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