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. 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. -- 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 -~----------~----~----~----~------~----~------~--~---
