMA.

Mathematica 7

Wolfram released a new release of Mathematica: 7. More here.

One of the new features is Finite Group Theory. I'll have a look how it compares to GAP. More later.

Helpful people

Yesterday evening I ran out of ideas on how to implement something in GAP. Intuitively I knew it was possible. Within my understanding of mathematics, anything must bepossible with GAP, I mean GAP is used by professional researchers who speak GAP. Today, when I took a short break at work, I posted my question in the GAP forum. I immediately received two replies and that happened before. There is always someone who helps you out. Somehow that means a lot to me, it's why I believe in the future despite the tremendous problems we have in our world.

Dear Nilo,
You have a ring R, and you want to treat it as a group. This group is
written additively in GAP; so you must convert it to a multiplicative
group first.

Here is a simple example of what you (probably) want to do. Hope it helps:

gap> r := GroupRing(GF(2),SymmetricGroup(3));

gap> rplus := Group(List(Basis(r),AdditiveElementAsMultiplicativeElement));

gap> StructureDescription(rplus);
"C2 x C2 x C2 x C2 x C2 x C2"
--
Laurent Bartholdi \ laurent.bartholdi*****com
EPFL SB SMA IMB MAD \ Téléphone: +41 21-*******
Station 8 \ Secrétaire: +41 21-*******
CH-1015 Lausanne, Switzerland \ Fax: +41 21-*******

Thanks again Mr. Bartholdi.