def as_permutation_group(self, limit=4096000):
"""
Return an isomorphic permutation group.
The generators of the resulting group correspond to the images
by the isomorphism of the generators of the given group.
INPUT:
- ``limit`` -- integer (default: 4096000). The maximal number
of cosets before the computation is aborted.
OUTPUT:
A Sage
:func:`~sage.groups.perm_gps.permgroup.PermutationGroup`. If
the number of cosets exceeds the given ``limit``, a
``ValueError`` is returned.
EXAMPLES::
sage: G.<a,b> = FreeGroup()
sage: H = G / (a^2, b^3, a*b*~a*~b)
sage: H.as_permutation_group()
Permutation Group with generators [(1,2)(3,5)(4,6), (1,3,4)(2,5,6)]
sage: G.<a,b> = FreeGroup()
sage: H = G / [a^3*b]
sage: H.as_permutation_group(limit=1000)
Traceback (most recent call last):
...
ValueError: Coset enumeration exceeded limit, is the group finite?
ALGORITHM:
Uses GAP's coset enumeration on the trivial subgroup.
.. WARNING::
This is in general not a decidable problem (in fact, it is
not even posible to check if the group is finite or
not). If the group is infinite, or too big, you should be
prepared for a long computation that consumes all the
memory without finishing if you do not set a sensible
``limit``.
"""
with libgap.global_context('CosetTableDefaultMaxLimit', limit):
try:
trivial_subgroup = self.gap().TrivialSubgroup()
coset_table = self.gap().CosetTable(trivial_subgroup).sage()
except ValueError:
raise ValueError(
'Coset enumeration exceeded limit, is the group finite?')
from sage.combinat.permutation import Permutation
from sage.groups.perm_gps.permgroup import PermutationGroup
return PermutationGroup([
Permutation(coset_table[2 * i])
for i in range(len(coset_table) / 2)
])
def as_permutation_group(self, limit=4096000):
"""
Return an isomorphic permutation group.
The generators of the resulting group correspond to the images
by the isomorphism of the generators of the given group.
INPUT:
- ``limit`` -- integer (default: 4096000). The maximal number
of cosets before the computation is aborted.
OUTPUT:
A Sage
:func:`~sage.groups.perm_gps.permgroup.PermutationGroup`. If
the number of cosets exceeds the given ``limit``, a
``ValueError`` is returned.
EXAMPLES::
sage: G.<a,b> = FreeGroup()
sage: H = G / (a^2, b^3, a*b*~a*~b)
sage: H.as_permutation_group()
Permutation Group with generators [(1,2)(3,5)(4,6), (1,3,4)(2,5,6)]
sage: G.<a,b> = FreeGroup()
sage: H = G / [a^3*b]
sage: H.as_permutation_group(limit=1000)
Traceback (most recent call last):
...
ValueError: Coset enumeration exceeded limit, is the group finite?
ALGORITHM:
Uses GAP's coset enumeration on the trivial subgroup.
.. WARNING::
This is in general not a decidable problem (in fact, it is
not even posible to check if the group is finite or
not). If the group is infinite, or too big, you should be
prepared for a long computation that consumes all the
memory without finishing if you do not set a sensible
``limit``.
"""
with libgap.global_context('CosetTableDefaultMaxLimit', limit):
try:
trivial_subgroup = self.gap().TrivialSubgroup()
coset_table = self.gap().CosetTable(trivial_subgroup).sage()
except ValueError:
raise ValueError('Coset enumeration exceeded limit, is the group finite?')
from sage.combinat.permutation import Permutation
from sage.groups.perm_gps.permgroup import PermutationGroup
return PermutationGroup([
Permutation(coset_table[2*i]) for i in range(len(coset_table)/2)])
def cardinality(self, limit=4096000):
"""
Compute the cardinality of ``self``.
INPUT:
- ``limit`` -- integer (default: 4096000). The maximal number
of cosets before the computation is aborted.
OUTPUT:
Integer or ``Infinity``. The number of elements in the group.
EXAMPLES::
sage: G.<a,b> = FreeGroup('a, b')
sage: H = G / (a^2, b^3, a*b*~a*~b)
sage: H.cardinality()
6
sage: F.<a,b,c> = FreeGroup()
sage: J = F / (F([1]), F([2, 2, 2]))
sage: J.cardinality()
+Infinity
ALGORITHM:
Uses GAP.
.. WARNING::
This is in general not a decidable problem, so it is not
guaranteed to give an answer. If the group is infinite, or
too big, you should be prepared for a long computation
that consumes all the memory without finishing if you do
not set a sensible ``limit``.
"""
with libgap.global_context('CosetTableDefaultMaxLimit', limit):
if not libgap.IsFinite(self.gap()):
from sage.rings.infinity import Infinity
return Infinity
try:
size = self.gap().Size()
except ValueError:
raise ValueError(
'Coset enumeration ran out of memory, is the group finite?'
)
return size.sage()
def cardinality(self, limit=4096000):
"""
Compute the cardinality of ``self``.
INPUT:
- ``limit`` -- integer (default: 4096000). The maximal number
of cosets before the computation is aborted.
OUTPUT:
Integer or ``Infinity``. The number of elements in the group.
EXAMPLES::
sage: G.<a,b> = FreeGroup('a, b')
sage: H = G / (a^2, b^3, a*b*~a*~b)
sage: H.cardinality()
6
sage: F.<a,b,c> = FreeGroup()
sage: J = F / (F([1]), F([2, 2, 2]))
sage: J.cardinality()
+Infinity
ALGORITHM:
Uses GAP.
.. WARNING::
This is in general not a decidable problem, so it is not
guaranteed to give an answer. If the group is infinite, or
too big, you should be prepared for a long computation
that consumes all the memory without finishing if you do
not set a sensible ``limit``.
"""
with libgap.global_context("CosetTableDefaultMaxLimit", limit):
if not libgap.IsFinite(self.gap()):
from sage.rings.infinity import Infinity
return Infinity
try:
size = self.gap().Size()
except ValueError:
raise ValueError("Coset enumeration ran out of memory, is the group finite?")
return size.sage()
