Helloooooooooo everybody !
Today I needed to build a cyclic group Z/nZ (and products of them),
and to be able to add elements together.
I screamed for help, Aladin offered his, we tried a few things and
Hell followed.
I report those bugs, as I know next to nothing about groups and
categories, and I have no idea how to fix those myself.... Without
destroying something else :-P
1) I did not know at first how to create Z/nZ in Sage.
* I tried groups.<tab> and found "groups.permutation.Cyclic" but
its elements are permutations, not integers. Which is normal.
* I tried g=groups.presentation.Cyclic(5) and though I do not know
what "presentation" means it seems to be what I need. Thouuuugh g(2)
raises an exception, and I can't make a group element from an integer.
And... Well :
sage: g = groups.presentation.Cyclic(5)
sage: a=g.group_generators()[0]
sage: a*a
a^2
sage: a**80
a^80
* Aladin remembered the existence of IntegerModRing, equivalent to
Integers, both in the global namespace. They seem to do the job,
though none of them is exactly the most natural thing to try when you
want to build Z/nZ. And they are rings while I just want a group. But
they seem to do the job :
sage: G=Integers(5)
sage: G(1), G(6), G(3) + G(3)
(1, 1, 1)
Note : there is also an IntegerMod in the global namespace,
whose docstring reads : "Create an integer modulo n with the given
parent. This is mainly for internal use."
2) Using IntegerModRing, or Integers. Let's begin :
sage: groups.matrix.GL(3,2) in Groups
True
sage: Integers(5) in Groups
False
Then, I tried to create (Z/5Z)^2 from G=Integers(5). But how ?
a) G.cartesian_product()
sage: G2 = G.cartesian_product(G)
sage: G2((3,3))
(3, 3)
sage: G2((3,3))+G2((3,3))
TypeError: unsupported operand type(s) for +:
'CartesianProduct_with_category.element_class' and
'CartesianProduct_with_category.element_class'
sage: G2((3,3))*G2((3,4))
(9, 12)
12 is an unusual thing in Z/5Z. And anyway :
sage: G2((9,12)) == G2((4,2))
False
In particular, I don't get how computing a product LOST the
addition defined on the elements of my two groups/rings/whatever.
b) G.CartesianProduct() (yes, it has a .cartesian_product method
AND a CartesianProduct method)
sage: G.CartesianProduct(G)
TypeError: __init__() takes at least 3 arguments (2 given)
sage: G.CartesianProduct(G,G)
AttributeError: 'IntegerModRing_generic_with_category' object
has no attribute 'parent_class'
c) The global CartesianProduct function
sage: G2=CartesianProduct(G,G)
sage: G2((1,1))
[1, 1]
sage: G2((1,1))+G2((1,1))
TypeError: can only concatenate list (not "CombinatorialObject") to list
sage: G2((1,1))*G2((1,1))
TypeError: unsupported operand type(s) for *:
'CombinatorialObject' and 'CombinatorialObject'
Sooooooo that was my attempt at "creating Z/nZ in Sage". And I gave
up, but I still to do this for my next patch ^^;
Have fuuuuuuuuuuuuuuuuun everybody !!!
Nathann
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