abelian_group_element / dual_abelian_group_element classes need rewrite

[boothby]

Implementation uses slow algorithms, where O(1) algorithms are easy.

[boothby]

Problems are more serious than I originally thought. I would now describe AbelianGroup? and AbelianGroupElement? as "severely broken".

sage: A = AbelianGroup(1,[0],names="a")
sage: a = A.gens()[0]
sage: B = A.subgroup([a,a,a,a]); B
Multiplicative Abelian Group isomorphic to Z x Z x Z x Z, which is the 
subgroup of
Multiplicative Abelian Group isomorphic to Z 
generated by [a, a, a, a]

Z⁴ is a subgroup of Z???

sage: a^-1
Exception (click to the left for traceback):
...
TypeError: unsupported operand type(s) for %: 'int' and 'PlusInfinity'
sage: ~a
Exception (click to the left for traceback):
...
AttributeError: 'AbelianGroup_class' object has no attribute 'fraction_field'

You can't compute the inverse of a group element?

[mabshoff]

#1849 is a rewrite of this code, so please check what is going on over there.

Cheers,

Michael

[wdj]

Unfortunately, there is still (in sage 3.0) a serious lack of functionality for infinite abelian groups. I don't see an easy fix for this bug. Currently, it is AbelianSemiGroupExceptWhenFiniteThenItIsAbelianGroup_class:-)

[mabshoff]

I am closing this ticket as a duplicate of #1849.

Cheers,

Michael