def test_homomorphism(): F, a, b = free_group("a, b") G = FpGroup(F, [a**3, b**3, (a * b)**2]) c = Permutation(3)(0, 1, 2) d = Permutation(3)(1, 2, 3) A = AlternatingGroup(4) T = homomorphism(G, A, [a, b], [c, d]) assert T(a * b**2 * a**-1) == c * d**2 * c**-1 assert T.is_isomorphism() assert T(T.invert(Permutation(3)(0, 2, 3))) == Permutation(3)(0, 2, 3) T = homomorphism(G, AlternatingGroup(4), G.generators) assert T.is_trivial() assert T.kernel().order() == G.order() F, a = free_group("a") G = FpGroup(F, [a**8]) P = PermutationGroup([Permutation(0, 1, 2, 3), Permutation(0, 2)]) T = homomorphism(G, P, [a], [Permutation(0, 1, 2, 3)]) assert T.image().order() == 4 assert T(T.invert(Permutation(0, 2)(1, 3))) == Permutation(0, 2)(1, 3) T = homomorphism(F, AlternatingGroup(4), F.generators, [c]) assert T.invert(c**2) == a**2
def _to_perm_group(self): ''' Return an isomorphic permutation group and the isomorphism. The implementation is dependent on coset enumeration so will only terminate for finite groups. ''' from sympy.combinatorics import Permutation, PermutationGroup from sympy.combinatorics.homomorphisms import homomorphism from sympy import S if self.order() == S.Infinity: raise NotImplementedError("Permutation presentation of infinite " "groups is not implemented") if self._perm_isomorphism: T = self._perm_isomorphism P = T.image() else: C = self.coset_table([]) gens = self.generators images = [[C[i][2 * gens.index(g)] for i in range(len(C))] for g in gens] images = [Permutation(i) for i in images] P = PermutationGroup(images) T = homomorphism(self, P, gens, images, check=False) self._perm_isomorphism = T return P, T
def _to_perm_group(self): ''' Return an isomorphic permutation group and the isomorphism. The implementation is dependent on coset enumeration so will only terminate for finite groups. ''' from sympy.combinatorics import Permutation, PermutationGroup from sympy.combinatorics.homomorphisms import homomorphism from sympy import S if self.order() == S.Infinity: raise NotImplementedError("Permutation presentation of infinite " "groups is not implemented") if self._perm_isomorphism: T = self._perm_isomorphism P = T.image() else: C = self.coset_table([]) gens = self.generators images = [[C[i][2*gens.index(g)] for i in range(len(C))] for g in gens] images = [Permutation(i) for i in images] P = PermutationGroup(images) T = homomorphism(self, P, gens, images, check=False) self._perm_isomorphism = T return P, T
def test_homomorphism(): # FpGroup -> PermutationGroup F, a, b = free_group("a, b") G = FpGroup(F, [a**3, b**3, (a*b)**2]) c = Permutation(3)(0, 1, 2) d = Permutation(3)(1, 2, 3) A = AlternatingGroup(4) T = homomorphism(G, A, [a, b], [c, d]) assert T(a*b**2*a**-1) == c*d**2*c**-1 assert T.is_isomorphism() assert T(T.invert(Permutation(3)(0, 2, 3))) == Permutation(3)(0, 2, 3) T = homomorphism(G, AlternatingGroup(4), G.generators) assert T.is_trivial() assert T.kernel().order() == G.order() E, e = free_group("e") G = FpGroup(E, [e**8]) P = PermutationGroup([Permutation(0, 1, 2, 3), Permutation(0, 2)]) T = homomorphism(G, P, [e], [Permutation(0, 1, 2, 3)]) assert T.image().order() == 4 assert T(T.invert(Permutation(0, 2)(1, 3))) == Permutation(0, 2)(1, 3) T = homomorphism(E, AlternatingGroup(4), E.generators, [c]) assert T.invert(c**2) == e**-1 #order(c) == 3 so c**2 == c**-1 # FreeGroup -> FreeGroup T = homomorphism(F, E, [a], [e]) assert T(a**-2*b**4*a**2).is_identity # FreeGroup -> FpGroup G = FpGroup(F, [a*b*a**-1*b**-1]) T = homomorphism(F, G, F.generators, G.generators) assert T.invert(a**-1*b**-1*a**2) == a*b**-1 # PermutationGroup -> PermutationGroup D = DihedralGroup(8) p = Permutation(0, 1, 2, 3, 4, 5, 6, 7) P = PermutationGroup(p) T = homomorphism(P, D, [p], [p]) assert T.is_injective() assert not T.is_isomorphism() assert T.invert(p**3) == p**3 T2 = homomorphism(F, P, [F.generators[0]], P.generators) T = T.compose(T2) assert T.domain == F assert T.codomain == D assert T(a*b) == p
def subgroup(self, gens, C=None, homomorphism=False): """ Return the subgroup generated by `gens` using the Reidemeister-Schreier algorithm homomorphism -- When set to True, return a dictionary containing the images of the presentation generators in the original group. Examples ======== >>> from sympy.combinatorics.fp_groups import (FpGroup, FpSubgroup) >>> from sympy.combinatorics.free_groups import free_group >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x**3, y**5, (x*y)**2]) >>> H = [x*y, x**-1*y**-1*x*y*x] >>> K, T = f.subgroup(H, homomorphism=True) >>> T(K.generators) [x*y, x**-1*y**2*x**-1] """ if not all([isinstance(g, FreeGroupElement) for g in gens]): raise ValueError("Generators must be `FreeGroupElement`s") if not all([g.group == self.free_group for g in gens]): raise ValueError("Given generators are not members of the group") if homomorphism: g, rels, _gens = reidemeister_presentation(self, gens, C=C, homomorphism=True) else: g, rels = reidemeister_presentation(self, gens, C=C) if g: g = FpGroup(g[0].group, rels) else: g = FpGroup(free_group("")[0], []) if homomorphism: from sympy.combinatorics.homomorphisms import homomorphism return g, homomorphism(g, self, g.generators, _gens, check=False) return g
def subgroup(self, gens, C=None, homomorphism=False): ''' Return the subgroup generated by `gens` using the Reidemeister-Schreier algorithm homomorphism -- When set to True, return a dictionary containing the images of the presentation generators in the original group. Examples ======== >>> from sympy.combinatorics.fp_groups import (FpGroup, FpSubgroup) >>> from sympy.combinatorics.free_groups import free_group >>> F, x, y = free_group("x, y") >>> f = FpGroup(F, [x**3, y**5, (x*y)**2]) >>> H = [x*y, x**-1*y**-1*x*y*x] >>> K, T = f.subgroup(H, homomorphism=True) >>> T(K.generators) [x*y, x**-1*y**2*x**-1] ''' if not all([isinstance(g, FreeGroupElement) for g in gens]): raise ValueError("Generators must be `FreeGroupElement`s") if not all([g.group == self.free_group for g in gens]): raise ValueError("Given generators are not members of the group") if homomorphism: g, rels, _gens = reidemeister_presentation(self, gens, C=C, homomorphism=True) else: g, rels = reidemeister_presentation(self, gens, C=C) if g: g = FpGroup(g[0].group, rels) else: g = FpGroup(free_group('')[0], []) if homomorphism: from sympy.combinatorics.homomorphisms import homomorphism return g, homomorphism(g, self, g.generators, _gens, check=False) return g
def test_check_homomorphism(): a = Permutation(1,2,3,4) b = Permutation(1,3) G = PermutationGroup([a, b]) raises(ValueError, lambda: homomorphism(G, G, [a], [a]))