550 lines
18 KiB
Python
550 lines
18 KiB
Python
|
import itertools
|
||
|
|
from sympy.combinatorics.fp_groups import FpGroup, FpSubgroup, simplify_presentation
|
||
|
|
from sympy.combinatorics.free_groups import FreeGroup
|
||
|
|
from sympy.combinatorics.perm_groups import PermutationGroup
|
||
|
|
from sympy.core.numbers import igcd
|
||
|
|
from sympy.ntheory.factor_ import totient
|
||
|
|
from sympy.core.singleton import S
|
||
|
|
|
||
|
|
class GroupHomomorphism:
|
||
|
|
'''
|
||
|
|
A class representing group homomorphisms. Instantiate using `homomorphism()`.
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] Holt, D., Eick, B. and O'Brien, E. (2005). Handbook of computational group theory.
|
||
|
|
|
||
|
|
'''
|
||
|
|
|
||
|
|
def __init__(self, domain, codomain, images):
|
||
|
|
self.domain = domain
|
||
|
|
self.codomain = codomain
|
||
|
|
self.images = images
|
||
|
|
self._inverses = None
|
||
|
|
self._kernel = None
|
||
|
|
self._image = None
|
||
|
|
|
||
|
|
def _invs(self):
|
||
|
|
'''
|
||
|
|
Return a dictionary with `{gen: inverse}` where `gen` is a rewriting
|
||
|
|
generator of `codomain` (e.g. strong generator for permutation groups)
|
||
|
|
and `inverse` is an element of its preimage
|
||
|
|
|
||
|
|
'''
|
||
|
|
image = self.image()
|
||
|
|
inverses = {}
|
||
|
|
for k in list(self.images.keys()):
|
||
|
|
v = self.images[k]
|
||
|
|
if not (v in inverses
|
||
|
|
or v.is_identity):
|
||
|
|
inverses[v] = k
|
||
|
|
if isinstance(self.codomain, PermutationGroup):
|
||
|
|
gens = image.strong_gens
|
||
|
|
else:
|
||
|
|
gens = image.generators
|
||
|
|
for g in gens:
|
||
|
|
if g in inverses or g.is_identity:
|
||
|
|
continue
|
||
|
|
w = self.domain.identity
|
||
|
|
if isinstance(self.codomain, PermutationGroup):
|
||
|
|
parts = image._strong_gens_slp[g][::-1]
|
||
|
|
else:
|
||
|
|
parts = g
|
||
|
|
for s in parts:
|
||
|
|
if s in inverses:
|
||
|
|
w = w*inverses[s]
|
||
|
|
else:
|
||
|
|
w = w*inverses[s**-1]**-1
|
||
|
|
inverses[g] = w
|
||
|
|
|
||
|
|
return inverses
|
||
|
|
|
||
|
|
def invert(self, g):
|
||
|
|
'''
|
||
|
|
Return an element of the preimage of ``g`` or of each element
|
||
|
|
of ``g`` if ``g`` is a list.
|
||
|
|
|
||
|
|
Explanation
|
||
|
|
===========
|
||
|
|
|
||
|
|
If the codomain is an FpGroup, the inverse for equal
|
||
|
|
elements might not always be the same unless the FpGroup's
|
||
|
|
rewriting system is confluent. However, making a system
|
||
|
|
confluent can be time-consuming. If it's important, try
|
||
|
|
`self.codomain.make_confluent()` first.
|
||
|
|
|
||
|
|
'''
|
||
|
|
from sympy.combinatorics import Permutation
|
||
|
|
from sympy.combinatorics.free_groups import FreeGroupElement
|
||
|
|
if isinstance(g, (Permutation, FreeGroupElement)):
|
||
|
|
if isinstance(self.codomain, FpGroup):
|
||
|
|
g = self.codomain.reduce(g)
|
||
|
|
if self._inverses is None:
|
||
|
|
self._inverses = self._invs()
|
||
|
|
image = self.image()
|
||
|
|
w = self.domain.identity
|
||
|
|
if isinstance(self.codomain, PermutationGroup):
|
||
|
|
gens = image.generator_product(g)[::-1]
|
||
|
|
else:
|
||
|
|
gens = g
|
||
|
|
# the following can't be "for s in gens:"
|
||
|
|
# because that would be equivalent to
|
||
|
|
# "for s in gens.array_form:" when g is
|
||
|
|
# a FreeGroupElement. On the other hand,
|
||
|
|
# when you call gens by index, the generator
|
||
|
|
# (or inverse) at position i is returned.
|
||
|
|
for i in range(len(gens)):
|
||
|
|
s = gens[i]
|
||
|
|
if s.is_identity:
|
||
|
|
continue
|
||
|
|
if s in self._inverses:
|
||
|
|
w = w*self._inverses[s]
|
||
|
|
else:
|
||
|
|
w = w*self._inverses[s**-1]**-1
|
||
|
|
return w
|
||
|
|
elif isinstance(g, list):
|
||
|
|
return [self.invert(e) for e in g]
|
||
|
|
|
||
|
|
def kernel(self):
|
||
|
|
'''
|
||
|
|
Compute the kernel of `self`.
|
||
|
|
|
||
|
|
'''
|
||
|
|
if self._kernel is None:
|
||
|
|
self._kernel = self._compute_kernel()
|
||
|
|
return self._kernel
|
||
|
|
|
||
|
|
def _compute_kernel(self):
|
||
|
|
G = self.domain
|
||
|
|
G_order = G.order()
|
||
|
|
if G_order is S.Infinity:
|
||
|
|
raise NotImplementedError(
|
||
|
|
"Kernel computation is not implemented for infinite groups")
|
||
|
|
gens = []
|
||
|
|
if isinstance(G, PermutationGroup):
|
||
|
|
K = PermutationGroup(G.identity)
|
||
|
|
else:
|
||
|
|
K = FpSubgroup(G, gens, normal=True)
|
||
|
|
i = self.image().order()
|
||
|
|
while K.order()*i != G_order:
|
||
|
|
r = G.random()
|
||
|
|
k = r*self.invert(self(r))**-1
|
||
|
|
if k not in K:
|
||
|
|
gens.append(k)
|
||
|
|
if isinstance(G, PermutationGroup):
|
||
|
|
K = PermutationGroup(gens)
|
||
|
|
else:
|
||
|
|
K = FpSubgroup(G, gens, normal=True)
|
||
|
|
return K
|
||
|
|
|
||
|
|
def image(self):
|
||
|
|
'''
|
||
|
|
Compute the image of `self`.
|
||
|
|
|
||
|
|
'''
|
||
|
|
if self._image is None:
|
||
|
|
values = list(set(self.images.values()))
|
||
|
|
if isinstance(self.codomain, PermutationGroup):
|
||
|
|
self._image = self.codomain.subgroup(values)
|
||
|
|
else:
|
||
|
|
self._image = FpSubgroup(self.codomain, values)
|
||
|
|
return self._image
|
||
|
|
|
||
|
|
def _apply(self, elem):
|
||
|
|
'''
|
||
|
|
Apply `self` to `elem`.
|
||
|
|
|
||
|
|
'''
|
||
|
|
if elem not in self.domain:
|
||
|
|
if isinstance(elem, (list, tuple)):
|
||
|
|
return [self._apply(e) for e in elem]
|
||
|
|
raise ValueError("The supplied element does not belong to the domain")
|
||
|
|
if elem.is_identity:
|
||
|
|
return self.codomain.identity
|
||
|
|
else:
|
||
|
|
images = self.images
|
||
|
|
value = self.codomain.identity
|
||
|
|
if isinstance(self.domain, PermutationGroup):
|
||
|
|
gens = self.domain.generator_product(elem, original=True)
|
||
|
|
for g in gens:
|
||
|
|
if g in self.images:
|
||
|
|
value = images[g]*value
|
||
|
|
else:
|
||
|
|
value = images[g**-1]**-1*value
|
||
|
|
else:
|
||
|
|
i = 0
|
||
|
|
for _, p in elem.array_form:
|
||
|
|
if p < 0:
|
||
|
|
g = elem[i]**-1
|
||
|
|
else:
|
||
|
|
g = elem[i]
|
||
|
|
value = value*images[g]**p
|
||
|
|
i += abs(p)
|
||
|
|
return value
|
||
|
|
|
||
|
|
def __call__(self, elem):
|
||
|
|
return self._apply(elem)
|
||
|
|
|
||
|
|
def is_injective(self):
|
||
|
|
'''
|
||
|
|
Check if the homomorphism is injective
|
||
|
|
|
||
|
|
'''
|
||
|
|
return self.kernel().order() == 1
|
||
|
|
|
||
|
|
def is_surjective(self):
|
||
|
|
'''
|
||
|
|
Check if the homomorphism is surjective
|
||
|
|
|
||
|
|
'''
|
||
|
|
im = self.image().order()
|
||
|
|
oth = self.codomain.order()
|
||
|
|
if im is S.Infinity and oth is S.Infinity:
|
||
|
|
return None
|
||
|
|
else:
|
||
|
|
return im == oth
|
||
|
|
|
||
|
|
def is_isomorphism(self):
|
||
|
|
'''
|
||
|
|
Check if `self` is an isomorphism.
|
||
|
|
|
||
|
|
'''
|
||
|
|
return self.is_injective() and self.is_surjective()
|
||
|
|
|
||
|
|
def is_trivial(self):
|
||
|
|
'''
|
||
|
|
Check is `self` is a trivial homomorphism, i.e. all elements
|
||
|
|
are mapped to the identity.
|
||
|
|
|
||
|
|
'''
|
||
|
|
return self.image().order() == 1
|
||
|
|
|
||
|
|
def compose(self, other):
|
||
|
|
'''
|
||
|
|
Return the composition of `self` and `other`, i.e.
|
||
|
|
the homomorphism phi such that for all g in the domain
|
||
|
|
of `other`, phi(g) = self(other(g))
|
||
|
|
|
||
|
|
'''
|
||
|
|
if not other.image().is_subgroup(self.domain):
|
||
|
|
raise ValueError("The image of `other` must be a subgroup of "
|
||
|
|
"the domain of `self`")
|
||
|
|
images = {g: self(other(g)) for g in other.images}
|
||
|
|
return GroupHomomorphism(other.domain, self.codomain, images)
|
||
|
|
|
||
|
|
def restrict_to(self, H):
|
||
|
|
'''
|
||
|
|
Return the restriction of the homomorphism to the subgroup `H`
|
||
|
|
of the domain.
|
||
|
|
|
||
|
|
'''
|
||
|
|
if not isinstance(H, PermutationGroup) or not H.is_subgroup(self.domain):
|
||
|
|
raise ValueError("Given H is not a subgroup of the domain")
|
||
|
|
domain = H
|
||
|
|
images = {g: self(g) for g in H.generators}
|
||
|
|
return GroupHomomorphism(domain, self.codomain, images)
|
||
|
|
|
||
|
|
def invert_subgroup(self, H):
|
||
|
|
'''
|
||
|
|
Return the subgroup of the domain that is the inverse image
|
||
|
|
of the subgroup ``H`` of the homomorphism image
|
||
|
|
|
||
|
|
'''
|
||
|
|
if not H.is_subgroup(self.image()):
|
||
|
|
raise ValueError("Given H is not a subgroup of the image")
|
||
|
|
gens = []
|
||
|
|
P = PermutationGroup(self.image().identity)
|
||
|
|
for h in H.generators:
|
||
|
|
h_i = self.invert(h)
|
||
|
|
if h_i not in P:
|
||
|
|
gens.append(h_i)
|
||
|
|
P = PermutationGroup(gens)
|
||
|
|
for k in self.kernel().generators:
|
||
|
|
if k*h_i not in P:
|
||
|
|
gens.append(k*h_i)
|
||
|
|
P = PermutationGroup(gens)
|
||
|
|
return P
|
||
|
|
|
||
|
|
def homomorphism(domain, codomain, gens, images=(), check=True):
|
||
|
|
'''
|
||
|
|
Create (if possible) a group homomorphism from the group ``domain``
|
||
|
|
to the group ``codomain`` defined by the images of the domain's
|
||
|
|
generators ``gens``. ``gens`` and ``images`` can be either lists or tuples
|
||
|
|
of equal sizes. If ``gens`` is a proper subset of the group's generators,
|
||
|
|
the unspecified generators will be mapped to the identity. If the
|
||
|
|
images are not specified, a trivial homomorphism will be created.
|
||
|
|
|
||
|
|
If the given images of the generators do not define a homomorphism,
|
||
|
|
an exception is raised.
|
||
|
|
|
||
|
|
If ``check`` is ``False``, do not check whether the given images actually
|
||
|
|
define a homomorphism.
|
||
|
|
|
||
|
|
'''
|
||
|
|
if not isinstance(domain, (PermutationGroup, FpGroup, FreeGroup)):
|
||
|
|
raise TypeError("The domain must be a group")
|
||
|
|
if not isinstance(codomain, (PermutationGroup, FpGroup, FreeGroup)):
|
||
|
|
raise TypeError("The codomain must be a group")
|
||
|
|
|
||
|
|
generators = domain.generators
|
||
|
|
if not all(g in generators for g in gens):
|
||
|
|
raise ValueError("The supplied generators must be a subset of the domain's generators")
|
||
|
|
if not all(g in codomain for g in images):
|
||
|
|
raise ValueError("The images must be elements of the codomain")
|
||
|
|
|
||
|
|
if images and len(images) != len(gens):
|
||
|
|
raise ValueError("The number of images must be equal to the number of generators")
|
||
|
|
|
||
|
|
gens = list(gens)
|
||
|
|
images = list(images)
|
||
|
|
|
||
|
|
images.extend([codomain.identity]*(len(generators)-len(images)))
|
||
|
|
gens.extend([g for g in generators if g not in gens])
|
||
|
|
images = dict(zip(gens,images))
|
||
|
|
|
||
|
|
if check and not _check_homomorphism(domain, codomain, images):
|
||
|
|
raise ValueError("The given images do not define a homomorphism")
|
||
|
|
return GroupHomomorphism(domain, codomain, images)
|
||
|
|
|
||
|
|
def _check_homomorphism(domain, codomain, images):
|
||
|
|
"""
|
||
|
|
Check that a given mapping of generators to images defines a homomorphism.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
domain : PermutationGroup, FpGroup, FreeGroup
|
||
|
|
codomain : PermutationGroup, FpGroup, FreeGroup
|
||
|
|
images : dict
|
||
|
|
The set of keys must be equal to domain.generators.
|
||
|
|
The values must be elements of the codomain.
|
||
|
|
|
||
|
|
"""
|
||
|
|
pres = domain if hasattr(domain, 'relators') else domain.presentation()
|
||
|
|
rels = pres.relators
|
||
|
|
gens = pres.generators
|
||
|
|
symbols = [g.ext_rep[0] for g in gens]
|
||
|
|
symbols_to_domain_generators = dict(zip(symbols, domain.generators))
|
||
|
|
identity = codomain.identity
|
||
|
|
|
||
|
|
def _image(r):
|
||
|
|
w = identity
|
||
|
|
for symbol, power in r.array_form:
|
||
|
|
g = symbols_to_domain_generators[symbol]
|
||
|
|
w *= images[g]**power
|
||
|
|
return w
|
||
|
|
|
||
|
|
for r in rels:
|
||
|
|
if isinstance(codomain, FpGroup):
|
||
|
|
s = codomain.equals(_image(r), identity)
|
||
|
|
if s is None:
|
||
|
|
# only try to make the rewriting system
|
||
|
|
# confluent when it can't determine the
|
||
|
|
# truth of equality otherwise
|
||
|
|
success = codomain.make_confluent()
|
||
|
|
s = codomain.equals(_image(r), identity)
|
||
|
|
if s is None and not success:
|
||
|
|
raise RuntimeError("Can't determine if the images "
|
||
|
|
"define a homomorphism. Try increasing "
|
||
|
|
"the maximum number of rewriting rules "
|
||
|
|
"(group._rewriting_system.set_max(new_value); "
|
||
|
|
"the current value is stored in group._rewriting"
|
||
|
|
"_system.maxeqns)")
|
||
|
|
else:
|
||
|
|
s = _image(r).is_identity
|
||
|
|
if not s:
|
||
|
|
return False
|
||
|
|
return True
|
||
|
|
|
||
|
|
def orbit_homomorphism(group, omega):
|
||
|
|
'''
|
||
|
|
Return the homomorphism induced by the action of the permutation
|
||
|
|
group ``group`` on the set ``omega`` that is closed under the action.
|
||
|
|
|
||
|
|
'''
|
||
|
|
from sympy.combinatorics import Permutation
|
||
|
|
from sympy.combinatorics.named_groups import SymmetricGroup
|
||
|
|
codomain = SymmetricGroup(len(omega))
|
||
|
|
identity = codomain.identity
|
||
|
|
omega = list(omega)
|
||
|
|
images = {g: identity*Permutation([omega.index(o^g) for o in omega]) for g in group.generators}
|
||
|
|
group._schreier_sims(base=omega)
|
||
|
|
H = GroupHomomorphism(group, codomain, images)
|
||
|
|
if len(group.basic_stabilizers) > len(omega):
|
||
|
|
H._kernel = group.basic_stabilizers[len(omega)]
|
||
|
|
else:
|
||
|
|
H._kernel = PermutationGroup([group.identity])
|
||
|
|
return H
|
||
|
|
|
||
|
|
def block_homomorphism(group, blocks):
|
||
|
|
'''
|
||
|
|
Return the homomorphism induced by the action of the permutation
|
||
|
|
group ``group`` on the block system ``blocks``. The latter should be
|
||
|
|
of the same form as returned by the ``minimal_block`` method for
|
||
|
|
permutation groups, namely a list of length ``group.degree`` where
|
||
|
|
the i-th entry is a representative of the block i belongs to.
|
||
|
|
|
||
|
|
'''
|
||
|
|
from sympy.combinatorics import Permutation
|
||
|
|
from sympy.combinatorics.named_groups import SymmetricGroup
|
||
|
|
|
||
|
|
n = len(blocks)
|
||
|
|
|
||
|
|
# number the blocks; m is the total number,
|
||
|
|
# b is such that b[i] is the number of the block i belongs to,
|
||
|
|
# p is the list of length m such that p[i] is the representative
|
||
|
|
# of the i-th block
|
||
|
|
m = 0
|
||
|
|
p = []
|
||
|
|
b = [None]*n
|
||
|
|
for i in range(n):
|
||
|
|
if blocks[i] == i:
|
||
|
|
p.append(i)
|
||
|
|
b[i] = m
|
||
|
|
m += 1
|
||
|
|
for i in range(n):
|
||
|
|
b[i] = b[blocks[i]]
|
||
|
|
|
||
|
|
codomain = SymmetricGroup(m)
|
||
|
|
# the list corresponding to the identity permutation in codomain
|
||
|
|
identity = range(m)
|
||
|
|
images = {g: Permutation([b[p[i]^g] for i in identity]) for g in group.generators}
|
||
|
|
H = GroupHomomorphism(group, codomain, images)
|
||
|
|
return H
|
||
|
|
|
||
|
|
def group_isomorphism(G, H, isomorphism=True):
|
||
|
|
'''
|
||
|
|
Compute an isomorphism between 2 given groups.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
G : A finite ``FpGroup`` or a ``PermutationGroup``.
|
||
|
|
First group.
|
||
|
|
|
||
|
|
H : A finite ``FpGroup`` or a ``PermutationGroup``
|
||
|
|
Second group.
|
||
|
|
|
||
|
|
isomorphism : bool
|
||
|
|
This is used to avoid the computation of homomorphism
|
||
|
|
when the user only wants to check if there exists
|
||
|
|
an isomorphism between the groups.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
If isomorphism = False -- Returns a boolean.
|
||
|
|
If isomorphism = True -- Returns a boolean and an isomorphism between `G` and `H`.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.combinatorics import free_group, Permutation
|
||
|
|
>>> from sympy.combinatorics.perm_groups import PermutationGroup
|
||
|
|
>>> from sympy.combinatorics.fp_groups import FpGroup
|
||
|
|
>>> from sympy.combinatorics.homomorphisms import group_isomorphism
|
||
|
|
>>> from sympy.combinatorics.named_groups import DihedralGroup, AlternatingGroup
|
||
|
|
|
||
|
|
>>> D = DihedralGroup(8)
|
||
|
|
>>> p = Permutation(0, 1, 2, 3, 4, 5, 6, 7)
|
||
|
|
>>> P = PermutationGroup(p)
|
||
|
|
>>> group_isomorphism(D, P)
|
||
|
|
(False, None)
|
||
|
|
|
||
|
|
>>> F, a, b = free_group("a, b")
|
||
|
|
>>> G = FpGroup(F, [a**3, b**3, (a*b)**2])
|
||
|
|
>>> H = AlternatingGroup(4)
|
||
|
|
>>> (check, T) = group_isomorphism(G, H)
|
||
|
|
>>> check
|
||
|
|
True
|
||
|
|
>>> T(b*a*b**-1*a**-1*b**-1)
|
||
|
|
(0 2 3)
|
||
|
|
|
||
|
|
Notes
|
||
|
|
=====
|
||
|
|
|
||
|
|
Uses the approach suggested by Robert Tarjan to compute the isomorphism between two groups.
|
||
|
|
First, the generators of ``G`` are mapped to the elements of ``H`` and
|
||
|
|
we check if the mapping induces an isomorphism.
|
||
|
|
|
||
|
|
'''
|
||
|
|
if not isinstance(G, (PermutationGroup, FpGroup)):
|
||
|
|
raise TypeError("The group must be a PermutationGroup or an FpGroup")
|
||
|
|
if not isinstance(H, (PermutationGroup, FpGroup)):
|
||
|
|
raise TypeError("The group must be a PermutationGroup or an FpGroup")
|
||
|
|
|
||
|
|
if isinstance(G, FpGroup) and isinstance(H, FpGroup):
|
||
|
|
G = simplify_presentation(G)
|
||
|
|
H = simplify_presentation(H)
|
||
|
|
# Two infinite FpGroups with the same generators are isomorphic
|
||
|
|
# when the relators are same but are ordered differently.
|
||
|
|
if G.generators == H.generators and (G.relators).sort() == (H.relators).sort():
|
||
|
|
if not isomorphism:
|
||
|
|
return True
|
||
|
|
return (True, homomorphism(G, H, G.generators, H.generators))
|
||
|
|
|
||
|
|
# `_H` is the permutation group isomorphic to `H`.
|
||
|
|
_H = H
|
||
|
|
g_order = G.order()
|
||
|
|
h_order = H.order()
|
||
|
|
|
||
|
|
if g_order is S.Infinity:
|
||
|
|
raise NotImplementedError("Isomorphism methods are not implemented for infinite groups.")
|
||
|
|
|
||
|
|
if isinstance(H, FpGroup):
|
||
|
|
if h_order is S.Infinity:
|
||
|
|
raise NotImplementedError("Isomorphism methods are not implemented for infinite groups.")
|
||
|
|
_H, h_isomorphism = H._to_perm_group()
|
||
|
|
|
||
|
|
if (g_order != h_order) or (G.is_abelian != H.is_abelian):
|
||
|
|
if not isomorphism:
|
||
|
|
return False
|
||
|
|
return (False, None)
|
||
|
|
|
||
|
|
if not isomorphism:
|
||
|
|
# Two groups of the same cyclic numbered order
|
||
|
|
# are isomorphic to each other.
|
||
|
|
n = g_order
|
||
|
|
if (igcd(n, totient(n))) == 1:
|
||
|
|
return True
|
||
|
|
|
||
|
|
# Match the generators of `G` with subsets of `_H`
|
||
|
|
gens = list(G.generators)
|
||
|
|
for subset in itertools.permutations(_H, len(gens)):
|
||
|
|
images = list(subset)
|
||
|
|
images.extend([_H.identity]*(len(G.generators)-len(images)))
|
||
|
|
_images = dict(zip(gens,images))
|
||
|
|
if _check_homomorphism(G, _H, _images):
|
||
|
|
if isinstance(H, FpGroup):
|
||
|
|
images = h_isomorphism.invert(images)
|
||
|
|
T = homomorphism(G, H, G.generators, images, check=False)
|
||
|
|
if T.is_isomorphism():
|
||
|
|
# It is a valid isomorphism
|
||
|
|
if not isomorphism:
|
||
|
|
return True
|
||
|
|
return (True, T)
|
||
|
|
|
||
|
|
if not isomorphism:
|
||
|
|
return False
|
||
|
|
return (False, None)
|
||
|
|
|
||
|
|
def is_isomorphic(G, H):
|
||
|
|
'''
|
||
|
|
Check if the groups are isomorphic to each other
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
G : A finite ``FpGroup`` or a ``PermutationGroup``
|
||
|
|
First group.
|
||
|
|
|
||
|
|
H : A finite ``FpGroup`` or a ``PermutationGroup``
|
||
|
|
Second group.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
=======
|
||
|
|
|
||
|
|
boolean
|
||
|
|
'''
|
||
|
|
return group_isomorphism(G, H, isomorphism=False)
|