def __init__(self, degree, ring):
"""
Initialize ``self``.
INPUT:
- ``degree`` -- integer. The degree of the affine group, that
is, the dimension of the affine space the group is acting on
naturally.
- ``ring`` -- a ring. The base ring of the affine space.
EXAMPLES::
sage: Aff6 = AffineGroup(6, QQ)
sage: Aff6 == Aff6
True
sage: Aff6 != Aff6
False
TESTS::
sage: G = AffineGroup(2, GF(5)); G
Affine Group of degree 2 over Finite Field of size 5
sage: TestSuite(G).run()
"""
self._degree = degree
Group.__init__(self, base=ring)
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, surface, base):
if not surface.is_finite():
raise ValueError("the method only work for finite surfaces")
self._s = surface
self._b = base
Group.__init__(self, category=Groups().Infinite())
def __init__(self, degree, ring):
"""
Initialize ``self``.
INPUT:
- ``degree`` -- integer. The degree of the affine group, that
is, the dimension of the affine space the group is acting on
naturally.
- ``ring`` -- a ring. The base ring of the affine space.
EXAMPLES::
sage: Aff6 = AffineGroup(6, QQ)
sage: Aff6 == Aff6
True
sage: Aff6 != Aff6
False
TESTS::
sage: G = AffineGroup(2, GF(5)); G
Affine Group of degree 2 over Finite Field of size 5
sage: TestSuite(G).run()
"""
self._degree = degree
Group.__init__(self, base=ring)
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, surface, base):
if not surface.is_finite():
raise ValueError("the method only work for finite surfaces")
self._s = surface
self._b = base
Group.__init__(self, category=Groups().Infinite())
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, base_ring):
r"""
TESTS::
sage: from flatsurf.geometry.similarity import SimilarityGroup
sage: TestSuite(SimilarityGroup(QQ)).run()
sage: TestSuite(SimilarityGroup(AA)).run()
"""
self._ring = base_ring
Group.__init__(self, category=Groups().Infinite())
def __init__(self, base_field):
r"""
TESTS::
sage: from flatsurf.geometry.similarity import SimilarityGroup
sage: TestSuite(SimilarityGroup(QQ)).run() # not tested
sage: TestSuite(SimilarityGroup(AA)).run() # not tested
"""
self._field = base_field
# The vector space of vectors
Group.__init__(self, category=Groups().Infinite())
def __init__(self, base_field):
r"""
TESTS::
sage: from flatsurf.geometry.similarity import SimilarityGroup
sage: TestSuite(SimilarityGroup(QQ)).run()
sage: TestSuite(SimilarityGroup(AA)).run()
"""
self._field = base_field
# The vector space of vectors
Group.__init__(self, category=Groups().Infinite())
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 __init__(self, pos, neg=None):
r"""
INPUT:
- ``pos`` - the alphabet of positive letters
- ``neg`` - the alphabet of negative letters
- ``bij`` - bijection between positive and negative letters
"""
Group.__init__(self, category=(Groups(),InfiniteEnumeratedSets()))
self._pos = pos
self._neg = neg
self._invert = {}
self._invert.update(zip(pos,neg))
self._invert.update(zip(neg,pos))
self._alphabet = build_alphabet(pos.list() + neg.list())
def __init__(self, degree, ring):
"""
Initialize ``self``.
INPUT:
- ``degree`` -- integer. The degree of the affine group, that
is, the dimension of the affine space the group is acting on
naturally.
- ``ring`` -- a ring. The base ring of the affine space.
EXAMPLES::
sage: Aff6 = AffineGroup(6, QQ)
sage: Aff6 == Aff6
True
sage: Aff6 != Aff6
False
TESTS::
sage: G = AffineGroup(2, GF(5)); G
Affine Group of degree 2 over Finite Field of size 5
sage: TestSuite(G).run()
sage: G.category()
Category of finite groups
sage: Aff6 = AffineGroup(6, QQ)
sage: Aff6.category()
Category of infinite groups
"""
self._degree = degree
cat = Groups()
if degree == 0 or ring in Rings().Finite():
cat = cat.Finite()
elif ring in Rings().Infinite():
cat = cat.Infinite()
self._GL = GL(degree, ring)
Group.__init__(self, base=ring, category=cat)
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 __call__(self, *args, **kwds):
r"""
Call method. This exists *purely* to override the old-fashioned
behaviour of the parent AbelianGroup class and ensure that
:meth:`element_constructor` gets called.
EXAMPLE::
sage: K.<b> = NumberField(x^2 + 389)
sage: C = K.class_group()
sage: C(K.ideal(b))
Trivial principal fractional ideal class
"""
return Group.__call__(self, *args, **kwds)
def __call__(self, *args, **kwds):
r"""
Call method. This exists *purely* to override the old-fashioned
behaviour of the parent AbelianGroup class and ensure that
:meth:`element_constructor` gets called.
EXAMPLE::
sage: K.<b> = NumberField(x^2 + 389)
sage: C = K.class_group()
sage: C(K.ideal(b))
Trivial principal fractional ideal class
"""
return Group.__call__(self, *args, **kwds)
def __call__(self, *args, **kwds):
r"""
Call method.
EXAMPLES::
sage: K.<a> = QuadraticField(-14)
sage: I = K.ideal(2,a)
sage: S = (I,)
sage: CS = K.S_class_group(S)
sage: J = K.ideal(7,a)
sage: G = K.ideal(3,a+1)
sage: CS(I)
Trivial S-ideal class
sage: CS(J)
Trivial S-ideal class
sage: CS(G)
Fractional S-ideal class (3, a + 1)
"""
return Group.__call__(self, *args, **kwds)
def __call__(self, *args, **kwds):
r"""
Call method.
EXAMPLES::
sage: K.<a> = QuadraticField(-14)
sage: I = K.ideal(2,a)
sage: S = (I,)
sage: CS = K.S_class_group(S)
sage: J = K.ideal(7,a)
sage: G = K.ideal(3,a+1)
sage: CS(I)
Trivial S-ideal class
sage: CS(J)
Trivial S-ideal class
sage: CS(G)
Fractional S-ideal class (3, a + 1)
"""
return Group.__call__(self, *args, **kwds)
def __init__(self):
Group.__init__(self, category=Groups().Infinite())
def __init__(self, base_field):
self._f=base_field
# The vector space of vectors
self._vs = VectorSpace(self._f,2)
Group.__init__(self, category=Groups().Infinite())
def __init__(self):
Group.__init__(self, category=Groups().Infinite())
