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
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
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()
])