Esempio n. 1
0
 def multiplicative_subgroups(self):
     r"""
     Return generators for each subgroup of
     `(\ZZ/N\ZZ)^*`.
     
     EXAMPLES::
     
         sage: Integers(5).multiplicative_subgroups()
         ([2], [4], [])
         sage: Integers(15).multiplicative_subgroups()
         ([11, 7], [4, 11], [8], [11], [14], [7], [4], [])
         sage: Integers(2).multiplicative_subgroups()
         ([],)
         sage: len(Integers(341).multiplicative_subgroups())
         80
     """
     from sage.groups.abelian_gps.abelian_group import AbelianGroup
     from sage.misc.misc import mul
     U = self.unit_gens()
     G = AbelianGroup([x.multiplicative_order() for x in U])
     rawsubs = G.subgroups()
     mysubs = []
     for G in rawsubs:
         mysubs.append([])
         for s in G.gens():
             mysubs[-1].append(mul([U[i] ** s.list()[i] for i in xrange(len(U))]))
     return tuple(mysubs) # make it immutable, so that we can cache
Esempio n. 2
0
 def multiplicative_subgroups(self):
     r"""
     Return generators for each subgroup of
     `(\ZZ/N\ZZ)^*`.
     
     EXAMPLES::
     
         sage: Integers(5).multiplicative_subgroups()
         ([2], [4], [])
         sage: Integers(15).multiplicative_subgroups()
         ([11, 7], [4, 11], [8], [11], [14], [7], [4], [])
         sage: Integers(2).multiplicative_subgroups()
         ([],)
         sage: len(Integers(341).multiplicative_subgroups())
         80
     """
     from sage.groups.abelian_gps.abelian_group import AbelianGroup
     from sage.misc.misc import mul
     U = self.unit_gens()
     G = AbelianGroup([x.multiplicative_order() for x in U])
     rawsubs = G.subgroups()
     mysubs = []
     for G in rawsubs:
         mysubs.append([])
         for s in G.gens():
             mysubs[-1].append(
                 mul([U[i]**s.list()[i] for i in xrange(len(U))]))
     return tuple(mysubs)  # make it immutable, so that we can cache
Esempio n. 3
0
def subgroups_of_finite_abelian_group(A):
    assert len(A.invariants()) == len(A.gens())
    B = AbelianGroup(
        A.invariants()
    )  # A is an instance of FGP_Module, which is not a subclass of AbelianGroup
    # The Sage function AbelianGroup.subgroups is *really, really* slow (as of version 8.1)
    for BB in reversed(
            B.subgroups()):  # I prefer to get smaller subgroups first
        yield A.submodule([
            A.sum((gen.exponents()[i] * A.gens()[i]
                   for i in range(len(A.gens())))) for gen in BB.gens()
        ])