[Tutor] determining whether a set is a group

[Daniel Yoo]

On Sat, 28 Apr 2001, Sheila King wrote:

> The axioms (basic rules) for a group are:=20
> ( where the dot (=95) symbolizes some operation...)
>=20
>    1.CLOSURE: If a and b are in the group then a =95 b is also in the gro=
up.=20
>    2.ASSOCIATIVITY: If a, b and c are in the group then (a =95 b) =95 c =
=3D a =95 (b =95
> c).=20
>    3.IDENTITY: There is an element e of the group such that for any eleme=
nt a
> of the group
>      a =95 e =3D e =95 a =3D a.=20
>    4.INVERSES: For any element a of the group there is an element a^(-1) =
such
> that=20
>           a =95 a^(-1) =3D e=20
>           and=20
>           a^(-1) =95 a =3D e=20

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 =3D ([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 =3D 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!