2009/5/6 William Stein <wst...@gmail.com>: > > On Wed, May 6, 2009 at 11:33 AM, davidloeffler <dave.loeff...@gmail.com> > wrote: >> >> On May 6, 7.10pm, William Stein wrote: >>>Crap. Thanks for spotting this. Fortunately this is used in only >>> one place in Sage; this one line in congroup_gamma0.py: >>> >>> return [GammaH(N, H) for H in R.multiplicative_subgroups()] >> >> Yes, that was how I noticed this -- my fix for 5250 caused >> multiplicative_subgroups to give a different answer when n = 2, which >> broke the doctest for that function in congroup_gamma0. >> >> On May 6, 7:13 pm, John Cremona <john.crem...@gmail.com> wrote: >>> Do we have a function which returns (Z/NZ)* as an abelian group? >> >> Somehow secretly we do, because it's embedded in the DirichletGroup >> machinery; but nothing that explicitly creates an AbelianGroup object. >> > > Just some clarification. The functionality to do that is not embedded > in the DirichletGroup machinery -- what's embedded is that the > functionality is *used* there, so David knows it must be in Sage > somewhere :-). > > Here's an example of how to get an explicit description of (Z/NZ)^* as > a product of cyclic abelian groups: > > sage: R = Zmod(341) > sage: R.unit_gens() > [156, 34] > sage: [a.multiplicative_order() for a in R.unit_gens()] > [10, 30] > sage: euler_phi(341) > 300 > > Thus (Z/341Z)* = Z/10Z + Z/30Z with the map sending (1,0) to 156 and > (0,1) to 34.
OK, so if anyone did want to implement a class for multiplicative-group-mod-N then all the needed machinery is already there. I had tried r.<tab> ut had not see the unit_gens() function. > > Being able to work with abelian groups in all kinds of different > "arrangements" and with subtle relations between them is one of the > most critical and central tools needed in implementing computational > number theory algorithms. That's why I've put a lot of work into > http://trac.sagemath.org/sage_trac/ticket/5882 lately. I did not know of that work, which looks very interesting (and will perhaps make AbelianGroup, as currently implemented, redundant?) John > > -- William > > > > --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
