On Wed, May 6, 2009 at 10:58 AM, daveloeffler <dave.loeff...@gmail.com> wrote:
>
> Just now I was doing some tinkering in sage.rings.integer_mod_ring
> with the aim of fixing ticket #5250, where Sage wrongly claims that
> (Z / 162Z)^* is non-cyclic when it is. That turned out to be easy to
> fix, but in the process I discovered something more nasty: the method
> "multiplicative_subgroups" for the IntegerModRing class returns
> answers that are wrong.
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()]
But of course that is a potentially important line.
> For example, take N = 341 = 11 * 31. Then (Z / NZ)^* is isomorphic to
> C_10 x C_30. The current method just calculates all subgroups of C_10
> and all subgroups of C_30, and returns the products of those. This is
> clearly wrong, since not all subgroups of a direct product are
> products of subgroups of the factors. In fact in this example there
> are 80 subgroups -- I checked in GAP -- and Sage only finds 32 of
> them.
>
> Does anyone know a good, fast algorithm for solving this problem
> (correctly)? Alternatively, can someone give me a hint on how to use
> Gap for this? It has the right command (SubgroupsSolvableGroup) but
> I'm struggling to work out how to convert Gap's output back into Sage.
If possible, I would greatly prefer not to use GAP for this. As it is,
none of the modular forms code in Sage depends on GAP or any other
external *pexpect* library. This can be really important when doing
parallel computing, which I do a lot when doing research computations.
Of course now the onus is on me to give an algorithm (that works), and
I don't have time right now.
William
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