def __init__(self, generator_names, libgap_free_group=None):
"""
Python constructor.
INPUT:
- ``generator_names`` -- a tuple of strings. The names of the
generators.
- ``libgap_free_group`` -- a LibGAP free group or ``None``
(default). The LibGAP free group to wrap. If ``None``, a
suitable group will be constructed.
TESTS::
sage: G.<a,b> = FreeGroup() # indirect doctest
sage: G
Free Group on generators {a, b}
sage: G.variable_names()
('a', 'b')
"""
n = len(generator_names)
self._assign_names(generator_names)
if libgap_free_group is None:
libgap_free_group = libgap.FreeGroup(generator_names)
ParentLibGAP.__init__(self, libgap_free_group)
Group.__init__(self)
def __init__(self, degree, base_ring, libgap_group, ambient=None, category=None):
"""
Base class for matrix groups that implements GAP interface.
INPUT:
- ``degree`` -- integer. The degree (matrix size) of the
matrix group.
- ``base_ring`` -- ring. The base ring of the matrices.
- ``libgap_group`` -- the defining libgap group.
- ``ambient`` -- A derived class of :class:`ParentLibGAP` or
``None`` (default). The ambient class if ``libgap_group``
has been defined as a subgroup.
TESTS::
sage: from sage.groups.matrix_gps.matrix_group import MatrixGroup_gap
sage: MatrixGroup_gap(2, ZZ, libgap.eval('GL(2, Integers)'))
Matrix group over Integer Ring with 3 generators (
[0 1] [-1 0] [1 1]
[1 0], [ 0 1], [0 1]
)
"""
ParentLibGAP.__init__(self, libgap_group, ambient=ambient)
MatrixGroup_generic.__init__(self, degree, base_ring, category=category)
def __init__(self, generator_names, libgap_free_group=None):
"""
Python constructor.
INPUT:
- ``generator_names`` -- a tuple of strings. The names of the
generators.
- ``libgap_free_group`` -- a LibGAP free group or ``None``
(default). The LibGAP free group to wrap. If ``None``, a
suitable group will be constructed.
TESTS::
sage: G.<a,b> = FreeGroup() # indirect doctest
sage: G
Free Group on generators {a, b}
sage: G.variable_names()
('a', 'b')
"""
n = len(generator_names)
self._assign_names(generator_names)
if libgap_free_group is None:
libgap_free_group = libgap.FreeGroup(generator_names)
ParentLibGAP.__init__(self, libgap_free_group)
Group.__init__(self)
def __init__(self, free_group, relations):
"""
The Python constructor
TESTS::
sage: G = FreeGroup('a, b')
sage: H = G / (G([1]), G([2])^3)
sage: H
Finitely presented group < a, b | a, b^3 >
sage: F = FreeGroup('a, b')
sage: J = F / (F([1]), F([2, 2, 2]))
sage: J is H
True
sage: TestSuite(H).run()
sage: TestSuite(J).run()
"""
from sage.groups.free_group import is_FreeGroup
assert is_FreeGroup(free_group)
assert isinstance(relations, tuple)
self._free_group = free_group
self._relations = relations
self._assign_names(free_group.variable_names())
parent_gap = free_group.gap() / libgap([ rel.gap() for rel in relations])
ParentLibGAP.__init__(self, parent_gap)
Group.__init__(self)
def __init__(self, degree, base_ring, libgap_group, ambient=None, category=None):
"""
Base class for matrix groups that implements GAP interface.
INPUT:
- ``degree`` -- integer. The degree (matrix size) of the
matrix group.
- ``base_ring`` -- ring. The base ring of the matrices.
- ``libgap_group`` -- the defining libgap group.
- ``ambient`` -- A derived class of :class:`ParentLibGAP` or
``None`` (default). The ambient class if ``libgap_group``
has been defined as a subgroup.
TESTS::
sage: from sage.groups.matrix_gps.matrix_group import MatrixGroup_gap
sage: MatrixGroup_gap(2, ZZ, libgap.eval('GL(2, Integers)'))
Matrix group over Integer Ring with 3 generators (
[0 1] [-1 0] [1 1]
[1 0], [ 0 1], [0 1]
)
"""
ParentLibGAP.__init__(self, libgap_group, ambient=ambient)
MatrixGroup_generic.__init__(self, degree, base_ring, category=category)
def __init__(self, free_group, relations):
"""
The Python constructor.
TESTS::
sage: G = FreeGroup('a, b')
sage: H = G / (G([1]), G([2])^3)
sage: H
Finitely presented group < a, b | a, b^3 >
sage: F = FreeGroup('a, b')
sage: J = F / (F([1]), F([2, 2, 2]))
sage: J is H
True
sage: TestSuite(H).run()
sage: TestSuite(J).run()
"""
from sage.groups.free_group import is_FreeGroup
assert is_FreeGroup(free_group)
assert isinstance(relations, tuple)
self._free_group = free_group
self._relations = relations
self._assign_names(free_group.variable_names())
parent_gap = free_group.gap() / libgap(
[rel.gap() for rel in relations])
ParentLibGAP.__init__(self, parent_gap)
Group.__init__(self)
def __init__(self, G, category, ambient=None):
r"""
Create an instance of this class.
See :class:`AbelianGroup_gap` for details
TESTS::
sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap
sage: G = AbelianGroupGap([3,2,5])
sage: TestSuite(G).run()
"""
AbelianGroupBase.__init__(self, category=category)
ParentLibGAP.__init__(self, G, ambient=ambient)
def __init__(self, domain, gap_group, category, ambient=None):
"""
Constructor.
EXAMPLES::
sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap
sage: G = AbelianGroupGap([2,3,4,5])
sage: G.aut()
Full group of automorphisms of Abelian group with gap, generator orders (2, 3, 4, 5)
"""
self._domain = domain
n = len(self._domain.gens())
self._covering_matrix_ring = MatrixSpace(ZZ, n)
ParentLibGAP.__init__(self, gap_group, ambient=ambient)
Group.__init__(self, category=category)
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()
"""
# 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)
subgroup._ambient = self
return subgroup
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 __init__(self, *args, **kwds):
"""
Group interface for LibGAP-based groups.
INPUT:
Same as :class:`~sage.groups.libgap_wrapper.ParentLibGAP`.
TESTS::
sage: F.<a,b> = FreeGroup()
sage: G_gap = libgap.Group([ (a*b^2).gap() ])
sage: from sage.groups.libgap_group import GroupLibGAP
sage: G = GroupLibGAP(G_gap); G
Group([ a*b^2 ])
sage: g = G.gen(0); g
a*b^2
sage: TestSuite(G).run(skip=['_test_pickling', '_test_elements'])
sage: TestSuite(g).run(skip=['_test_pickling'])
"""
ParentLibGAP.__init__(self, *args, **kwds)
Group.__init__(self)
def __init__(self, *args, **kwds):
"""
Group interface for LibGAP-based groups.
INPUT:
Same as :class:`~sage.groups.libgap_wrapper.ParentLibGAP`.
TESTS::
sage: F.<a,b> = FreeGroup()
sage: G_gap = libgap.Group([ (a*b^2).gap() ])
sage: from sage.groups.libgap_group import GroupLibGAP
sage: G = GroupLibGAP(G_gap); G
Group([ a*b^2 ])
sage: g = G.gen(0); g
a*b^2
sage: TestSuite(G).run(skip=['_test_pickling', '_test_elements'])
sage: TestSuite(g).run(skip=['_test_pickling'])
"""
ParentLibGAP.__init__(self, *args, **kwds)
Group.__init__(self)
def __init__(self,
degree,
base_ring,
libgap_group,
ambient=None,
category=None):
"""
Base class for matrix groups that implements GAP interface.
INPUT:
- ``degree`` -- integer. The degree (matrix size) of the
matrix group.
- ``base_ring`` -- ring. The base ring of the matrices.
- ``libgap_group`` -- the defining libgap group.
- ``ambient`` -- A derived class of :class:`ParentLibGAP` or
``None`` (default). The ambient class if ``libgap_group``
has been defined as a subgroup.
TESTS:
::
sage: from sage.groups.matrix_gps.matrix_group import MatrixGroup_gap
sage: MatrixGroup_gap(2, ZZ, libgap.eval('GL(2, Integers)'))
Matrix group over Integer Ring with 3 generators (
[0 1] [-1 0] [1 1]
[1 0], [ 0 1], [0 1]
)
Check that the slowness of GAP iterators and enumerators for matrix groups
(cf. http://tracker.gap-system.org/issues/369) has been fixed::
sage: i = iter(GL(6,5))
sage: [ next(i) for j in range(8) ]
[
[1 0 0 0 0 0] [4 0 0 0 0 1] [0 4 0 0 0 0] [0 4 0 0 0 0]
[0 1 0 0 0 0] [4 0 0 0 0 0] [0 0 4 0 0 0] [0 0 4 0 0 0]
[0 0 1 0 0 0] [0 4 0 0 0 0] [0 0 0 4 0 0] [0 0 0 4 0 0]
[0 0 0 1 0 0] [0 0 4 0 0 0] [0 0 0 0 4 0] [0 0 0 0 4 0]
[0 0 0 0 1 0] [0 0 0 4 0 0] [0 0 0 0 0 4] [0 0 0 0 0 4]
[0 0 0 0 0 1], [0 0 0 0 4 0], [1 4 0 0 0 0], [2 4 0 0 0 0],
[3 0 0 0 0 1] [4 0 0 1 3 3] [0 0 0 2 0 0] [1 0 0 0 4 4]
[3 0 0 0 0 0] [4 0 0 0 3 3] [0 0 0 0 4 0] [1 0 0 0 0 4]
[0 4 0 0 0 0] [3 0 0 0 0 1] [2 2 0 0 0 2] [1 0 0 0 0 0]
[0 0 4 0 0 0] [3 0 0 0 0 0] [1 4 0 0 0 0] [0 1 0 0 0 0]
[0 0 0 4 0 0] [0 4 0 0 0 0] [0 2 4 0 0 0] [0 0 1 0 0 0]
[4 0 0 0 2 3], [2 0 3 4 4 4], [0 0 1 4 0 0], [0 0 0 1 0 0]
]
And the same for listing the group elements, as well as few other issues::
sage: F = GF(3)
sage: gens = [matrix(F,2, [1,0, -1,1]), matrix(F, 2, [1,1,0,1])]
sage: G = MatrixGroup(gens)
sage: G.cardinality()
24
sage: v = G.list()
sage: len(v)
24
sage: v[:5]
(
[0 1] [0 1] [0 1] [0 2] [0 2]
[2 0], [2 1], [2 2], [1 0], [1 1]
)
sage: all(g in G for g in G.list())
True
An example over a ring (see :trac:`5241`)::
sage: M1 = matrix(ZZ,2,[[-1,0],[0,1]])
sage: M2 = matrix(ZZ,2,[[1,0],[0,-1]])
sage: M3 = matrix(ZZ,2,[[-1,0],[0,-1]])
sage: MG = MatrixGroup([M1, M2, M3])
sage: MG.list()
(
[-1 0] [-1 0] [ 1 0] [1 0]
[ 0 -1], [ 0 1], [ 0 -1], [0 1]
)
sage: MG.list()[1]
[-1 0]
[ 0 1]
sage: MG.list()[1].parent()
Matrix group over Integer Ring with 3 generators (
[-1 0] [ 1 0] [-1 0]
[ 0 1], [ 0 -1], [ 0 -1]
)
An example over a field (see :trac:`10515`)::
sage: gens = [matrix(QQ,2,[1,0,0,1])]
sage: MatrixGroup(gens).list()
(
[1 0]
[0 1]
)
Another example over a ring (see :trac:`9437`)::
sage: len(SL(2, Zmod(4)).list())
48
An error is raised if the group is not finite::
sage: GL(2,ZZ).list()
Traceback (most recent call last):
...
NotImplementedError: group must be finite
"""
ParentLibGAP.__init__(self, libgap_group, ambient=ambient)
MatrixGroup_generic.__init__(self,
degree,
base_ring,
category=category)
def __init__(self, degree, base_ring, libgap_group, ambient=None, category=None):
"""
Base class for matrix groups that implements GAP interface.
INPUT:
- ``degree`` -- integer. The degree (matrix size) of the
matrix group.
- ``base_ring`` -- ring. The base ring of the matrices.
- ``libgap_group`` -- the defining libgap group.
- ``ambient`` -- A derived class of :class:`ParentLibGAP` or
``None`` (default). The ambient class if ``libgap_group``
has been defined as a subgroup.
TESTS:
::
sage: from sage.groups.matrix_gps.matrix_group import MatrixGroup_gap
sage: MatrixGroup_gap(2, ZZ, libgap.eval('GL(2, Integers)'))
Matrix group over Integer Ring with 3 generators (
[0 1] [-1 0] [1 1]
[1 0], [ 0 1], [0 1]
)
Check that the slowness of GAP iterators and enumerators for matrix groups
(cf. http://tracker.gap-system.org/issues/369) has been fixed::
sage: i = iter(GL(6,5))
sage: [ next(i) for j in range(8) ]
[
[1 0 0 0 0 0] [4 0 0 0 0 1] [0 4 0 0 0 0] [0 4 0 0 0 0]
[0 1 0 0 0 0] [4 0 0 0 0 0] [0 0 4 0 0 0] [0 0 4 0 0 0]
[0 0 1 0 0 0] [0 4 0 0 0 0] [0 0 0 4 0 0] [0 0 0 4 0 0]
[0 0 0 1 0 0] [0 0 4 0 0 0] [0 0 0 0 4 0] [0 0 0 0 4 0]
[0 0 0 0 1 0] [0 0 0 4 0 0] [0 0 0 0 0 4] [0 0 0 0 0 4]
[0 0 0 0 0 1], [0 0 0 0 4 0], [1 4 0 0 0 0], [2 4 0 0 0 0],
[3 0 0 0 0 1] [4 0 0 1 3 3] [0 0 0 2 0 0] [1 0 0 0 4 4]
[3 0 0 0 0 0] [4 0 0 0 3 3] [0 0 0 0 4 0] [1 0 0 0 0 4]
[0 4 0 0 0 0] [3 0 0 0 0 1] [2 2 0 0 0 2] [1 0 0 0 0 0]
[0 0 4 0 0 0] [3 0 0 0 0 0] [1 4 0 0 0 0] [0 1 0 0 0 0]
[0 0 0 4 0 0] [0 4 0 0 0 0] [0 2 4 0 0 0] [0 0 1 0 0 0]
[4 0 0 0 2 3], [2 0 3 4 4 4], [0 0 1 4 0 0], [0 0 0 1 0 0]
]
And the same for listing the group elements, as well as few other issues::
sage: F = GF(3)
sage: gens = [matrix(F,2, [1,0, -1,1]), matrix(F, 2, [1,1,0,1])]
sage: G = MatrixGroup(gens)
sage: G.cardinality()
24
sage: v = G.list()
sage: len(v)
24
sage: v[:5]
(
[0 1] [0 1] [0 1] [0 2] [0 2]
[2 0], [2 1], [2 2], [1 0], [1 1]
)
sage: all(g in G for g in G.list())
True
An example over a ring (see :trac:`5241`)::
sage: M1 = matrix(ZZ,2,[[-1,0],[0,1]])
sage: M2 = matrix(ZZ,2,[[1,0],[0,-1]])
sage: M3 = matrix(ZZ,2,[[-1,0],[0,-1]])
sage: MG = MatrixGroup([M1, M2, M3])
sage: MG.list()
(
[-1 0] [-1 0] [ 1 0] [1 0]
[ 0 -1], [ 0 1], [ 0 -1], [0 1]
)
sage: MG.list()[1]
[-1 0]
[ 0 1]
sage: MG.list()[1].parent()
Matrix group over Integer Ring with 3 generators (
[-1 0] [ 1 0] [-1 0]
[ 0 1], [ 0 -1], [ 0 -1]
)
An example over a field (see :trac:`10515`)::
sage: gens = [matrix(QQ,2,[1,0,0,1])]
sage: MatrixGroup(gens).list()
(
[1 0]
[0 1]
)
Another example over a ring (see :trac:`9437`)::
sage: len(SL(2, Zmod(4)).list())
48
An error is raised if the group is not finite::
sage: GL(2,ZZ).list()
Traceback (most recent call last):
...
NotImplementedError: group must be finite
"""
ParentLibGAP.__init__(self, libgap_group, ambient=ambient)
MatrixGroup_generic.__init__(self, degree, base_ring, category=category)
