On Sat, 28 Apr 2001, Sheila King wrote:

>  The axioms (basic rules) for a group are: 

>  ( where the dot (•) symbolizes some operation...)

>  

>     1.CLOSURE: If a and b are in the group then a • b is also in the group. 

>     2.ASSOCIATIVITY: If a, b and c are in the group then (a • b) • c = a • (b •

>  c). 

>     3.IDENTITY: There is an element e of the group such that for any element a

>  of the group

>       a • e = e • a = a. 

>     4.INVERSES: For any element a of the group there is an element a^(-1) such

>  that 

>            a • a^(-1) = e 

>            and 

>            a^(-1) • a = e 


Doing stuff with infinite sets looks messy, so let's try working with
finite groups for now.  It appears that a group is a combination of:

    1.  a set of elements.
    2.  some operation that combines any two elements.

There's no built-in that perfectly represents a set in Python, but in a
pinch, the list structure works very well.  Also, Python functions seem
like a great way to represent a group operator.  For example, we can say
that:

###
def addModFour(x, y):
    return (x + y) % 4

mygroup = ([0, 1, 2, 3], addModFour)
###

would be one way to represent a group in Python, as a 2-tuple.  What's
nice is that this representation allows us to test for all four cases
fairly nicely.  Here's a little snippet that hints at a possible way to
test for closure:

###
elements, operator = mygroup
if operator(elements[0], elements[1]) in elements:
    print "Our group might not be closed,"
    print "but at least %s dotted with %s is in our set" % (elements[0],
                                                            elements[1]")
else:
    print "There's no way this is a group, because it's not closed!"
###

Once you get through the for-loops and lists chapter in Teach Yourself
Python, try writing the closure checking function; I think you'll be
pleasantly surprised.


Hope this helps!


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