def example(self, G = None):
"""
Return an example of group algebra.
EXAMPLES::
sage: GroupAlgebras(QQ['x']).example()
Group algebra of Dihedral group of order 8 as a permutation group over Univariate Polynomial Ring in x over Rational Field
An other group can be specified as optional argument::
sage: GroupAlgebras(QQ).example(AlternatingGroup(4))
Group algebra of Alternating group of order 4!/2 as a permutation group over Rational Field
"""
from sage.groups.perm_gps.permgroup_named import DihedralGroup
if G is None:
G = DihedralGroup(4)
return G.algebra(self.base_ring())
def example(self, G=None):
"""
Return an example of group algebra.
EXAMPLES::
sage: GroupAlgebras(QQ[x]).example()
Group algebra of Dihedral group of order 8 as a permutation group over Univariate Polynomial Ring in x over Rational Field
An other group can be specified as optional argument::
sage: GroupAlgebras(QQ).example(SymmetricGroup(4))
Group algebra of Symmetric group of order 4! as a permutation group over Rational Field
"""
from sage.groups.perm_gps.permgroup_named import DihedralGroup
if G is None:
G = DihedralGroup(4)
return G.algebra(self.base_ring())
def example(self):
"""
Returns an example of finite permutation group, as per
:meth:`Category.example`.
EXAMPLES::
sage: G = FinitePermutationGroups().example(); G
Dihedral group of order 6 as a permutation group
"""
from sage.groups.perm_gps.permgroup_named import DihedralGroup
return DihedralGroup(3)
def example(self, G = None):
"""
Returns an example of algebra with basis::
sage: HopfAlgebrasWithBasis(QQ['x']).example()
An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Univariate Polynomial Ring in x over Rational Field
An other group can be specified as optional argument::
sage: HopfAlgebrasWithBasis(QQ).example(SymmetricGroup(4))
An example of Hopf algebra with basis: the group algebra of the Symmetric group of order 4! as a permutation group over Rational Field
"""
from sage.categories.examples.hopf_algebras_with_basis import MyGroupAlgebra
from sage.groups.perm_gps.permgroup_named import DihedralGroup
if G is None:
G = DihedralGroup(3)
return MyGroupAlgebra(self.base_ring(), G)
print('Draw the flower with 6 sides out to layer 4.')
f.render_flower((elems[1], elems[1]), 6, 4, 'flower')
print('')
print('The default colors aren\'t the best, so we override them.')
f.render_flower((elems[1], elems[1]),
6,
4,
'flower_better_colors',
initial_colors=('blue', 'red', 'green', 'purple', 'orange'))
print('')
print(
'Let\'s examine a batch of \'flower\' covering space images for the dihedral group on 4 vertices.'
)
b = GroupStructure(DihedralGroup(4))
print(b)
print('')
g = OperationComplex(b, '*')
elems = b.canonical_order
x = elems[2]
y = elems[5]
print('We will start our complexes with the product of {} and {}.'.format(
x, y))
print('We let the number of \'petals\' on our complex vary from 4 to 29.')
print(
'We also adjust the number of \'layers\' we draw in order to avoid overlapping triangles in the plane.'
)
cols = ('blue', 'red', 'green', 'purple', 'orange')
for i in range(4, 30):
g.render_flower((x, y), i, i, 'dihedral{}'.format(i), initial_colors=cols)
