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 RealProjectiveSpace(n):
"""
Return real `n`-dimensional projective space, as a simplicial set.
This is constructed as the `n`-skeleton of the nerve of the group
of order 2, and therefore has a single non-degenerate simplex in
each dimension up to `n`.
EXAMPLES::
sage: simplicial_sets.RealProjectiveSpace(7)
RP^7
sage: RP5 = simplicial_sets.RealProjectiveSpace(5)
sage: RP5.homology()
{0: 0, 1: C2, 2: 0, 3: C2, 4: 0, 5: Z}
sage: RP5
RP^5
sage: latex(RP5)
RP^{5}
sage: BC2 = simplicial_sets.RealProjectiveSpace(Infinity)
sage: latex(BC2)
RP^{\infty}
"""
if n == Infinity:
X = AbelianGroup([2]).nerve()
X.rename('RP^oo')
X.rename_latex('RP^{\\infty}')
else:
X = RealProjectiveSpace(Infinity).n_skeleton(n)
X.rename('RP^{}'.format(n))
X.rename_latex('RP^{{{}}}'.format(n))
return X
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()
])
def free(index_set=None, names=None, **kwds):
r"""
Return the free commutative group.
INPUT:
- ``index_set`` -- (optional) an index set for the generators; if
an integer, then this represents `\{0, 1, \ldots, n-1\}`
- ``names`` -- a string or list/tuple/iterable of strings
(default: ``'x'``); the generator names or name prefix
EXAMPLES::
sage: Groups.Commutative.free(index_set=ZZ)
Free abelian group indexed by Integer Ring
sage: Groups().Commutative().free(ZZ)
Free abelian group indexed by Integer Ring
sage: Groups().Commutative().free(5)
Multiplicative Abelian group isomorphic to Z x Z x Z x Z x Z
sage: F.<x,y,z> = Groups().Commutative().free(); F
Multiplicative Abelian group isomorphic to Z x Z x Z
"""
from sage.rings.all import ZZ
if names is not None:
if isinstance(names, str):
if ',' not in names and index_set in ZZ:
names = [names + repr(i) for i in range(index_set)]
else:
names = names.split(',')
names = tuple(names)
if index_set is None:
index_set = ZZ(len(names))
if index_set in ZZ:
from sage.groups.abelian_gps.abelian_group import AbelianGroup
return AbelianGroup(index_set, names=names, **kwds)
if index_set in ZZ:
from sage.groups.abelian_gps.abelian_group import AbelianGroup
return AbelianGroup(index_set, **kwds)
from sage.groups.indexed_free_group import IndexedFreeAbelianGroup
return IndexedFreeAbelianGroup(index_set, names=names, **kwds)
def homotopy_group(self, n):
"""
Return the n'th homotopy group of ``self``
INPUT:
- ``n`` - the dimension of the homotopy group to be computed
EXAMPLES::
sage: from sage.interfaces.kenzo import Sphere # optional - kenzo
sage: s2 = Sphere(2) # optional - kenzo
sage: p = s2.cartesian_product(s2) # optional - kenzo
sage: p.homotopy_group(3) # optional - kenzo
Multiplicative Abelian group isomorphic to Z x Z
"""
if n not in ZZ or n < 2:
raise ValueError(
"homotopy groups can only be computed for dimensions greater than 1"
)
lgens = homotopy_list(self._kenzo, n).python()
if lgens is not None:
trgens = [0 if i == 1 else i for i in sorted(lgens)]
return AbelianGroup(trgens)
else:
return AbelianGroup([])
def dft(self, chi = lambda x: x):
"""
A discrete Fourier transform "over `\QQ`" using exact
`N`-th roots of unity.
EXAMPLES::
sage: J = range(6)
sage: A = [ZZ(1) for i in J]
sage: s = IndexedSequence(A,J)
sage: s.dft(lambda x:x^2)
Indexed sequence: [6, 0, 0, 6, 0, 0]
indexed by [0, 1, 2, 3, 4, 5]
sage: s.dft()
Indexed sequence: [6, 0, 0, 0, 0, 0]
indexed by [0, 1, 2, 3, 4, 5]
sage: G = SymmetricGroup(3)
sage: J = G.conjugacy_classes_representatives()
sage: s = IndexedSequence([1,2,3],J) # 1,2,3 are the values of a class fcn on G
sage: s.dft() # the "scalar-valued Fourier transform" of this class fcn
Indexed sequence: [8, 2, 2]
indexed by [(), (1,2), (1,2,3)]
sage: J = AbelianGroup(2,[2,3],names='ab')
sage: s = IndexedSequence([1,2,3,4,5,6],J)
sage: s.dft() # the precision of output is somewhat random and architecture dependent.
Indexed sequence: [21.0000000000000, -2.99999999999997 - 1.73205080756885*I, -2.99999999999999 + 1.73205080756888*I, -9.00000000000000 + 0.0000000000000485744257349999*I, -0.00000000000000976996261670137 - 0.0000000000000159872115546022*I, -0.00000000000000621724893790087 - 0.0000000000000106581410364015*I]
indexed by Multiplicative Abelian group isomorphic to C2 x C3
sage: J = CyclicPermutationGroup(6)
sage: s = IndexedSequence([1,2,3,4,5,6],J)
sage: s.dft() # the precision of output is somewhat random and architecture dependent.
Indexed sequence: [21.0000000000000, -2.99999999999997 - 1.73205080756885*I, -2.99999999999999 + 1.73205080756888*I, -9.00000000000000 + 0.0000000000000485744257349999*I, -0.00000000000000976996261670137 - 0.0000000000000159872115546022*I, -0.00000000000000621724893790087 - 0.0000000000000106581410364015*I]
indexed by Cyclic group of order 6 as a permutation group
sage: p = 7; J = range(p); A = [kronecker_symbol(j,p) for j in J]
sage: s = IndexedSequence(A,J)
sage: Fs = s.dft()
sage: c = Fs.list()[1]; [x/c for x in Fs.list()]; s.list()
[0, 1, 1, -1, 1, -1, -1]
[0, 1, 1, -1, 1, -1, -1]
The DFT of the values of the quadratic residue symbol is itself, up to
a constant factor (denoted c on the last line above).
.. TODO::
Read the parent of the elements of S; if `\QQ` or `\CC` leave as
is; if AbelianGroup, use abelian_group_dual; if some other
implemented Group (permutation, matrix), call .characters()
and test if the index list is the set of conjugacy classes.
"""
J = self.index_object() ## index set of length N
N = len(J)
S = self.list()
F = self.base_ring() ## elements must be coercible into QQ(zeta_N)
if not(J[0] in ZZ):
G = J[0].parent() ## if J is not a range it is a group G
if J[0] in ZZ and F.base_ring().fraction_field()==QQ:
## assumes J is range(N)
zeta = CyclotomicField(N).gen()
FT = [sum([S[i]*chi(zeta**(i*j)) for i in J]) for j in J]
elif not(J[0] in ZZ) and G.is_abelian() and F == ZZ or (F.is_field() and F.base_ring()==QQ):
if is_PermutationGroupElement(J[0]):
## J is a CyclicPermGp
n = G.order()
a = list(factor(n))
invs = [x[0]**x[1] for x in a]
G = AbelianGroup(len(a),invs)
## assumes J is AbelianGroup(...)
Gd = G.dual_group()
FT = [sum([S[i]*chid(G.list()[i]) for i in range(N)])
for chid in Gd]
elif not(J[0] in ZZ) and G.is_finite() and F == ZZ or (F.is_field() and F.base_ring()==QQ):
## assumes J is the list of conj class representatives of a
## PermuationGroup(...) or Matrixgroup(...)
chi = G.character_table()
FT = [sum([S[i]*chi[i,j] for i in range(N)]) for j in range(N)]
else:
raise ValueError("list elements must be in QQ(zeta_"+str(N)+")")
return IndexedSequence(FT,J)
def dft(self, chi = lambda x: x):
"""
A discrete Fourier transform "over `\QQ`" using exact
`N`-th roots of unity.
EXAMPLES::
sage: J = range(6)
sage: A = [ZZ(1) for i in J]
sage: s = IndexedSequence(A,J)
sage: s.dft(lambda x:x^2)
Indexed sequence: [6, 0, 0, 6, 0, 0]
indexed by [0, 1, 2, 3, 4, 5]
sage: s.dft()
Indexed sequence: [6, 0, 0, 0, 0, 0]
indexed by [0, 1, 2, 3, 4, 5]
sage: G = SymmetricGroup(3)
sage: J = G.conjugacy_classes_representatives()
sage: s = IndexedSequence([1,2,3],J) # 1,2,3 are the values of a class fcn on G
sage: s.dft() # the "scalar-valued Fourier transform" of this class fcn
Indexed sequence: [8, 2, 2]
indexed by [(), (1,2), (1,2,3)]
sage: J = AbelianGroup(2,[2,3],names='ab')
sage: s = IndexedSequence([1,2,3,4,5,6],J)
sage: s.dft() # the precision of output is somewhat random and architecture dependent.
Indexed sequence: [21.0000000000000, -2.99999999999997 - 1.73205080756885*I, -2.99999999999999 + 1.73205080756888*I, -9.00000000000000 + 0.0000000000000485744257349999*I, -0.00000000000000976996261670137 - 0.0000000000000159872115546022*I, -0.00000000000000621724893790087 - 0.0000000000000106581410364015*I]
indexed by Multiplicative Abelian group isomorphic to C2 x C3
sage: J = CyclicPermutationGroup(6)
sage: s = IndexedSequence([1,2,3,4,5,6],J)
sage: s.dft() # the precision of output is somewhat random and architecture dependent.
Indexed sequence: [21.0000000000000, -2.99999999999997 - 1.73205080756885*I, -2.99999999999999 + 1.73205080756888*I, -9.00000000000000 + 0.0000000000000485744257349999*I, -0.00000000000000976996261670137 - 0.0000000000000159872115546022*I, -0.00000000000000621724893790087 - 0.0000000000000106581410364015*I]
indexed by Cyclic group of order 6 as a permutation group
sage: p = 7; J = range(p); A = [kronecker_symbol(j,p) for j in J]
sage: s = IndexedSequence(A,J)
sage: Fs = s.dft()
sage: c = Fs.list()[1]; [x/c for x in Fs.list()]; s.list()
[0, 1, 1, -1, 1, -1, -1]
[0, 1, 1, -1, 1, -1, -1]
The DFT of the values of the quadratic residue symbol is itself, up to
a constant factor (denoted c on the last line above).
.. TODO::
Read the parent of the elements of S; if `\QQ` or `\CC` leave as
is; if AbelianGroup, use abelian_group_dual; if some other
implemented Group (permutation, matrix), call .characters()
and test if the index list is the set of conjugacy classes.
"""
J = self.index_object() ## index set of length N
N = len(J)
S = self.list()
F = self.base_ring() ## elements must be coercible into QQ(zeta_N)
if not(J[0] in ZZ):
G = J[0].parent() ## if J is not a range it is a group G
if J[0] in ZZ and F.base_ring().fraction_field()==QQ:
## assumes J is range(N)
zeta = CyclotomicField(N).gen()
FT = [sum([S[i]*chi(zeta**(i*j)) for i in J]) for j in J]
elif not(J[0] in ZZ) and G.is_abelian() and F == ZZ or (F.is_field() and F.base_ring()==QQ):
if is_PermutationGroupElement(J[0]):
## J is a CyclicPermGp
n = G.order()
a = list(factor(n))
invs = [x[0]**x[1] for x in a]
G = AbelianGroup(len(a),invs)
## assumes J is AbelianGroup(...)
Gd = G.dual_group()
FT = [sum([S[i]*chid(G.list()[i]) for i in range(N)])
for chid in Gd]
elif not(J[0] in ZZ) and G.is_finite() and F == ZZ or (F.is_field() and F.base_ring()==QQ):
## assumes J is the list of conj class representatives of a
## PermuationGroup(...) or Matrixgroup(...)
chi = G.character_table()
FT = [sum([S[i]*chi[i,j] for i in range(N)]) for j in range(N)]
else:
raise ValueError("list elements must be in QQ(zeta_"+str(N)+")")
return IndexedSequence(FT,J)
def RealProjectiveSpace(n):
r"""
Return real `n`-dimensional projective space, as a simplicial set.
This is constructed as the `n`-skeleton of the nerve of the group
of order 2, and therefore has a single non-degenerate simplex in
each dimension up to `n`.
EXAMPLES::
sage: simplicial_sets.RealProjectiveSpace(7)
RP^7
sage: RP5 = simplicial_sets.RealProjectiveSpace(5)
sage: RP5.homology()
{0: 0, 1: C2, 2: 0, 3: C2, 4: 0, 5: Z}
sage: RP5
RP^5
sage: latex(RP5)
RP^{5}
sage: BC2 = simplicial_sets.RealProjectiveSpace(Infinity)
sage: latex(BC2)
RP^{\infty}
"""
if n == Infinity:
X = AbelianGroup([2]).nerve()
X.rename('RP^oo')
X.rename_latex('RP^{\\infty}')
else:
X = RealProjectiveSpace(Infinity).n_skeleton(n)
X.rename('RP^{}'.format(n))
X.rename_latex('RP^{{{}}}'.format(n))
return X