def _minimize_level(G):
r"""
Utility function. Given a matrix group `G` contained in `SL(2, \ZZ / N\ZZ)`
for some `N`, test whether or not `G` is the preimage of a subgroup of
smaller level, and if so, return that subgroup.
The trivial group is handled specially: instead of returning a group, it
returns an integer `N`, representing the trivial subgroup of `SL(2, \ZZ /
N\ZZ)`.
EXAMPLE::
sage: M = MatrixSpace(Zmod(9), 2, 2)
sage: G = MatrixGroup([M(x) for x in [[1,1,0,1],[1,3,0,1],[1,0,3,1],[4,0,0,7]]]); G
Matrix group over Ring of integers modulo 9 with 4 generators (
[1 1] [1 3] [1 0] [4 0]
[0 1], [0 1], [3 1], [0 7]
)
sage: sage.modular.arithgroup.congroup_generic._minimize_level(G)
Matrix group over Ring of integers modulo 3 with 1 generators (
[1 1]
[0 1]
)
sage: G = MatrixGroup([M(x) for x in [[1,3,0,1],[1,0,3,1],[4,0,0,7]]]); G
Matrix group over Ring of integers modulo 9 with 3 generators (
[1 3] [1 0] [4 0]
[0 1], [3 1], [0 7]
)
sage: sage.modular.arithgroup.congroup_generic._minimize_level(G)
3
"""
from congroup_gamma import Gamma_constructor as Gamma
Glist = list(G)
N = G.base_ring().characteristic()
i = Gamma(N).index()
for p in N.prime_divisors():
j = Gamma(N // p).index()
k = len(
[g for g in Glist if g.matrix().change_ring(Zmod(N // p)) == 1])
if k == i // j:
if N // p == 1:
return ZZ(1)
H = MatrixGroup(
[g.matrix().change_ring(Zmod(N // p)) for g in G.gens()])
return _minimize_level(H)
# now sanitize the generators (remove duplicates and copies of the identity)
new_gens = [x.matrix() for x in G.gens() if x.matrix() != 1]
all([x.set_immutable() for x in new_gens])
new_gens = list(Set(new_gens))
if new_gens == []:
return ZZ(G.base_ring().characteristic())
return MatrixGroup(new_gens)
def _minimize_level(G):
r"""
Utility function. Given a matrix group `G` contained in `SL(2, \ZZ / N\ZZ)`
for some `N`, test whether or not `G` is the preimage of a subgroup of
smaller level, and if so, return that subgroup.
The trivial group is handled specially: instead of returning a group, it
returns an integer `N`, representing the trivial subgroup of `SL(2, \ZZ /
N\ZZ)`.
EXAMPLE::
sage: M = MatrixSpace(Zmod(9), 2, 2)
sage: G = MatrixGroup([M(x) for x in [[1,1,0,1],[1,3,0,1],[1,0,3,1],[4,0,0,7]]]); G
Matrix group over Ring of integers modulo 9 with 4 generators (
[1 1] [1 3] [1 0] [4 0]
[0 1], [0 1], [3 1], [0 7]
)
sage: sage.modular.arithgroup.congroup_generic._minimize_level(G)
Matrix group over Ring of integers modulo 3 with 1 generators (
[1 1]
[0 1]
)
sage: G = MatrixGroup([M(x) for x in [[1,3,0,1],[1,0,3,1],[4,0,0,7]]]); G
Matrix group over Ring of integers modulo 9 with 3 generators (
[1 3] [1 0] [4 0]
[0 1], [3 1], [0 7]
)
sage: sage.modular.arithgroup.congroup_generic._minimize_level(G)
3
"""
from .congroup_gamma import Gamma_constructor as Gamma
Glist = list(G)
N = G.base_ring().characteristic()
i = Gamma(N).index()
for d in N.divisors()[:-1]:
j = Gamma(d).index()
k = len([g for g in Glist if g.matrix().change_ring(Zmod(d)) == 1])
if k == i // j:
if d == 1:
return ZZ(1)
G = MatrixGroup([g.matrix().change_ring(Zmod(d)) for g in G.gens()])
N = d
break
# now sanitize the generators (remove duplicates and copies of the identity)
new_gens = [x.matrix() for x in G.gens() if x.matrix() != 1]
all([x.set_immutable() for x in new_gens])
new_gens = list(Set(new_gens))
if new_gens == []:
return ZZ(N)
return MatrixGroup(new_gens)
def pushforward(self, J, *args, **kwds):
"""
The image of an element or a subgroup.
INPUT:
``J`` -- a subgroup or an element of the domain of ``self``
OUTPUT:
The image of ``J`` under ``self``.
.. NOTE::
``pushforward`` is the method that is used when a map is called
on anything that is not an element of its domain. For historical
reasons, we keep the alias ``image()`` for this method.
EXAMPLES::
sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: F.multiplicative_generator()
3
sage: G = MatrixGroup([MS([3,0,0,1])])
sage: a = G.gens()[0]^2
sage: phi = G.hom([a])
sage: phi.image(G.gens()[0]) # indirect doctest
[2 0]
[0 1]
sage: H = MatrixGroup([MS(a.list())])
sage: H
Matrix group over Finite Field of size 7 with 1 generators (
[2 0]
[0 1]
)
The following tests against :trac:`10659`::
sage: phi(H) # indirect doctest
Matrix group over Finite Field of size 7 with 1 generators (
[4 0]
[0 1]
)
"""
phi = self.gap()
F = self.codomain().base_ring()
gapJ = to_libgap(J)
if gapJ.IsGroup():
from sage.groups.matrix_gps.all import MatrixGroup
img_gens = [
x.matrix(F) for x in phi.Image(gapJ).GeneratorsOfGroup()
]
return MatrixGroup(img_gens)
C = self.codomain()
return C(phi.Image(gapJ).matrix(F))
def to_even_subgroup(self):
r"""
Return the smallest even subgroup of `SL(2, \ZZ)` containing self.
EXAMPLES::
sage: G = Gamma(3)
sage: G.to_even_subgroup()
Congruence subgroup of SL(2,Z) of level 3, preimage of:
Matrix group over Ring of integers modulo 3 with 1 generators (
[2 0]
[0 2]
)
"""
if self.is_even():
return self
else:
G = self.image_mod_n()
H = MatrixGroup([ g.matrix() for g in G.gens()] + [G.matrix_space()(-1)])
return CongruenceSubgroup_constructor(H)
def kernel(self):
"""
Return the kernel of ``self``, i.e., a matrix group.
EXAMPLES::
sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: F.multiplicative_generator()
3
sage: G = MatrixGroup([MS([3,0,0,1])])
sage: a = G.gens()[0]^2
sage: phi = G.hom([a])
sage: phi.kernel()
Matrix group over Finite Field of size 7 with 1 generators (
[6 0]
[0 1]
)
"""
gap_ker = self.gap().Kernel()
F = self.domain().base_ring()
from sage.groups.matrix_gps.all import MatrixGroup
return MatrixGroup([x.matrix(F) for x in gap_ker.GeneratorsOfGroup()])
def CongruenceSubgroup_constructor(*args):
r"""
Attempt to create a congruence subgroup from the given data.
The allowed inputs are as follows:
- A :class:`~sage.groups.matrix_gps.matrix_group.MatrixGroup` object. This
must be a group of matrices over `\ZZ / N\ZZ` for some `N`, with
determinant 1, in which case the function will return the group of
matrices in `SL(2, \ZZ)` whose reduction mod `N` is in the given group.
- A list of matrices over `\ZZ / N\ZZ` for some `N`. The function will then
compute the subgroup of `SL(2, \ZZ)` generated by these matrices, and
proceed as above.
- An integer `N` and a list of matrices (over any ring coercible to `\ZZ /
N\ZZ`, e.g. over `\ZZ`). The matrices will then be coerced to `\ZZ /
N\ZZ`.
The function checks that the input G is valid. It then tests to see if
`G` is the preimage mod `N` of some group of matrices modulo a proper
divisor `M` of `N`, in which case it replaces `G` with this group before
continuing.
EXAMPLES::
sage: from sage.modular.arithgroup.congroup_generic import CongruenceSubgroup_constructor as CS
sage: CS(2, [[1,1,0,1]])
Congruence subgroup of SL(2,Z) of level 2, preimage of:
Matrix group over Ring of integers modulo 2 with 1 generators (
[1 1]
[0 1]
)
sage: CS([matrix(Zmod(2), 2, [1,1,0,1])])
Congruence subgroup of SL(2,Z) of level 2, preimage of:
Matrix group over Ring of integers modulo 2 with 1 generators (
[1 1]
[0 1]
)
sage: CS(MatrixGroup([matrix(Zmod(2), 2, [1,1,0,1])]))
Congruence subgroup of SL(2,Z) of level 2, preimage of:
Matrix group over Ring of integers modulo 2 with 1 generators (
[1 1]
[0 1]
)
sage: CS(SL(2, 2))
Modular Group SL(2,Z)
Some invalid inputs::
sage: CS(SU(2, 7))
Traceback (most recent call last):
...
TypeError: Ring of definition must be Z / NZ for some N
"""
from sage.groups.matrix_gps.matrix_group import is_MatrixGroup
if is_MatrixGroup(args[0]):
G = args[0]
elif type(args[0]) == type([]):
G = MatrixGroup(args[0])
elif args[0] in ZZ:
M = MatrixSpace(Zmod(args[0]), 2)
G = MatrixGroup([M(x) for x in args[1]])
R = G.matrix_space().base_ring()
if not hasattr(R, "cover_ring") or R.cover_ring() != ZZ:
raise TypeError, "Ring of definition must be Z / NZ for some N"
if not all([x.matrix().det() == 1 for x in G.gens()]):
raise ValueError, "Group must be contained in SL(2, Z / N)"
GG = _minimize_level(G)
if GG in ZZ:
from all import Gamma
return Gamma(GG)
else:
return CongruenceSubgroupFromGroup(GG)
def CongruenceSubgroup_constructor(*args):
r"""
Attempt to create a congruence subgroup from the given data.
The allowed inputs are as follows:
- A :class:`~sage.groups.matrix_gps.matrix_group.MatrixGroup` object. This
must be a group of matrices over `\ZZ / N\ZZ` for some `N`, with
determinant 1, in which case the function will return the group of
matrices in `SL(2, \ZZ)` whose reduction mod `N` is in the given group.
- A list of matrices over `\ZZ / N\ZZ` for some `N`. The function will then
compute the subgroup of `SL(2, \ZZ)` generated by these matrices, and
proceed as above.
- An integer `N` and a list of matrices (over any ring coercible to `\ZZ /
N\ZZ`, e.g. over `\ZZ`). The matrices will then be coerced to `\ZZ /
N\ZZ`.
The function checks that the input G is valid. It then tests to see if
`G` is the preimage mod `N` of some group of matrices modulo a proper
divisor `M` of `N`, in which case it replaces `G` with this group before
continuing.
EXAMPLES::
sage: from sage.modular.arithgroup.congroup_generic import CongruenceSubgroup_constructor as CS
sage: CS(2, [[1,1,0,1]])
Congruence subgroup of SL(2,Z) of level 2, preimage of:
Matrix group over Ring of integers modulo 2 with 1 generators (
[1 1]
[0 1]
)
sage: CS([matrix(Zmod(2), 2, [1,1,0,1])])
Congruence subgroup of SL(2,Z) of level 2, preimage of:
Matrix group over Ring of integers modulo 2 with 1 generators (
[1 1]
[0 1]
)
sage: CS(MatrixGroup([matrix(Zmod(2), 2, [1,1,0,1])]))
Congruence subgroup of SL(2,Z) of level 2, preimage of:
Matrix group over Ring of integers modulo 2 with 1 generators (
[1 1]
[0 1]
)
sage: CS(SL(2, 2))
Modular Group SL(2,Z)
Some invalid inputs::
sage: CS(SU(2, 7))
Traceback (most recent call last):
...
TypeError: Ring of definition must be Z / NZ for some N
"""
from sage.groups.matrix_gps.matrix_group import is_MatrixGroup
if is_MatrixGroup(args[0]):
G = args[0]
elif isinstance(args[0], list):
G = MatrixGroup(args[0])
elif args[0] in ZZ:
M = MatrixSpace(Zmod(args[0]), 2)
G = MatrixGroup([M(x) for x in args[1]])
R = G.matrix_space().base_ring()
if not hasattr(R, "cover_ring") or R.cover_ring() != ZZ:
raise TypeError("Ring of definition must be Z / NZ for some N")
if not all([x.matrix().det() == 1 for x in G.gens()]):
raise ValueError("Group must be contained in SL(2, Z / N)")
GG = _minimize_level(G)
if GG in ZZ:
from .all import Gamma
return Gamma(GG)
else:
return CongruenceSubgroupFromGroup(GG)
def QuaternionMatrixGroupGF3():
r"""
The quaternion group as a set of `2\times 2` matrices over `GF(3)`.
OUTPUT:
A matrix group consisting of `2\times 2` matrices with
elements from the finite field of order 3. The group is
the quaternion group, the nonabelian group of order 8 that
is not isomorphic to the group of symmetries of a square
(the dihedral group `D_4`).
.. note::
This group is most easily available via ``groups.matrix.QuaternionGF3()``.
EXAMPLES:
The generators are the matrix representations of the
elements commonly called `I` and `J`, while `K`
is the product of `I` and `J`. ::
sage: from sage.groups.misc_gps.misc_groups import QuaternionMatrixGroupGF3
sage: Q = QuaternionMatrixGroupGF3()
sage: Q.order()
8
sage: aye = Q.gens()[0]; aye
[1 1]
[1 2]
sage: jay = Q.gens()[1]; jay
[2 1]
[1 1]
sage: kay = aye*jay; kay
[0 2]
[1 0]
TESTS::
sage: groups.matrix.QuaternionGF3()
Matrix group over Finite Field of size 3 with 2 generators (
[1 1] [2 1]
[1 2], [1 1]
)
sage: Q = QuaternionMatrixGroupGF3()
sage: QP = Q.as_permutation_group()
sage: QP.is_isomorphic(QuaternionGroup())
True
sage: H = DihedralGroup(4)
sage: H.order()
8
sage: QP.is_abelian(), H.is_abelian()
(False, False)
sage: QP.is_isomorphic(H)
False
"""
from sage.rings.finite_rings.constructor import FiniteField
from sage.matrix.matrix_space import MatrixSpace
from sage.groups.matrix_gps.all import MatrixGroup
MS = MatrixSpace(FiniteField(3), 2)
aye = MS([1, 1, 1, 2])
jay = MS([2, 1, 1, 1])
return MatrixGroup([aye, jay])