Thank you for your answer.

A few years ago, iwas only doing mathematics and physics and the kind of exercice to
practice we got was like the following one:
a * b = a+b - a x b
is (Z, *) a group ?
Yes ? no ?
If no how can i modify it to make it a group ?

It would be interesting to make a computer program to solve this kinds of pb.
As you suggest it, i would use a subset, lets say -10 ; 10  (because 0, 1 are often
worth to try as a general rule ) and additionnally i would build a function, that
takes a variable and checks if it belongs to Z


Answer:
if b= 1,   a * b = a+ b - a x b = a + 1 - a x 1 = 1
so a * 1 = 1 whatever 'a' is. so 1 has no inverse, i have to remove it from Z to
have a group....

But to make a  computer think like that is not so easy... :o)
Benoit


Corran Webster wrote:

> 
> 
>  As far as I can see, the best solution to the original problem (given
>  a finite set and the Cayley matrix of an operation on the set, is it
>  a group?) is a brute force checking of the axioms.  For efficiency
>  it's probably best use a dictionary to represent the matrix, rather
>  than a list of lists, and for generality it's probably better to have
>  the algortihm work with the operation as a function, rather than
>  directly with the matrix, but I could be wrong.
> 
>  Eg. (untested code)
> 
>  def cayleyToOp(set, cayley):
>     """Given a set and a Cayley matrix, returns a function which
>  performs the operation on two elements."""
>     matrix = {}
>     for a in range(len(set)):
>       for b in range(len(set)):
>         matrix[(set[a], set[b])] = cayley[a][b]
>     return lambda a, b, matrix=matrix: matrix[(a,b)]
> 
>  Z3 = [0, 1, 2]
>  Z3Cayley = [[0,1,2], [1,2,0], [2,0,1]]
>  Z3op = cayleyToOp(Z3, Z3Cayley)
> 
>  print Z3op(1,1)
>  #should print 2
> 
>  of course Z3op could be more efficiently implimented as
> 
>  def Z3op(a, b):
>     return (a+b) % 3
> 
>  which is why the group checking stuff will be more flexible if you
>  allow functions.
> 
>  >Although i remember what a group is , i don't remember what groups are useful
>  >for....... any math teachers?
> 
>  As it so happens, I teach math at UNLV.  Groups typically come up in
>  situations where there is symmetry of some sort and so can be used to
>  describe the symmetry which is present.  Knowing symmetries can help
>  you find and classify solutions to concrete problems.
> 
>  They are also one of the fundamental building blocks of abstract
>  algebra and most interesting algebraic systems involve some sort of
>  group structure.  The standard addition of numbers is an example of a
>  group.
> 
>  Regards,
>  Corran
> 
>  _______________________________________________
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--
Benoit Dupire
Graduate Student
----------------
I'd like to buy a new Boomerang. How can i get rid of the old one?



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