def example(self):
"""
EXAMPLES::
sage: Groups().example()
General Linear Group of degree 4 over Rational Field
"""
from sage.rings.rational_field import QQ
from sage.groups.matrix_gps.linear import GL
return GL(4, QQ)
def example(self):
"""
Returns an example of finite group, as per
:meth:`Category.example`.
EXAMPLES::
sage: G = FiniteGroups().example(); G
General Linear Group of degree 2 over Finite Field of size 3
"""
from sage.groups.matrix_gps.linear import GL
return GL(2, 3)
def __setstate__(self, state):
"""
Restore from old pickle.
EXAMPLES::
sage: from sage.groups.group import Group
sage: from sage.groups.matrix_gps.pickling_overrides import LegacyGeneralLinearGroup
sage: state = dict()
sage: state['_MatrixGroup_gap__n'] = 2
sage: state['_MatrixGroup_gap__R'] = ZZ
sage: M = Group.__new__(LegacyGeneralLinearGroup)
sage: M.__setstate__(state)
sage: M
General Linear Group of degree 2 over Integer Ring
"""
ring = state['_MatrixGroup_gap__R']
n = state['_MatrixGroup_gap__n']
G = GL(n, ring)
self.__init__(G.degree(), G.base_ring(), G._special, G._name_string, G._latex_string)
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 ProjectiveLine(p, r=None, u=None):
r"""
Return the projective line with action given by the matrices ``r``
and ``u`` on the projective line over the field with ``p`` elements.
If ``r`` and ``u`` are None, then r and u are choosen to be
.. MATH::
r: z \mapsto z+1 \qquad u: z \mapsto 1/z
The monodromy of this origami is included in `PGL(2,\ZZ/p\ZZ)`.
EXAMPLES::
sage: from surface_dynamics.all import *
By default, the matrix used to generate the projective line origami are
given by `z -> z+1` and `z -> 1/z` which gives a family of origamis
which belongs to the strata ``H(2^k)``::
sage: o = origamis.ProjectiveLine(3); o
Projective line origami on GF(3)
r = [1 1] u = [0 1]
[0 1] [1 0]
sage: o.stratum()
H_2(2)
sage: o.veech_group().index()
9
sage: o.sum_of_lyapunov_exponents()
4/3
sage: o = origamis.ProjectiveLine(5); o
Projective line origami on GF(5)
r = [1 1] u = [0 1]
[0 1] [1 0]
sage: o.stratum()
H_3(2^2)
sage: o.veech_group().index()
9
sage: o.sum_of_lyapunov_exponents()
5/3
sage: o = origamis.ProjectiveLine(7); o
Projective line origami on GF(7)
r = [1 1] u = [0 1]
[0 1] [1 0]
sage: o.stratum()
H_3(2^2)
sage: o.veech_group().index()
45
sage: o.sum_of_lyapunov_exponents()
5/3
Any invertible matrix mod p can be choosed to generate the origami::
sage: r = matrix([[1,3],[0,1]])
sage: u = matrix([[1,0],[3,1]])
sage: o = origamis.ProjectiveLine(5,r,u); o
Projective line origami on GF(5)
r = [1 3] u = [1 0]
[0 1] [3 1]
sage: o.veech_group().index()
10
sage: o.sum_of_lyapunov_exponents()
9/5
sage: o.stratum_component()
H_3(4)^hyp
"""
from sage.arith.all import is_prime
from sage.rings.finite_rings.finite_field_constructor import GF
from sage.groups.matrix_gps.linear import GL
from sage.modules.free_module import VectorSpace
p = ZZ(p)
if not p.is_prime():
raise ValueError("p (={}) must be a prime number".format(p))
G = GL(2,GF(p))
V = VectorSpace(GF(p),2)
if r is None:
mr = G([[1,1],[0,1]])
else:
mr = G(r)
if u is None:
mu = G([[0,1],[1,0]])
else:
mu = G(u)
sr = str(mr).split('\n')
su = str(mu).split('\n')
mr = mr
mu = mu
if r is None and u is None:
positions = [(i,0) for i in xrange(p+1)]
else:
positions = None
r = []
u = []
for i in xrange(p):
v = V((i,1)) * mr
if v[1]:
r.append(int(v[0]/v[1]))
else:
r.append(p)
v = V((i,1)) * mu
if v[1]:
u.append(int(v[0]/v[1]))
else:
u.append(p)
v = V((1,0)) * mr
if v[1]:
r.append(int(v[0]/v[1]))
else:
r.append(p)
v = V((1,0)) * mu
if v[1]:
u.append(int(v[0]/v[1]))
else:
u.append(p)
o = Origami(r,u,
as_tuple=True,
positions=positions,
name="Projective line origami on GF(%d)\n r = %s u = %s\n %s %s"%(p,sr[0],su[0],sr[1],su[1]))
return o