def subgroup(self, generators, check=True):
"""
Return the subgroup generated by the given generators.
INPUT:
- ``generators`` -- a list/tuple/iterable of group elements of self
- ``check`` -- boolean (optional, default: ``True``). Whether to check that each matrix is invertible.
OUTPUT: The subgroup generated by ``generators`` as an instance of FinitelyGeneratedMatrixGroup_gap
EAMPLES::
sage: UCF = UniversalCyclotomicField()
sage: G = GL(3, UCF)
sage: e3 = UCF.gen(3); e5 =UCF.gen(5)
sage: m = matrix(UCF, 3,3, [[e3, 1, 0], [0, e5, 7],[4, 3, 2]])
sage: S = G.subgroup([m]); S
Subgroup with 1 generators (
[E(3) 1 0]
[ 0 E(5) 7]
[ 4 3 2]
) of General Linear Group of degree 3 over Universal Cyclotomic Field
sage: CF3 = CyclotomicField(3)
sage: G = GL(3, CF3)
sage: e3 = CF3.gen()
sage: m = matrix(CF3, 3,3, [[e3, 1, 0], [0, ~e3, 7],[4, 3, 2]])
sage: S = G.subgroup([m]); S
Subgroup with 1 generators (
[ zeta3 1 0]
[ 0 -zeta3 - 1 7]
[ 4 3 2]
) of General Linear Group of degree 3 over Cyclotomic Field of order 3 and degree 2
TESTS::
sage: TestSuite(G).run()
sage: TestSuite(S).run()
"""
# this method enlarges the method with same name of ParentLibGAP to cases where the ambient group is not inheritet from ParentLibGAP.
if isinstance(self, ParentLibGAP):
return ParentLibGAP.subgroup(self, generators)
for g in generators:
if g not in self:
raise ValueError("generator %s is not in the group"%(g))
from sage.groups.matrix_gps.finitely_generated import MatrixGroup
subgroup = MatrixGroup(generators, check=check)
subgroup._ambient = self
return subgroup
def image_mod_n(self):
r"""
Return the image of this group in `SL(2, \ZZ / N\ZZ)`.
EXAMPLE::
sage: Gamma0(3).image_mod_n()
Matrix group over Ring of integers modulo 3 with 2 generators (
[2 0] [1 1]
[0 2], [0 1]
)
TEST::
sage: for n in [2..20]:
... for g in Gamma0(n).gamma_h_subgroups():
... G = g.image_mod_n()
... assert G.order() == Gamma(n).index() / g.index()
"""
N = self.level()
if N == 1:
raise NotImplementedError(
"Matrix groups over ring of integers modulo 1 not implemented")
gens = [
matrix(Zmod(N), 2, 2, [x, 0, 0, Zmod(N)(1) / x])
for x in self._generators_for_H()
]
gens += [matrix(Zmod(N), 2, [1, 1, 0, 1])]
return MatrixGroup(gens)
def as_matrix_group(self):
"""
Return a new matrix group from the generators.
This will throw away any extra structure (encoded in a derived
class) that a group of special matrices has.
EXAMPLES::
sage: G = SU(4,GF(5))
sage: G.as_matrix_group()
Matrix group over Finite Field in a of size 5^2 with 2 generators (
[ a 0 0 0] [ 1 0 4*a + 3 0]
[ 0 2*a + 3 0 0] [ 1 0 0 0]
[ 0 0 4*a + 1 0] [ 0 2*a + 4 0 1]
[ 0 0 0 3*a], [ 0 3*a + 1 0 0]
)
sage: G = GO(3,GF(5))
sage: G.as_matrix_group()
Matrix group over Finite Field of size 5 with 2 generators (
[2 0 0] [0 1 0]
[0 3 0] [1 4 4]
[0 0 1], [0 2 1]
)
"""
from sage.groups.matrix_gps.finitely_generated import MatrixGroup
return MatrixGroup(self.gens())
def image_mod_n(self):
r"""
Return the image of this group modulo `N`, as a subgroup of `SL(2, \ZZ
/ N\ZZ)`. This is just the trivial subgroup.
EXAMPLE::
sage: Gamma(3).image_mod_n()
Matrix group over Ring of integers modulo 3 with 1 generators (
[1 0]
[0 1]
)
"""
return MatrixGroup([matrix(Zmod(self.level()), 2, 2, 1)])
def matrix_group(self):
"""
Return the matrix group corresponding to ``self``.
EXAMPLES::
sage: C = ColoredPermutations(4, 3)
sage: C.matrix_group()
Matrix group over Cyclotomic Field of order 4 and degree 2 with 3 generators (
[0 1 0] [1 0 0] [ 1 0 0]
[1 0 0] [0 0 1] [ 0 1 0]
[0 0 1], [0 1 0], [ 0 0 zeta4]
)
"""
from sage.groups.matrix_gps.finitely_generated import MatrixGroup
return MatrixGroup([g.to_matrix() for g in self.gens()])
def automorphism_group(self):
"""
Return the group of automorphisms of the quadratic form.
OUTPUT: a :class:`MatrixGroup`
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.automorphism_group()
Matrix group over Rational Field with 3 generators (
[-1 0 0] [0 0 1] [ 0 0 1]
[ 0 -1 0] [0 1 0] [-1 0 0]
[ 0 0 -1], [1 0 0], [ 0 1 0]
)
::
sage: DiagonalQuadraticForm(ZZ, [1,3,5,7]).automorphism_group()
Matrix group over Rational Field with 4 generators (
[-1 0 0 0] [ 1 0 0 0] [ 1 0 0 0] [ 1 0 0 0]
[ 0 -1 0 0] [ 0 -1 0 0] [ 0 1 0 0] [ 0 1 0 0]
[ 0 0 -1 0] [ 0 0 1 0] [ 0 0 -1 0] [ 0 0 1 0]
[ 0 0 0 -1], [ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 -1]
)
The smallest possible automorphism group has order two, since we
can always change all signs::
sage: Q = QuadraticForm(ZZ, 3, [2, 1, 2, 2, 1, 3])
sage: Q.automorphism_group()
Matrix group over Rational Field with 1 generators (
[-1 0 0]
[ 0 -1 0]
[ 0 0 -1]
)
"""
self._compute_automorphisms()
from sage.matrix.matrix_space import MatrixSpace
from sage.groups.matrix_gps.finitely_generated import MatrixGroup
MS = MatrixSpace(self.base_ring().fraction_field(), self.dim(), self.dim())
gens = [MS(x.sage()) for x in self.__automorphisms_pari]
return MatrixGroup(gens)
def subgroup(self, generators, check=True):
"""
Return the subgroup generated by the given generators.
INPUT:
- ``generators`` -- a list/tuple/iterable of group elements of self
- ``check`` -- boolean (optional, default: ``True``). Whether to check that each matrix is invertible.
OUTPUT: The subgroup generated by ``generators`` as an instance of FinitelyGeneratedMatrixGroup_gap
EXAMPLES::
sage: UCF = UniversalCyclotomicField()
sage: G = GL(3, UCF)
sage: e3 = UCF.gen(3); e5 =UCF.gen(5)
sage: m = matrix(UCF, 3,3, [[e3, 1, 0], [0, e5, 7],[4, 3, 2]])
sage: S = G.subgroup([m]); S
Subgroup with 1 generators (
[E(3) 1 0]
[ 0 E(5) 7]
[ 4 3 2]
) of General Linear Group of degree 3 over Universal Cyclotomic Field
sage: CF3 = CyclotomicField(3)
sage: G = GL(3, CF3)
sage: e3 = CF3.gen()
sage: m = matrix(CF3, 3,3, [[e3, 1, 0], [0, ~e3, 7],[4, 3, 2]])
sage: S = G.subgroup([m]); S
Subgroup with 1 generators (
[ zeta3 1 0]
[ 0 -zeta3 - 1 7]
[ 4 3 2]
) of General Linear Group of degree 3 over Cyclotomic Field of order 3 and degree 2
TESTS::
sage: TestSuite(G).run()
sage: TestSuite(S).run()
sage: W = CoxeterGroup(['I',7])
sage: s = W.simple_reflections()
sage: G = W.subgroup([s[1]])
sage: G.category()
Category of finite groups
sage: W = WeylGroup(['A',2])
sage: s = W.simple_reflections()
sage: G = W.subgroup([s[1]])
sage: G.category()
Category of finite groups
"""
try:
test = self.is_finite()
except NotImplementedError:
test = self in Groups().Finite()
cat = Groups().Finite() if test else Groups()
# this method enlarges the method with same name of
# ParentLibGAP to cases where the ambient group is not
# inherited from ParentLibGAP.
if isinstance(self, ParentLibGAP):
return ParentLibGAP.subgroup(self, generators)
for g in generators:
if g not in self:
raise ValueError("generator %s is not in the group" % (g))
from sage.groups.matrix_gps.finitely_generated import MatrixGroup
subgroup = MatrixGroup(generators, check=check, category=cat)
subgroup._ambient = self
return subgroup
