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Define a simple Group?

GROUPS:

I apologize for the trivial nature of this question but my early attempts to seek an answer via the Community search engine just takes me to topics about creating Mathematica User Groups.

I am starting to teach myself the elements of Group Theory and I wish to define a very simple group that comprises four elements under the operation of traditional multiplication. Those elements are 1, -1, i, -i.

The in-built Mathematica help has left me baffled so far as it talks of Cycles, Permutation Groups etc. which mean nothing to me at the moment.

Any help for a real beginner will be gratefully received.

Thank you, David

POSTED BY: David Mackay
Answer
5 months ago

What is it you want to do with this group? What exactly would you want that is not what you already have, to wit, the multiplicative cyclic group generated by I (or by -I)?

I ask this because it is difficult to offer direction without a better understanding of what exactly you wish to have or do.

POSTED BY: Daniel Lichtblau
Answer
5 months ago

Hello Daniel.

I am in the very early stages of trying to teach myself some basic Group Theory and I want to use MMA as much as possible in this exercise. I was hoping that I could define a simple group object by specifying the elements of the group and the operations that are relevant to it.

As far as I could get in the online help, all of the built-in functionality seems aimed a much more sophisticated level than mine.

POSTED BY: David Mackay
Answer
5 months ago

Maybe just give your generator(s) a name e.g. z, define relations e.g. z^4-1, and use, say, PolynomialMod to rewrite monomials. This will work in the case of commutative groups. For noncommutative one actually requires serious machinery; maybe most light-weight is to embed into a matrix group.

POSTED BY: Daniel Lichtblau
Answer
5 months ago

You can make a multiplication table:

TableForm[Table[I^n*I^m, {n, 0, 3}, {m, 0, 3}], 
 TableHeadings -> {Table[I^n, {n, 0, 3}], Table[I^n, {n, 0, 3}]}]
POSTED BY: Gianluca Gorni
Answer
5 months ago
POSTED BY: Richard Conrad
Answer
10 days ago

Richard,

Thanks for this. At a first look it seems like a great place for me to start. I got sidetracked onto some other topics since my last post but your response has encouraged me to restart my group theory learning.

Again, many thanks, David

POSTED BY: David Mackay
Answer
9 days ago

Perhaps my post

http://community.wolfram.com/groups/-/m/t/1143419

could be helpful. Regards, Hans

POSTED BY: Hans Dolhaine
Answer
9 days ago

Group Abstract Group Abstract