# HG changeset patch # User Mike Hansen # Date 1307563431 25200 # Node ID 048c24f4556305b322f88fba8cff9c35933e97d6 # Parent 6c965fff4a5aaa3996d72985e35ee0c4170cbd44 #10335: Add support for domains of permutation groups. This includes the functionality from #8929 by Jason Hill. diff --git a/doc/en/bordeaux_2008/nf_galois_groups.rst b/doc/en/bordeaux_2008/nf_galois_groups.rst --- a/doc/en/bordeaux_2008/nf_galois_groups.rst +++ b/doc/en/bordeaux_2008/nf_galois_groups.rst @@ -41,8 +41,8 @@ sage: P = K.primes_above(2)[0] sage: G.inertia_group(P) Subgroup [(), (1,4,6)(2,5,3), (1,6,4)(2,3,5)] of Galois group of Number Field in alpha with defining polynomial x^6 + 40*x^3 + 1372 - sage: sorted([G.artin_symbol(Q) for Q in K.primes_above(5)]) # (order is platform-dependent) - [(1,2)(3,4)(5,6), (1,3)(2,6)(4,5), (1,5)(2,4)(3,6)] + sage: sorted([G.artin_symbol(Q) for Q in K.primes_above(5)]) + [(1,3)(2,6)(4,5), (1,2)(3,4)(5,6), (1,5)(2,4)(3,6)] If the number field is not Galois over `\QQ`, then the ``galois_group`` command will construct its Galois closure and return the Galois group of that; diff --git a/doc/en/constructions/groups.rst b/doc/en/constructions/groups.rst --- a/doc/en/constructions/groups.rst +++ b/doc/en/constructions/groups.rst @@ -232,7 +232,7 @@ sage: G = PermutationGroup(['(1,2,3)(4,5)', '(3,4)']) sage: G.center() - Permutation Group with generators [()] + Subgroup of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]) generated by [()] A similar syntax for matrix groups also works: @@ -244,7 +244,7 @@ [[[4, 0], [0, 4]]] sage: G = PSL(2, 5 ) sage: G.center() - Permutation Group with generators [()] + Subgroup of (The projective special linear group of degree 2 over Finite Field of size 5) generated by [()] Note: ``center`` can be spelled either way in GAP, not so in Sage. diff --git a/doc/en/thematic_tutorials/group_theory.rst b/doc/en/thematic_tutorials/group_theory.rst --- a/doc/en/thematic_tutorials/group_theory.rst +++ b/doc/en/thematic_tutorials/group_theory.rst @@ -569,7 +569,7 @@ sage: C20 = CyclicPermutationGroup(20) sage: C20.conjugacy_classes_subgroups() - [Permutation Group with generators [()], Permutation Group with generators [(1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)], Permutation Group with generators [(1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)], Permutation Group with generators [(1,5,9,13,17)(2,6,10,14,18)(3,7,11,15,19)(4,8,12,16,20)], Permutation Group with generators [(1,3,5,7,9,11,13,15,17,19)(2,4,6,8,10,12,14,16,18,20)], Permutation Group with generators [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)]] + [Subgroup of (Cyclic group of order 20 as a permutation group) generated by [()], Subgroup of (Cyclic group of order 20 as a permutation group) generated by [(1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)], Subgroup of (Cyclic group of order 20 as a permutation group) generated by [(1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)], Subgroup of (Cyclic group of order 20 as a permutation group) generated by [(1,5,9,13,17)(2,6,10,14,18)(3,7,11,15,19)(4,8,12,16,20)], Subgroup of (Cyclic group of order 20 as a permutation group) generated by [(1,3,5,7,9,11,13,15,17,19)(2,4,6,8,10,12,14,16,18,20)], Subgroup of (Cyclic group of order 20 as a permutation group) generated by [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)]] Be careful, this command uses some more advanced ideas and will not usually list *all* of the subgroups of a group. Here we are relying on @@ -623,7 +623,7 @@ sage: sg = K.conjugacy_classes_subgroups() sage: print "sg:\n", sg sg: - [Permutation Group with generators [()], Permutation Group with generators [(1,2)(3,12)(4,11)(5,10)(6,9)(7,8)], Permutation Group with generators [(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)], Permutation Group with generators [(2,12)(3,11)(4,10)(5,9)(6,8)], Permutation Group with generators [(1,5,9)(2,6,10)(3,7,11)(4,8,12)], Permutation Group with generators [(2,12)(3,11)(4,10)(5,9)(6,8), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)], Permutation Group with generators [(1,2)(3,12)(4,11)(5,10)(6,9)(7,8), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)], Permutation Group with generators [(1,10,7,4)(2,11,8,5)(3,12,9,6)], Permutation Group with generators [(1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,3,5,7,9,11)(2,4,6,8,10,12)], Permutation Group with generators [(1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,2)(3,12)(4,11)(5,10)(6,9)(7,8)], Permutation Group with generators [(1,5,9)(2,6,10)(3,7,11)(4,8,12), (2,12)(3,11)(4,10)(5,9)(6,8)], Permutation Group with generators [(2,12)(3,11)(4,10)(5,9)(6,8), (1,10,7,4)(2,11,8,5)(3,12,9,6)], Permutation Group with generators [(1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,3,5,7,9,11)(2,4,6,8,10,12), (2,12)(3,11)(4,10)(5,9)(6,8)], Permutation Group with generators [(1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,3,5,7,9,11)(2,4,6,8,10,12), (1,2)(3,12)(4,11)(5,10)(6,9)(7,8)], Permutation Group with generators [(1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,3,5,7,9,11)(2,4,6,8,10,12), (1,2,3,4,5,6,7,8,9,10,11,12)], Permutation Group with generators [(1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,3,5,7,9,11)(2,4,6,8,10,12), (2,12)(3,11)(4,10)(5,9)(6,8), (1,2,3,4,5,6,7,8,9,10,11,12)]] + [Subgroup of (Dihedral group of order 24 as a permutation group) generated by [()], Subgroup of (Dihedral group of order 24 as a permutation group) generated by [(1,2)(3,12)(4,11)(5,10)(6,9)(7,8)], Subgroup of (Dihedral group of order 24 as a permutation group) generated by [(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)], Subgroup of (Dihedral group of order 24 as a permutation group) generated by [(2,12)(3,11)(4,10)(5,9)(6,8)], Subgroup of (Dihedral group of order 24 as a permutation group) generated by [(1,5,9)(2,6,10)(3,7,11)(4,8,12)], Subgroup of (Dihedral group of order 24 as a permutation group) generated by [(2,12)(3,11)(4,10)(5,9)(6,8), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)], Subgroup of (Dihedral group of order 24 as a permutation group) generated by [(1,2)(3,12)(4,11)(5,10)(6,9)(7,8), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)], Subgroup of (Dihedral group of order 24 as a permutation group) generated by [(1,10,7,4)(2,11,8,5)(3,12,9,6)], Subgroup of (Dihedral group of order 24 as a permutation group) generated by [(1,3,5,7,9,11)(2,4,6,8,10,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12)], Subgroup of (Dihedral group of order 24 as a permutation group) generated by [(1,2)(3,12)(4,11)(5,10)(6,9)(7,8), (1,5,9)(2,6,10)(3,7,11)(4,8,12)], Subgroup of (Dihedral group of order 24 as a permutation group) generated by [(2,12)(3,11)(4,10)(5,9)(6,8), (1,5,9)(2,6,10)(3,7,11)(4,8,12)], Subgroup of (Dihedral group of order 24 as a permutation group) generated by [(2,12)(3,11)(4,10)(5,9)(6,8), (1,10,7,4)(2,11,8,5)(3,12,9,6)], Subgroup of (Dihedral group of order 24 as a permutation group) generated by [(2,12)(3,11)(4,10)(5,9)(6,8), (1,3,5,7,9,11)(2,4,6,8,10,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12)], Subgroup of (Dihedral group of order 24 as a permutation group) generated by [(1,2)(3,12)(4,11)(5,10)(6,9)(7,8), (1,3,5,7,9,11)(2,4,6,8,10,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12)], Subgroup of (Dihedral group of order 24 as a permutation group) generated by [(1,2,3,4,5,6,7,8,9,10,11,12), (1,3,5,7,9,11)(2,4,6,8,10,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12)], Subgroup of (Dihedral group of order 24 as a permutation group) generated by [(2,12)(3,11)(4,10)(5,9)(6,8), (1,2,3,4,5,6,7,8,9,10,11,12), (1,3,5,7,9,11)(2,4,6,8,10,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12)]] sage: print "\nAn order two subgroup:\n", sg[1].list() An order two subgroup: diff --git a/doc/en/tutorial/interfaces.rst b/doc/en/tutorial/interfaces.rst --- a/doc/en/tutorial/interfaces.rst +++ b/doc/en/tutorial/interfaces.rst @@ -174,7 +174,7 @@ sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]]) sage: G.center() - Permutation Group with generators [()] + Subgroup of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]) generated by [()] sage: G.group_id() # requires optional database_gap package [120, 34] sage: n = G.order(); n diff --git a/doc/en/tutorial/tour_groups.rst b/doc/en/tutorial/tour_groups.rst --- a/doc/en/tutorial/tour_groups.rst +++ b/doc/en/tutorial/tour_groups.rst @@ -22,7 +22,7 @@ [Permutation Group with generators [(1,2,3)(4,5), (3,4)], Permutation Group with generators [(1,5)(3,4), (1,5)(2,4), (1,3,5)]] sage: G.center() - Permutation Group with generators [()] + Subgroup of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]) generated by [()] sage: G.random_element() # random output (1,5,3)(2,4) sage: print latex(G) diff --git a/doc/fr/tutorial/interfaces.rst b/doc/fr/tutorial/interfaces.rst --- a/doc/fr/tutorial/interfaces.rst +++ b/doc/fr/tutorial/interfaces.rst @@ -175,7 +175,7 @@ sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]]) sage: G.center() - Permutation Group with generators [()] + Subgroup of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]) generated by [()] sage: G.group_id() # nécessite le paquet facultatif database_gap (optional) [120, 34] sage: n = G.order(); n diff --git a/doc/fr/tutorial/tour_groups.rst b/doc/fr/tutorial/tour_groups.rst --- a/doc/fr/tutorial/tour_groups.rst +++ b/doc/fr/tutorial/tour_groups.rst @@ -23,7 +23,7 @@ [Permutation Group with generators [(1,2,3)(4,5), (3,4)], Permutation Group with generators [(1,5)(3,4), (1,5)(2,4), (1,3,5)]] sage: G.center() - Permutation Group with generators [()] + Subgroup of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]) generated by [()] sage: G.random_element() # sortie aléatoire (random) (1,5,3)(2,4) sage: print latex(G) diff --git a/sage/categories/finite_groups.py b/sage/categories/finite_groups.py --- a/sage/categories/finite_groups.py +++ b/sage/categories/finite_groups.py @@ -82,7 +82,7 @@ sage: A = AlternatingGroup(4) sage: A.semigroup_generators() - Family ((1,2,3), (2,3,4)) + Family ((2,3,4), (1,2,3)) """ return self.group_generators() diff --git a/sage/categories/g_sets.py b/sage/categories/g_sets.py --- a/sage/categories/g_sets.py +++ b/sage/categories/g_sets.py @@ -25,7 +25,7 @@ sage: S = SymmetricGroup(3) sage: GSets(S) - Category of G-sets for SymmetricGroup(3) + Category of G-sets for Symmetric group of order 3! as a permutation group TODO: should this derive from Category_over_base? """ @@ -44,7 +44,7 @@ EXAMPLES:: sage: GSets(SymmetricGroup(8)) # indirect doctests - Category of G-sets for SymmetricGroup(8) + Category of G-sets for Symmetric group of order 8! as a permutation group """ return "G-sets for %s"%self.__G @@ -70,7 +70,7 @@ EXAMPLES:: sage: GSets.an_instance() # indirect doctest - Category of G-sets for SymmetricGroup(8) + Category of G-sets for Symmetric group of order 8! as a permutation group """ from sage.groups.perm_gps.permgroup_named import SymmetricGroup G = SymmetricGroup(8) diff --git a/sage/categories/groupoid.py b/sage/categories/groupoid.py --- a/sage/categories/groupoid.py +++ b/sage/categories/groupoid.py @@ -50,7 +50,7 @@ sage: S8 = SymmetricGroup(8) sage: Groupoid(S8) - Groupoid with underlying set SymmetricGroup(8) + Groupoid with underlying set Symmetric group of order 8! as a permutation group """ return "Groupoid with underlying set %s"%self.__G @@ -76,7 +76,7 @@ EXAMPLES:: sage: Groupoid.an_instance() # indirect doctest - Groupoid with underlying set SymmetricGroup(8) + Groupoid with underlying set Symmetric group of order 8! as a permutation group """ from sage.groups.perm_gps.permgroup_named import SymmetricGroup G = SymmetricGroup(8) diff --git a/sage/categories/groups.py b/sage/categories/groups.py --- a/sage/categories/groups.py +++ b/sage/categories/groups.py @@ -68,7 +68,7 @@ sage: A = AlternatingGroup(4) sage: A.group_generators() - Family ((1,2,3), (2,3,4)) + Family ((2,3,4), (1,2,3)) """ return Family(self.gens()) diff --git a/sage/categories/homset.py b/sage/categories/homset.py --- a/sage/categories/homset.py +++ b/sage/categories/homset.py @@ -63,7 +63,7 @@ Category of vector spaces over Rational Field sage: G = AlternatingGroup(3) sage: Hom(G, G) - Set of Morphisms from AlternatingGroup(3) to AlternatingGroup(3) in Category of finite permutation groups + Set of Morphisms from Alternating group of order 3!/2 as a permutation group to Alternating group of order 3!/2 as a permutation group in Category of finite permutation groups sage: Hom(ZZ, QQ, Sets()) Set of Morphisms from Integer Ring to Rational Field in Category of sets @@ -196,7 +196,7 @@ sage: G = AlternatingGroup(3) sage: S = End(G); S - Set of Morphisms from AlternatingGroup(3) to AlternatingGroup(3) in Category of finite permutation groups + Set of Morphisms from Alternating group of order 3!/2 as a permutation group to Alternating group of order 3!/2 as a permutation group in Category of finite permutation groups sage: from sage.categories.homset import is_Endset sage: is_Endset(S) True @@ -361,26 +361,26 @@ sage: phi = Hom(SymmetricGroup(5), SymmetricGroup(6)).natural_map() sage: phi Coercion morphism: - From: SymmetricGroup(5) - To: SymmetricGroup(6) + From: Symmetric group of order 5! as a permutation group + To: Symmetric group of order 6! as a permutation group sage: H(phi) Composite map: - From: SymmetricGroup(4) - To: SymmetricGroup(7) + From: Symmetric group of order 4! as a permutation group + To: Symmetric group of order 7! as a permutation group Defn: Composite map: - From: SymmetricGroup(4) - To: SymmetricGroup(6) + From: Symmetric group of order 4! as a permutation group + To: Symmetric group of order 6! as a permutation group Defn: Call morphism: - From: SymmetricGroup(4) - To: SymmetricGroup(5) + From: Symmetric group of order 4! as a permutation group + To: Symmetric group of order 5! as a permutation group then Coercion morphism: - From: SymmetricGroup(5) - To: SymmetricGroup(6) + From: Symmetric group of order 5! as a permutation group + To: Symmetric group of order 6! as a permutation group then Call morphism: - From: SymmetricGroup(6) - To: SymmetricGroup(7) + From: Symmetric group of order 6! as a permutation group + To: Symmetric group of order 7! as a permutation group sage: H = Hom(ZZ, ZZ, Sets()) sage: f = H( lambda x: x + 1 ) diff --git a/sage/categories/hopf_algebras_with_basis.py b/sage/categories/hopf_algebras_with_basis.py --- a/sage/categories/hopf_algebras_with_basis.py +++ b/sage/categories/hopf_algebras_with_basis.py @@ -123,7 +123,7 @@ 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 SymmetricGroup(4) over Rational Field + 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 diff --git a/sage/categories/map.pyx b/sage/categories/map.pyx --- a/sage/categories/map.pyx +++ b/sage/categories/map.pyx @@ -115,7 +115,7 @@ sage: Map(QQ['x'], SymmetricGroup(6)) Generic map: From: Univariate Polynomial Ring in x over Rational Field - To: SymmetricGroup(6) + To: Symmetric group of order 6! as a permutation group """ if codomain is not None: if PY_TYPE_CHECK(parent, type): diff --git a/sage/categories/pushout.py b/sage/categories/pushout.py --- a/sage/categories/pushout.py +++ b/sage/categories/pushout.py @@ -2710,16 +2710,17 @@ rank = 10 - def __init__(self, gens): + def __init__(self, gens, domain): """ EXAMPLES:: sage: from sage.categories.pushout import PermutationGroupFunctor - sage: PF = PermutationGroupFunctor([PermutationGroupElement([(1,2)])]); PF + sage: PF = PermutationGroupFunctor([PermutationGroupElement([(1,2)])], [1,2]); PF PermutationGroupFunctor[(1,2)] """ Functor.__init__(self, Groups(), Groups()) self._gens = gens + self._domain = domain def __repr__(self): """ @@ -2742,7 +2743,8 @@ Permutation Group with generators [(1,2)] """ from sage.groups.perm_gps.permgroup import PermutationGroup - return PermutationGroup([g for g in (R.gens() + self.gens()) if not g.is_one()]) + return PermutationGroup([g for g in (R.gens() + self.gens()) if not g.is_one()], + domain=self._domain) def gens(self): """ @@ -2770,7 +2772,12 @@ """ if self.__class__ != other.__class__: return None - return PermutationGroupFunctor(self.gens() + other.gens()) + from sage.sets.all import FiniteEnumeratedSet + + new_domain = set(self._domain).union(set(other._domain)) + new_domain = FiniteEnumeratedSet(sorted(new_domain)) + return PermutationGroupFunctor(self.gens() + other.gens(), + new_domain) class BlackBoxConstructionFunctor(ConstructionFunctor): """ diff --git a/sage/combinat/permutation.py b/sage/combinat/permutation.py --- a/sage/combinat/permutation.py +++ b/sage/combinat/permutation.py @@ -562,6 +562,8 @@ cycles.append(tuple(cycle)) return cycles + cycle_tuples = to_cycles + def _to_cycles_orig(self, singletons=True): r""" Returns the permutation p as a list of disjoint cycles. @@ -3423,6 +3425,8 @@ p = range(1,n+1) for cycle in cycles: + if not cycle: + continue first = cycle[0] for i in range(len(cycle)-1): p[cycle[i]-1] = cycle[i+1] diff --git a/sage/combinat/species/cycle_species.py b/sage/combinat/species/cycle_species.py --- a/sage/combinat/species/cycle_species.py +++ b/sage/combinat/species/cycle_species.py @@ -100,10 +100,10 @@ sage: [a.transport(perm) for perm in a.automorphism_group()] [(1, 2, 3, 4), (1, 2, 3, 4), (1, 2, 3, 4), (1, 2, 3, 4)] """ - from sage.groups.all import SymmetricGroup + from sage.groups.all import SymmetricGroup, PermutationGroup S = SymmetricGroup(len(self._labels)) p = self.permutation_group_element() - return S.centralizer(p) + return PermutationGroup(S.centralizer(p).gens()) @accept_size @cached_function diff --git a/sage/combinat/species/permutation_species.py b/sage/combinat/species/permutation_species.py --- a/sage/combinat/species/permutation_species.py +++ b/sage/combinat/species/permutation_species.py @@ -98,10 +98,10 @@ ['a', 'c', 'b', 'd'], ['a', 'c', 'b', 'd']] """ - from sage.groups.all import SymmetricGroup + from sage.groups.all import SymmetricGroup, PermutationGroup S = SymmetricGroup(len(self._labels)) p = self.permutation_group_element() - return S.centralizer(p) + return PermutationGroup(S.centralizer(p).gens()) @accept_size @cached_function diff --git a/sage/groups/abelian_gps/abelian_group.py b/sage/groups/abelian_gps/abelian_group.py --- a/sage/groups/abelian_gps/abelian_group.py +++ b/sage/groups/abelian_gps/abelian_group.py @@ -830,7 +830,7 @@ sage: G = AbelianGroup(2,[2,3]); G Multiplicative Abelian Group isomorphic to C2 x C3 sage: G.permutation_group() - Permutation Group with generators [(1,4)(2,5)(3,6), (1,2,3)(4,5,6)] + Permutation Group with generators [(1,2,3)(4,5,6), (1,4)(2,5)(3,6)] """ from sage.groups.perm_gps.permgroup import PermutationGroup s = 'Image(IsomorphismPermGroup(%s))'%self._gap_init_() diff --git a/sage/groups/abelian_gps/abelian_group_element.py b/sage/groups/abelian_gps/abelian_group_element.py --- a/sage/groups/abelian_gps/abelian_group_element.py +++ b/sage/groups/abelian_gps/abelian_group_element.py @@ -227,7 +227,7 @@ Multiplicative Abelian Group isomorphic to C2 x C3 x C4 sage: a,b,c=G.gens() sage: Gp = G.permutation_group(); Gp - Permutation Group with generators [(1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (1,3,2,4)(5,7,6,8)(9,11,10,12)(13,15,14,16)(17,19,18,20)(21,23,22,24)] + Permutation Group with generators [(1,3,2,4)(5,7,6,8)(9,11,10,12)(13,15,14,16)(17,19,18,20)(21,23,22,24), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)] sage: a.as_permutation() (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24) sage: ap = a.as_permutation(); ap diff --git a/sage/groups/class_function.py b/sage/groups/class_function.py --- a/sage/groups/class_function.py +++ b/sage/groups/class_function.py @@ -407,7 +407,7 @@ Character of Symmetric group of order 5! as a permutation group sage: H = G.subgroup([(1,2,3), (1,2), (4,5)]) sage: chi.restrict(H) - Character of Subgroup of SymmetricGroup(5) generated by [(4,5), (1,2), (1,2,3)] + Character of Subgroup of (Symmetric group of order 5! as a permutation group) generated by [(4,5), (1,2), (1,2,3)] sage: chi.restrict(H).values() [3, -3, -3, -1, 0, 0] """ @@ -420,7 +420,7 @@ sage: G = SymmetricGroup(5) sage: H = G.subgroup([(1,2,3), (1,2), (4,5)]) sage: xi = H.trivial_character(); xi - Character of Subgroup of SymmetricGroup(5) generated by [(4,5), (1,2), (1,2,3)] + Character of Subgroup of (Symmetric group of order 5! as a permutation group) generated by [(4,5), (1,2), (1,2,3)] sage: xi.induct(G) Character of Symmetric group of order 5! as a permutation group sage: xi.induct(G).values() diff --git a/sage/groups/perm_gps/permgroup.py b/sage/groups/perm_gps/permgroup.py --- a/sage/groups/perm_gps/permgroup.py +++ b/sage/groups/perm_gps/permgroup.py @@ -125,11 +125,13 @@ from sage.rings.all import QQ, Integer from sage.interfaces.all import is_ExpectElement from sage.interfaces.gap import gap, GapElement -from sage.groups.perm_gps.permgroup_element import PermutationGroupElement +from sage.groups.perm_gps.permgroup_element import PermutationGroupElement, standardize_generator from sage.groups.abelian_gps.abelian_group import AbelianGroup from sage.misc.cachefunc import cached_method from sage.groups.class_function import ClassFunction from sage.misc.package import is_package_installed +from sage.sets.all import FiniteEnumeratedSet +from sage.categories.all import FiniteEnumeratedSets def load_hap(): """ @@ -231,7 +233,7 @@ srcs = map(G, srcs) return srcs -def PermutationGroup(gens=None, gap_group=None, canonicalize=True): +def PermutationGroup(gens=None, gap_group=None, domain=None, canonicalize=True, category=None): """ Return the permutation group associated to `x` (typically a list of generators). @@ -301,7 +303,8 @@ return gens._permgroup_() if gens is not None and not isinstance(gens, (tuple, list, GapElement)): raise TypeError, "gens must be a tuple, list, or GapElement" - return PermutationGroup_generic(gens=gens, gap_group=gap_group, canonicalize=canonicalize) + return PermutationGroup_generic(gens=gens, gap_group=gap_group, domain=domain, + canonicalize=canonicalize, category=category) class PermutationGroup_generic(group.Group): @@ -312,7 +315,7 @@ sage: G Permutation Group with generators [(3,4), (1,2,3)(4,5)] sage: G.center() - Permutation Group with generators [()] + Subgroup of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]) generated by [()] sage: G.group_id() # optional - database_gap [120, 34] sage: n = G.order(); n @@ -320,7 +323,7 @@ sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]]) sage: TestSuite(G).run() """ - def __init__(self, gens=None, gap_group=None, canonicalize=True): + def __init__(self, gens=None, gap_group=None, canonicalize=True, domain=None, category=None): r""" INPUT: @@ -347,7 +350,7 @@ sage: A4.__init__([[(1,2,3)],[(2,3,4)]]); A4 Permutation Group with generators [(2,3,4), (1,2,3)] sage: A4.center() - Permutation Group with generators [()] + Subgroup of (Permutation Group with generators [(2,3,4), (1,2,3)]) generated by [()] sage: A4.category() Category of finite permutation groups sage: TestSuite(A4).run() @@ -358,28 +361,44 @@ sage: TestSuite(PermutationGroup([])).run() """ from sage.categories.finite_permutation_groups import FinitePermutationGroups - super(PermutationGroup_generic, self).__init__(category = FinitePermutationGroups()) + super(PermutationGroup_generic, self).__init__(category = FinitePermutationGroups().or_subcategory(category)) if (gens is None and gap_group is None): raise ValueError, "you must specify gens or gap_group" #Handle the case where only the GAP group is specified. if gens is None: - from sage.interfaces.gap import gap - self._gap_string = gap_group if isinstance(gap_group, str) else str(gap_group) - self._gens = self._gens_from_gap() - self._deg = gap(gap_group).LargestMovedPoint() - self._set = range(1, self._deg+1) - else: - gens = [self._element_class()(x, check=False).list() for x in gens] - self._deg = max([0]+[max(g) for g in gens]) - self._set = range(1, self._deg+1) - if not gens: # length 0 - gens = [()] - gens = [self._element_class()(x, self, check=False) for x in gens] - if canonicalize: - gens = sorted(set(gens)) - self._gens = gens - self._gap_string = 'Group(%s)'%gens + if isinstance(gap_group, str): + gap_group = gap(gap_group) + gens = [gen for gen in gap_group.GeneratorsOfGroup()] + + if domain is None: + gens = [standardize_generator(x) for x in gens] + domain = set() + for x in gens: + for cycle in x: + domain = domain.union(cycle) + domain = sorted(domain) + + #Here we need to check if all of the points are integers + #to make the domain contain all integers up to the max. + #This is needed for backward compatibility + if all(isinstance(p, (Integer, int, long)) for p in domain): + domain = range(1, max([1] + domain)+1) + + if domain not in FiniteEnumeratedSets(): + domain = FiniteEnumeratedSet(domain) + + self._domain = domain + self._deg = len(self._domain) + self._domain_to_gap = dict((key, i+1) for i, key in enumerate(self._domain)) + self._domain_from_gap = dict((i+1, key) for i, key in enumerate(self._domain)) + + if not gens: # length 0 + gens = [()] + gens = [self._element_class()(x, self, check=False) for x in gens] + if canonicalize: + gens = sorted(set(gens)) + self._gens = gens def construction(self): """ @@ -400,36 +419,46 @@ (1,2,3) sage: p.parent() Permutation Group with generators [(1,2), (1,3)] + sage: p.parent().domain() + {1, 2, 3} + + Note that this will merge permutation groups with different + domains:: + + sage: g1 = PermutationGroupElement([(1,2),(3,4,5)]) + sage: g2 = PermutationGroup([('a','b')], domain=['a', 'b']).gens()[0] + sage: g2 + ('a','b') + sage: p = g1*g2; p + (1,2)(3,4,5)('a','b') """ gens = self.gens() if len(gens) == 1 and gens[0].is_one(): return None else: from sage.categories.pushout import PermutationGroupFunctor - return (PermutationGroupFunctor(gens), PermutationGroup([])) - - def _gens_from_gap(self): + return (PermutationGroupFunctor(gens, self.domain()), + PermutationGroup([])) + + @cached_method + def _has_natural_domain(self): """ - Returns the generators of the group by asking GAP for them. - + Returns True if the underlying domain is of the form (1,...,n) + EXAMPLES:: - - sage: S = SymmetricGroup(3) - sage: S._gap_string - 'SymmetricGroup(3)' - sage: S._gens_from_gap() - [(1,2,3), (1,2)] + + sage: SymmetricGroup(3)._has_natural_domain() + True + sage: SymmetricGroup((1,2,3))._has_natural_domain() + True + sage: SymmetricGroup((1,3))._has_natural_domain() + False + sage: SymmetricGroup((3,2,1))._has_natural_domain() + False """ - try: - gens = self._gap_().GeneratorsOfGroup() - except TypeError, s: - raise RuntimeError, "(It might be necessary to install the database_gap optional Sage package, if you haven't already.)\n%s"%s - gens = [self._element_class()(gens[n], self, check=False) - for n in range(1, int(gens.Length())+1)] - if gens == []: - gens = [()] - return gens - + domain = self.domain() + natural_domain = FiniteEnumeratedSet(range(1, len(domain)+1)) + return domain == natural_domain def _gap_init_(self): r""" @@ -445,14 +474,14 @@ sage: A4 = PermutationGroup([[(1,2,3)],[(2,3,4)]]); A4 Permutation Group with generators [(2,3,4), (1,2,3)] sage: A4._gap_init_() - 'Group([(2,3,4), (1,2,3)])' + 'Group([PermList([1, 3, 4, 2]), PermList([2, 3, 1, 4])])' """ - return self._gap_string + return 'Group([%s])'%(', '.join([g._gap_init_() for g in self.gens()])) def _magma_init_(self, magma): r""" Returns a string showing how to declare / initialize self in Magma. - + EXAMPLES: We explicitly construct the alternating group on four @@ -463,8 +492,12 @@ Permutation Group with generators [(2,3,4), (1,2,3)] sage: A4._magma_init_(magma) 'PermutationGroup<4 | (2,3,4), (1,2,3)>' + + sage: S = SymmetricGroup(['a', 'b', 'c']) + sage: S._magma_init_(magma) + 'PermutationGroup<3 | (1,2,3), (1,2)>' """ - g = str(self.gens())[1:-1] + g = ', '.join([g._gap_cycle_string() for g in self.gens()]) return 'PermutationGroup<%s | %s>'%(self.degree(), g) def __cmp__(self, right): @@ -500,9 +533,9 @@ True sage: H3 < H1 # since H3 is a subgroup of H1 True - """ - if not isinstance(right, PermutationGroup_generic): + if (not isinstance(right, PermutationGroup_generic) or + self.domain() != right.domain()): return -1 if self is right: return 0 @@ -581,7 +614,7 @@ sage: G([(1,2)]) Traceback (most recent call last): ... - TypeError: permutation (1, 2) not in Permutation Group with generators [(1,2,3,4)] + TypeError: permutation [(1, 2)] not in Permutation Group with generators [(1,2,3,4)] """ try: if x.parent() is self: @@ -630,9 +663,10 @@ x_parent = x.parent() if x_parent is self: return x - elif (x_parent.degree() <= self.degree() and - (self.__class__ == SymmetricGroup or x._gap_() in self._gap_())): - return self._element_class()(x.list(), self, check = False) + + compatible_domains = all(point in self._domain_to_gap for point in x_parent.domain()) + if compatible_domains and (self.__class__ == SymmetricGroup or x._gap_() in self._gap_()): + return self._element_class()(x.cycle_tuples(), self, check=False) raise TypeError, "no implicit coercion of element into permutation group" @cached_method @@ -645,12 +679,13 @@ sage: G = PermutationGroup([[(1,2,3,4)], [(1,2)]]) sage: G.list() [(), (3,4), (2,3), (2,3,4), (2,4,3), (2,4), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4), (1,2,4,3), (1,2,4), (1,3,2), (1,3,4,2), (1,3), (1,3,4), (1,3)(2,4), (1,3,2,4), (1,4,3,2), (1,4,2), (1,4,3), (1,4), (1,4,2,3), (1,4)(2,3)] + + sage: G = PermutationGroup([[('a','b')]], domain=('a', 'b')); G + Permutation Group with generators [('a','b')] + sage: G.list() + [(), ('a','b')] """ - X = self._gap_().Elements() - n = X.Length() - L = [self._element_class()(X[i], self, check = False) - for i in range(1,n+1)] - return L + return list(self.__iter__()) def __contains__(self, item): """ @@ -671,6 +706,10 @@ True sage: h in H False + + sage: G = PermutationGroup([[('a','b')]], domain=('a', 'b')) + sage: [('a', 'b')] in G + True """ try: item = self(item, check=True) @@ -709,11 +748,9 @@ sage: G = PermutationGroup([[(1,2,3)], [(1,2)]]) sage: [a for a in G] [(), (2,3), (1,2), (1,2,3), (1,3,2), (1,3)] - - TODO: this currently returns an iterator over the elements of - ``self.list()``. This should be made into a real iterator. """ - return iter(self.list()) + for g in self._gap_().Elements(): + yield self._element_class()(g, self, check=False) def gens(self): """ @@ -769,6 +806,10 @@ sage: G = PermutationGroup([R,L,U,F,B,D]) sage: len(G.gens_small()) 2 + + sage: G = PermutationGroup([[('a','b')], [('b', 'c')], [('a', 'c')]]) + sage: G.gens_small() + [('a','b'), ('a','b','c')] """ gens = self._gap_().SmallGeneratingSet() return [self._element_class()(x, self, check=False) for x in gens] @@ -825,6 +866,10 @@ (1,2,3)(4,5) sage: e*g (1,2,3)(4,5) + + sage: S = SymmetricGroup(['a','b','c']) + sage: S.identity() + () """ return self._element_class()([], self, check=True) @@ -858,61 +903,82 @@ :: - sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]]) + sage: G = PermutationGroup([[('a','b','c'),('d','e')]]) + sage: G.largest_moved_point() + 'e' + + .. warning:: + + The name of this function is not good; this function + should be deprecated in term of degree:: + + sage: P = PermutationGroup([[1,2,3,4]]) + sage: P.largest_moved_point() + 4 + sage: P.cardinality() + 1 + """ + try: + return self._domain_from_gap[Integer(self._gap_().LargestMovedPoint())] + except KeyError: + return self.degree() + + def degree(self): + """ + Returns the degree of this permutation group. + + EXAMPLES:: + + sage: S = SymmetricGroup(['a','b','c']) + sage: S.degree() + 3 + sage: G = PermutationGroup([(1,3),(4,5)]) sage: G.degree() 5 - TODO: the name of this function is not good; this function - should be deprecated in term of degree:: - - sage: P = PermutationGroup([[1,2,3,4]]) - sage: P.largest_moved_point() + Note that you can explicitly specify the domain to get a + permutation group of smaller degree:: + + sage: G = PermutationGroup([(1,3),(4,5)], domain=[1,3,4,5]) + sage: G.degree() 4 - sage: P.cardinality() - 1 """ - # This seems unndeeded since __init__ systematically sets - # self._deg This is actually needed because constructing the - # __init__ of PermutationGroupElement sometimes calls this - # group before is initialized. - try: - return self._deg - except AttributeError: - n = Integer(self._gap_().LargestMovedPoint()) - self._deg = n - return n - - degree = largest_moved_point - - def set(self): + return Integer(len(self._domain)) + + def domain(self): r""" Returns the underlying set that this permutation group acts - on. By default, the set is {1, ..., n} where n is the largest - moved point by the group. + on. EXAMPLES:: sage: P = PermutationGroup([(1,2),(3,5)]) - sage: P.set() - (1, 2, 3, 4, 5) - + sage: P.domain() + {1, 2, 3, 4, 5} + sage: S = SymmetricGroup(['a', 'b', 'c']) + sage: S.domain() + {'a', 'b', 'c'} """ - return tuple(self._set) - - domain = set - - def _set_gap(self): + return self._domain + + def _domain_gap(self, domain=None): """ Returns a GAP string representation of the underlying set - that this group acts on. See also :meth:`set`. + that this group acts on. See also :meth:`domain`. EXAMPLES:: sage: P = PermutationGroup([(1,2),(3,5)]) - sage: P._set_gap() + sage: P._domain_gap() '[1, 2, 3, 4, 5]' """ - return repr(list(self.set())) + if domain is None: + return repr(range(1, self.degree()+1)) + else: + try: + return repr([self._domain_to_gap[point] for point in domain]) + except KeyError: + raise ValueError, "domain must be a subdomain of self.domain()" @cached_method def smallest_moved_point(self): @@ -927,13 +993,20 @@ sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]]) sage: G.smallest_moved_point() 1 + + Note that this function uses the ordering from the domain:: + + sage: S = SymmetricGroup(['a','b','c']) + sage: S.smallest_moved_point() + 'a' """ - return Integer(self._gap_().SmallestMovedPoint()) + return self._domain_from_gap[Integer(self._gap_().SmallestMovedPoint())] @cached_method def orbits(self): """ - Returns the orbits of [1,2,...,degree] under the group action. + Returns the orbits of the elements of the domain under the + group action. EXAMPLES:: @@ -943,6 +1016,10 @@ sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]]) sage: G.orbits() [[1, 2, 3, 4, 10], [5], [6], [7], [8], [9]] + + sage: G = PermutationGroup([ [('c','d')], [('a','c')],[('b',)]]) + sage: G.orbits() + [['a', 'c', 'd'], ['b']] The answer is cached:: @@ -953,12 +1030,13 @@ - Nathan Dunfield """ - return self._gap_().Orbits(self._set_gap()).sage() + return [[self._domain_from_gap[x] for x in orbit] for orbit in + self._gap_().Orbits(self._domain_gap()).sage()] @cached_method - def orbit(self, integer): + def orbit(self, point): """ - Return the orbit of the given integer under the group action. + Return the orbit of the given point under the group action. EXAMPLES:: @@ -968,10 +1046,15 @@ sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]]) sage: G.orbit(3) [3, 4, 10, 1, 2] - """ - return self._gap_().Orbit(integer).sage() + + sage: G = PermutationGroup([ [('c','d')], [('a','c')] ]) + sage: G.orbit('a') + ['a', 'c', 'd'] + """ + point = self._domain_to_gap[point] + return [self._domain_from_gap[x] for x in self._gap_().Orbit(point).sage()] - def transversals(self, integer): + def transversals(self, point): """ If G is a permutation group acting on the set `X = \{1, 2, ...., n\}` and H is the stabilizer subgroup of , a right @@ -988,13 +1071,16 @@ sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]]) sage: G.transversals(1) [(), (1,2)(3,4), (1,3,2,10,4), (1,4,2,10,3), (1,10,4,3,2)] + + sage: G = PermutationGroup([ [('c','d')], [('a','c')] ]) + sage: G.transversals('a') + [(), ('a','c','d'), ('a','d','c')] """ - trans = [] - for i in self.orbit(integer): - trans.append(self(gap.RepresentativeAction(self._gap_(),integer,i))) - return trans - - def stabilizer(self, position): + G = self._gap_() + return [self(G.RepresentativeAction(self._domain_to_gap[point], self._domain_to_gap[i])) + for i in self.orbit(point)] + + def stabilizer(self, point): """ Return the subgroup of ``self`` which stabilize the given position. ``self`` and its stabilizers must have same degree. @@ -1003,35 +1089,39 @@ sage: G = PermutationGroup([ [(3,4)], [(1,3)] ]) sage: G.stabilizer(1) - Permutation Group with generators [(3,4)] + Subgroup of (Permutation Group with generators [(3,4), (1,3)]) generated by [(3,4)] sage: G.stabilizer(3) - Permutation Group with generators [(1,4)] + Subgroup of (Permutation Group with generators [(3,4), (1,3)]) generated by [(1,4)] sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]]) sage: G.stabilizer(10) - Permutation Group with generators [(1,2)(3,4), (2,3,4)] + Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4,10)]) generated by [(2,3,4), (1,2)(3,4)] sage: G.stabilizer(1) - Permutation Group with generators [(2,3)(4,10), (2,10,4)] + Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4,10)]) generated by [(2,3)(4,10), (2,10,4)] sage: G = PermutationGroup([[(2,3,4)],[(6,7)]]) sage: G.stabilizer(1) - Permutation Group with generators [(6,7), (2,3,4)] + Subgroup of (Permutation Group with generators [(6,7), (2,3,4)]) generated by [(6,7), (2,3,4)] sage: G.stabilizer(2) - Permutation Group with generators [(6,7)] + Subgroup of (Permutation Group with generators [(6,7), (2,3,4)]) generated by [(6,7)] sage: G.stabilizer(3) - Permutation Group with generators [(6,7)] + Subgroup of (Permutation Group with generators [(6,7), (2,3,4)]) generated by [(6,7)] sage: G.stabilizer(4) - Permutation Group with generators [(6,7)] + Subgroup of (Permutation Group with generators [(6,7), (2,3,4)]) generated by [(6,7)] sage: G.stabilizer(5) - Permutation Group with generators [(6,7), (2,3,4)] + Subgroup of (Permutation Group with generators [(6,7), (2,3,4)]) generated by [(6,7), (2,3,4)] sage: G.stabilizer(6) - Permutation Group with generators [(2,3,4)] + Subgroup of (Permutation Group with generators [(6,7), (2,3,4)]) generated by [(2,3,4)] sage: G.stabilizer(7) - Permutation Group with generators [(2,3,4)] + Subgroup of (Permutation Group with generators [(6,7), (2,3,4)]) generated by [(2,3,4)] sage: G.stabilizer(8) - Permutation Group with generators [(6,7), (2,3,4)] + Subgroup of (Permutation Group with generators [(6,7), (2,3,4)]) generated by [(6,7), (2,3,4)] """ - return PermutationGroup(gap_group=gap.Stabilizer(self, position)) - - def strong_generating_system(self, base_of_group = None): + try: + position = self._domain_to_gap[point] + except KeyError: + return self.subgroup(gens=self.gens()) + return self.subgroup(gap_group=gap.Stabilizer(self, point)) + + def strong_generating_system(self, base_of_group=None): """ Return a Strong Generating System of ``self`` according the given base for the right action of ``self`` on itself. @@ -1080,7 +1170,7 @@ [[(), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)], [(), (2,3,4), (2,4,3)], [(), (3,4)], [()]] sage: G = PermutationGroup([[(1,2,3)],[(4,5,7)],[(1,4,6)]]) sage: G.strong_generating_system() - [[(), (1,2,3), (1,4,6), (1,3,2), (1,5,7,4,6), (1,6,4), (1,7,5,4,6)], [(), (2,6,3), (2,3,6), (2,5,6,3)(4,7), (2,7,5,6,3), (2,4,5,6,3)], [(), (3,6)(5,7), (3,5,6), (3,7,4,5,6), (3,4,7,5,6)], [(), (4,5)(6,7), (4,7)(5,6), (4,6)(5,7)], [(), (5,6,7), (5,7,6)], [()], [()]] + [[(), (1,2,3), (1,4,6), (1,3,2), (1,5,7,4,6), (1,6,4), (1,7,5,4,6)], [(), (2,6,3), (2,5,7,6,3), (2,3,6), (2,7,5,6,3), (2,4,7,6,3)], [(), (3,6,7), (3,5,6), (3,7,6), (3,4,7,5,6)], [(), (4,5)(6,7), (4,7)(5,6), (4,6)(5,7)], [(), (5,7,6), (5,6,7)], [()], [()]] sage: G = PermutationGroup([[(1,2,3)],[(2,3,4)],[(3,4,5)]]) sage: G.strong_generating_system([5,4,3,2,1]) [[(), (1,5,3,4,2), (1,5,4,3,2), (1,5)(2,3), (1,5,2)], [(), (1,3)(2,4), (1,2)(3,4), (1,4)(2,3)], [(), (1,3,2), (1,2,3)], [()], [()]] @@ -1096,7 +1186,7 @@ sgs = [] stab = self if base_of_group is None: - base_of_group = self.set() + base_of_group = self.domain() for j in base_of_group: sgs.append(stab.transversals(j)) stab = stab.stabilizer(j) @@ -1140,6 +1230,10 @@ \langle (2,3,4), (1,2,3) \rangle sage: A4._latex_() '\\langle (2,3,4), (1,2,3) \\rangle' + + sage: S = SymmetricGroup(['a','b','c']) + sage: latex(S) + \langle (\texttt{a},\texttt{b},\texttt{c}), (\texttt{a},\texttt{b}) \rangle """ return '\\langle ' + \ ', '.join([x._latex_() for x in self.gens()]) + ' \\rangle' @@ -1218,12 +1312,82 @@ sage: G = PermutationGroup([[(1,2,3,4)]]) sage: G.center() - Permutation Group with generators [(1,2,3,4)] + Subgroup of (Permutation Group with generators [(1,2,3,4)]) generated by [(1,2,3,4)] sage: G = PermutationGroup([[(1,2,3,4)], [(1,2)]]) sage: G.center() - Permutation Group with generators [()] + Subgroup of (Permutation Group with generators [(1,2), (1,2,3,4)]) generated by [()] """ - return PermutationGroup(gap_group=self._gap_().Center()) + return self.subgroup(gap_group=self._gap_().Center()) + + def socle(self): + r""" + Returns the socle of ``self``. The socle of a group $G$ is + the subgroup generated by all minimal normal subgroups. + + EXAMPLES:: + + sage: G=SymmetricGroup(4) + sage: G.socle() + Subgroup of (Symmetric group of order 4! as a permutation group) generated by [(1,2)(3,4), (1,4)(2,3)] + sage: G.socle().socle() + Subgroup of (Subgroup of (Symmetric group of order 4! as a permutation group) generated by [(1,2)(3,4), (1,4)(2,3)]) generated by [(1,3)(2,4), (1,4)(2,3)] + + """ + return self.subgroup(gap_group=self._gap_().Socle()) + + def frattini_subgroup(self): + r""" + Returns the Frattini subgroup of ``self``. The Frattini + subgroup of a group $G$ is the intersection of all + maximal subgroups of $G$. + + EXAMPLES:: + + sage: G=PermutationGroup([[(1,2,3,4)],[(2,4)]]) + sage: G.frattini_subgroup() + Subgroup of (Permutation Group with generators [(2,4), (1,2,3,4)]) generated by [(1,3)(2,4)] + sage: G=SymmetricGroup(4) + sage: G.frattini_subgroup() + Subgroup of (Symmetric group of order 4! as a permutation group) generated by [()] + + """ + return self.subgroup(gap_group=self._gap_().FrattiniSubgroup()) + + def fitting_subgroup(self): + r""" + Returns the Fitting subgroup of ``self``. The Fitting + subgroup of a group $G$ is the largest nilpotent normal + subgroup of $G$. + + EXAMPLES:: + + sage: G=PermutationGroup([[(1,2,3,4)],[(2,4)]]) + sage: G.fitting_subgroup() + Subgroup of (Permutation Group with generators [(2,4), (1,2,3,4)]) generated by [(2,4), (1,2,3,4), (1,3)] + sage: G=PermutationGroup([[(1,2,3,4)],[(1,2)]]) + sage: G.fitting_subgroup() + Subgroup of (Permutation Group with generators [(1,2), (1,2,3,4)]) generated by [(1,2)(3,4), (1,3)(2,4)] + + """ + return self.subgroup(gap_group=self._gap_().FittingSubgroup()) + + def solvable_radical(self): + r""" + Returns the solvable radical of ``self``. The solvable + radical (or just radical) of a group $G$ is the largest + solvable normal subgroup of $G$. + + EXAMPLES:: + + sage: G=SymmetricGroup(4) + sage: G.solvable_radical() + Subgroup of (Symmetric group of order 4! as a permutation group) generated by [(1,2), (1,2,3,4)] + sage: G=SymmetricGroup(5) + sage: G.solvable_radical() + Subgroup of (Symmetric group of order 5! as a permutation group) generated by [()] + + """ + return self.subgroup(gap_group=self._gap_().RadicalGroup()) def intersection(self, other): r""" @@ -1310,7 +1474,7 @@ sage: G = DihedralGroup(6) sage: a = PermutationGroupElement("(1,2,3,4)") sage: G.conjugate(a) - Permutation Group with generators [(1,5,6,2,3,4), (1,4)(2,6)(3,5)] + Permutation Group with generators [(1,4)(2,6)(3,5), (1,5,6,2,3,4)] The element performing the conjugation can be specified in several ways. :: @@ -1318,15 +1482,15 @@ sage: G = DihedralGroup(6) sage: strng = "(1,2,3,4)" sage: G.conjugate(strng) - Permutation Group with generators [(1,5,6,2,3,4), (1,4)(2,6)(3,5)] + Permutation Group with generators [(1,4)(2,6)(3,5), (1,5,6,2,3,4)] sage: G = DihedralGroup(6) sage: lst = [2,3,4,1] sage: G.conjugate(lst) - Permutation Group with generators [(1,5,6,2,3,4), (1,4)(2,6)(3,5)] + Permutation Group with generators [(1,4)(2,6)(3,5), (1,5,6,2,3,4)] sage: G = DihedralGroup(6) sage: cycles = [(1,2,3,4)] sage: G.conjugate(cycles) - Permutation Group with generators [(1,5,6,2,3,4), (1,4)(2,6)(3,5)] + Permutation Group with generators [(1,4)(2,6)(3,5), (1,5,6,2,3,4)] Conjugation is a group automorphism, so conjugate groups will be isomorphic. :: @@ -1379,14 +1543,10 @@ INPUT: - - ``self, other`` - permutation groups - - OUTPUT: - - ``D`` - a direct product of the inputs, returned as a permutation group as well @@ -1400,30 +1560,29 @@ - ``pr2`` - the projection of ``D`` onto ``other`` (giving a splitting 1 - self - D - other - 1) - EXAMPLES:: sage: G = CyclicPermutationGroup(4) sage: D = G.direct_product(G,False) sage: D - Permutation Group with generators [(1,2,3,4), (5,6,7,8)] + Permutation Group with generators [(5,6,7,8), (1,2,3,4)] sage: D,iota1,iota2,pr1,pr2 = G.direct_product(G) sage: D; iota1; iota2; pr1; pr2 - Permutation Group with generators [(1,2,3,4), (5,6,7,8)] + Permutation Group with generators [(5,6,7,8), (1,2,3,4)] Permutation group morphism: From: Cyclic group of order 4 as a permutation group - To: Permutation Group with generators [(1,2,3,4), (5,6,7,8)] + To: Permutation Group with generators [(5,6,7,8), (1,2,3,4)] Defn: Embedding( Group( [ (1,2,3,4), (5,6,7,8) ] ), 1 ) Permutation group morphism: From: Cyclic group of order 4 as a permutation group - To: Permutation Group with generators [(1,2,3,4), (5,6,7,8)] + To: Permutation Group with generators [(5,6,7,8), (1,2,3,4)] Defn: Embedding( Group( [ (1,2,3,4), (5,6,7,8) ] ), 2 ) Permutation group morphism: - From: Permutation Group with generators [(1,2,3,4), (5,6,7,8)] + From: Permutation Group with generators [(5,6,7,8), (1,2,3,4)] To: Cyclic group of order 4 as a permutation group Defn: Projection( Group( [ (1,2,3,4), (5,6,7,8) ] ), 1 ) Permutation group morphism: - From: Permutation Group with generators [(1,2,3,4), (5,6,7,8)] + From: Permutation Group with generators [(5,6,7,8), (1,2,3,4)] To: Cyclic group of order 4 as a permutation group Defn: Projection( Group( [ (1,2,3,4), (5,6,7,8) ] ), 2 ) sage: g=D([(1,3),(2,4)]); g @@ -1436,7 +1595,7 @@ (1,4,3,2) (1,3)(2,4) """ - G = self._gap_().DirectProduct(other._gap_()) + G = self._gap_().DirectProduct(other) D = PermutationGroup(gap_group=G) if not maps: return D @@ -1448,7 +1607,7 @@ pr2 = PermutationGroupMorphism_from_gap(D, other, G.Projection(2)) return D, iota1, iota2, pr1, pr2 - def subgroup(self, gens): + def subgroup(self, gens=None, gap_group=None, domain=None, category=None, canonicalize=True, check=True): """ Wraps the ``PermutationGroup_subgroup`` constructor. The argument ``gens`` is a list of elements of ``self``. @@ -1458,10 +1617,10 @@ sage: G = PermutationGroup([(1,2,3),(3,4,5)]) sage: g = G((1,2,3)) sage: G.subgroup([g]) - Subgroup of Permutation Group with generators [(3,4,5), (1,2,3)] generated by [(1,2,3)] + Subgroup of (Permutation Group with generators [(3,4,5), (1,2,3)]) generated by [(1,2,3)] """ - return PermutationGroup_subgroup(self, gens) - + return PermutationGroup_subgroup(self, gens=gens, gap_group=gap_group, domain=None, + category=category, canonicalize=canonicalize, check=check) def quotient(self, N): """ @@ -1484,7 +1643,7 @@ # This is currently done using the right regular representation # FIXME: GAP certainly knows of a better way! phi = Q.RegularActionHomomorphism() - return PermutationGroup(gap_group = phi.Image()) + return PermutationGroup(gap_group=phi.Image()) def quotient_group(self, N): """ @@ -1921,7 +2080,13 @@ sage: G = SymmetricGroup(5) sage: G.conjugacy_classes_representatives() [(), (1,2), (1,2)(3,4), (1,2,3), (1,2,3)(4,5), (1,2,3,4), (1,2,3,4,5)] - + + :: + + sage: S = SymmetricGroup(['a','b','c']) + sage: S.conjugacy_classes_representatives() + [(), ('a','b'), ('a','b','c')] + AUTHORS: - David Joyner and William Stein (2006-01-04) @@ -1940,30 +2105,20 @@ sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]]) sage: cl = G.conjugacy_classes_subgroups() sage: cl - [Permutation Group with generators [()], - Permutation Group with generators [(1,2)(3,4)], - Permutation Group with generators [(1,3)(2,4)], - Permutation Group with generators [(2,4)], - Permutation Group with generators [(1,4)(2,3), (1,2)(3,4)], - Permutation Group with generators [(1,3)(2,4), (2,4)], - Permutation Group with generators [(1,3)(2,4), (1,2,3,4)], - Permutation Group with generators [(1,3)(2,4), (1,2)(3,4), (1,2,3,4)]] + [Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [()], Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(1,2)(3,4)], Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(1,3)(2,4)], Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(2,4)], Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(1,2)(3,4), (1,4)(2,3)], Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(2,4), (1,3)(2,4)], Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(1,2,3,4), (1,3)(2,4)], Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(1,2)(3,4), (1,2,3,4), (1,3)(2,4)]] :: sage: G = SymmetricGroup(3) sage: G.conjugacy_classes_subgroups() - [Permutation Group with generators [()], - Permutation Group with generators [(2,3)], - Permutation Group with generators [(1,2,3)], - Permutation Group with generators [(1,3,2), (1,2)]] + [Subgroup of (Symmetric group of order 3! as a permutation group) generated by [()], Subgroup of (Symmetric group of order 3! as a permutation group) generated by [(2,3)], Subgroup of (Symmetric group of order 3! as a permutation group) generated by [(1,2,3)], Subgroup of (Symmetric group of order 3! as a permutation group) generated by [(1,2), (1,3,2)]] AUTHORS: - David Joyner (2006-10) """ cl = self._gap_().ConjugacyClassesSubgroups() - return [PermutationGroup(gap_group=sub.Representative()) for sub in cl] + return [self.subgroup(gap_group=sub.Representative()) for sub in cl] def subgroups(self): r""" @@ -1990,7 +2145,9 @@ `30030 = 2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13` takes about twice as long. - For faster results, which still exhibit the structure of the possible subgroups, use :meth:`conjugacy_classes_subgroups`. + For faster results, which still exhibit the structure of + the possible subgroups, use + :meth:`conjugacy_classes_subgroups`. EXAMPLES:: @@ -2001,7 +2158,7 @@ Permutation Group with generators [(1,2)], Permutation Group with generators [(1,3)], Permutation Group with generators [(1,2,3)], - Permutation Group with generators [(1,3,2), (1,2)]] + Permutation Group with generators [(1,2), (1,3,2)]] sage: G = CyclicPermutationGroup(14) sage: G.subgroups() @@ -2160,7 +2317,7 @@ sage: A.cosets(S) Traceback (most recent call last): ... - ValueError: Subgroup of SymmetricGroup(3) generated by [(1,2)] is not a subgroup of AlternatingGroup(3) + ValueError: Subgroup of (Symmetric group of order 3! as a permutation group) generated by [(1,2)] is not a subgroup of Alternating group of order 3!/2 as a permutation group AUTHOR: @@ -2201,16 +2358,15 @@ sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]]) sage: g = G([(1,3)]) sage: G.normalizer(g) - Permutation Group with generators [(2,4), (1,3)] + Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(2,4), (1,3)] sage: g = G([(1,2,3,4)]) sage: G.normalizer(g) - Permutation Group with generators [(2,4), (1,2,3,4), (1,3)(2,4)] + Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(2,4), (1,2,3,4), (1,3)(2,4)] sage: H = G.subgroup([G([(1,2,3,4)])]) sage: G.normalizer(H) - Permutation Group with generators [(2,4), (1,2,3,4), (1,3)(2,4)] + Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(2,4), (1,2,3,4), (1,3)(2,4)] """ - N = self._gap_().Normalizer(g) - return PermutationGroup(N.GeneratorsOfGroup()) + return self.subgroup(gap_group=self._gap_().Normalizer(g)) def centralizer(self, g): """ @@ -2221,16 +2377,15 @@ sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]]) sage: g = G([(1,3)]) sage: G.centralizer(g) - Permutation Group with generators [(2,4), (1,3)] + Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(2,4), (1,3)] sage: g = G([(1,2,3,4)]) sage: G.centralizer(g) - Permutation Group with generators [(1,2,3,4)] + Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(1,2,3,4)] sage: H = G.subgroup([G([(1,2,3,4)])]) sage: G.centralizer(H) - Permutation Group with generators [(1,2,3,4)] + Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(1,2,3,4)] """ - N = self._gap_().Centralizer(g) - return PermutationGroup(N.GeneratorsOfGroup()) + return self.subgroup(gap_group=self._gap_().Centralizer(g)) def isomorphism_type_info_simple_group(self): """ @@ -2444,7 +2599,7 @@ """ if not(self.is_subgroup(other)): raise TypeError("%s must be a subgroup of %s"%(self, other)) - return other._gap_().IsNormal(self._gap_()).bool() + return other._gap_().IsNormal(self).bool() def is_perfect(self): """ @@ -2530,13 +2685,7 @@ sage: G.is_subgroup(H) True """ - G = other - gens = self.gens() - for i in range(len(gens)): - x = gens[i] - if not (x in G): - return False - return True + return all((x in other) for x in self.gens()) def is_supersolvable(self): """ @@ -2551,11 +2700,54 @@ """ return self._gap_().IsSupersolvableGroup().bool() - def is_transitive(self): + def non_fixed_points(self): + r""" + Returns the list of points not fixed by ``self``, i.e., the subset + of ``self.domain()`` moved by some element of ``self``. + + EXAMPLES:: + + sage: G = PermutationGroup([[(3,4,5)],[(7,10)]]) + sage: G.non_fixed_points() + [3, 4, 5, 7, 10] + sage: G = PermutationGroup([[(2,3,6)],[(9,)]]) # note: 9 is fixed + sage: G.non_fixed_points() + [2, 3, 6] """ - Return ``True`` if ``self`` is a transitive group, i.e., if the action - of self on [1..n] is transitive. + pnts = set() + for gens in self.gens(): + for cycles in gens.cycle_tuples(): + for thispnt in cycles: + if thispnt not in pnts: + pnts.add(thispnt) + return sorted(pnts) + def fixed_points(self): + r""" + Returns the list of points fixed by ``self``, i.e., the subset + of ``.domain()`` not moved by any element of ``self``. + + EXAMPLES:: + + sage: G=PermutationGroup([(1,2,3)]) + sage: G.fixed_points() + [] + sage: G=PermutationGroup([(1,2,3),(5,6)]) + sage: G.fixed_points() + [4] + sage: G=PermutationGroup([[(1,4,7)],[(4,3),(6,7)]]) + sage: G.fixed_points() + [2, 5] + """ + non_fixed_points = self.non_fixed_points() + return [i for i in self.domain() if i not in non_fixed_points] + + def is_transitive(self, domain=None): + """ + Returns ``True`` if ``self`` acts transitively on ``domain``. + A group $G$ acts transitively on set $S$ if for all $x,y\in S$ + there is some $g\in G$ such that $x^g=y$. + EXAMPLES:: sage: G = SymmetricGroup(5) @@ -2565,6 +2757,19 @@ sage: G.is_transitive() False + :: + + sage: G = PermutationGroup([[(1,2,3,4,5)],[(1,2)]]) #S_5 on [1..5] + sage: G.is_transitive([1,4,5]) + True + sage: G.is_transitive([2..6]) + False + sage: G.is_transitive(G.non_fixed_points()) + True + sage: H = PermutationGroup([[(1,2,3)],[(4,5,6)]]) + sage: H.is_transitive(H.non_fixed_points()) + False + Note that this differs from the definition in GAP, where ``IsTransitive`` returns whether the group is transitive on the set of points moved by the group. @@ -2577,8 +2782,126 @@ sage: gap(G).IsTransitive() true """ - return self._gap_().IsTransitive(self._set_gap()).bool() - + #If the domain is not a subset of self.domain(), then the + #action isn't transitive. + try: + domain = self._domain_gap(domain) + except ValueError: + return False + + return self._gap_().IsTransitive(domain).bool() + + + def is_primitive(self, domain=None): + r""" + Returns ``True`` if ``self`` acts primitively on ``domain``. + A group $G$ acts primitively on a set $S$ if + + 1. $G$ acts transitively on $S$ and + + 2. the action induces no non-trivial block system on $S$. + + INPUT: + + - ``domain`` (optional) + + EXAMPLES: + + By default, test for primitivity of ``self`` on its domain. + + sage: G = PermutationGroup([[(1,2,3,4)],[(1,2)]]) + sage: G.is_primitive() + True + sage: G = PermutationGroup([[(1,2,3,4)],[(2,4)]]) + sage: G.is_primitive() + False + + You can specify a domain on which to test primitivity:: + + sage: G = PermutationGroup([[(1,2,3,4)],[(2,4)]]) + sage: G.is_primitive([1..4]) + False + sage: G.is_primitive([1,2,3]) + True + sage: G = PermutationGroup([[(3,4,5,6)],[(3,4)]]) #S_4 on [3..6] + sage: G.is_primitive(G.non_fixed_points()) + True + + """ + #If the domain is not a subset of self.domain(), then the + #action isn't primitive. + try: + domain = self._domain_gap(domain) + except ValueError: + return False + + return self._gap_().IsPrimitive(domain).bool() + + def is_semi_regular(self, domain=None): + r""" + Returns ``True`` if ``self`` acts semi-regularly on ``domain``. + A group $G$ acts semi-regularly on a set $S$ if the point + stabilizers of $S$ in $G$ are trivial. + + ``domain`` is optional and may take several forms. See examples. + + EXAMPLES:: + + sage: G = PermutationGroup([[(1,2,3,4)]]) + sage: G.is_semi_regular() + True + sage: G = PermutationGroup([[(1,2,3,4)],[(5,6)]]) + sage: G.is_semi_regular() + False + + You can pass in a domain to test semi-regularity:: + + sage: G = PermutationGroup([[(1,2,3,4)],[(5,6)]]) + sage: G.is_semi_regular([1..4]) + True + sage: G.is_semi_regular(G.non_fixed_points()) + False + + """ + try: + domain = self._domain_gap(domain) + except ValueError: + return False + return self._gap_().IsSemiRegular(domain).bool() + + def is_regular(self, domain=None): + r""" + Returns ``True`` if ``self`` acts regularly on ``domain``. + A group $G$ acts regularly on a set $S$ if + + 1. $G$ acts transitively on $S$ and + 2. $G$ acts semi-regularly on $S$. + + EXAMPLES:: + + sage: G = PermutationGroup([[(1,2,3,4)]]) + sage: G.is_regular() + True + sage: G = PermutationGroup([[(1,2,3,4)],[(5,6)]]) + sage: G.is_regular() + False + + You can pass in a domain on which to test regularity:: + + sage: G = PermutationGroup([[(1,2,3,4)],[(5,6)]]) + sage: G.is_regular([1..4]) + True + sage: G.is_regular(G.non_fixed_points()) + False + + """ + try: + domain = self._domain_gap(domain) + except ValueError: + return False + return self._gap_().IsRegular(domain).bool() + + def normalizes(self, other): r""" Returns ``True`` if the group ``other`` is normalized by ``self``. @@ -2604,7 +2927,7 @@ In the last example, `G` and `H` are disjoint, so each normalizes the other. """ - return self._gap_().IsNormal(other._gap_()).bool() + return self._gap_().IsNormal(other).bool() ############## Series ###################### @@ -2627,11 +2950,11 @@ sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]]) sage: CS = G.composition_series() sage: CS[3] - Permutation Group with generators [()] + Subgroup of (Permutation Group with generators [(1,2), (1,2,3)(4,5)]) generated by [()] """ current_randstate().set_seed_gap() CS = self._gap_().CompositionSeries() - return [PermutationGroup(gap_group=group) for group in CS] + return [self.subgroup(gap_group=group) for group in CS] def derived_series(self): """ @@ -2652,7 +2975,7 @@ """ current_randstate().set_seed_gap() DS = self._gap_().DerivedSeries() - return [PermutationGroup(gap_group=group) for group in DS] + return [self.subgroup(gap_group=group) for group in DS] def lower_central_series(self): """ @@ -2673,7 +2996,7 @@ """ current_randstate().set_seed_gap() LCS = self._gap_().LowerCentralSeriesOfGroup() - return [PermutationGroup(gap_group=group) for group in LCS] + return [self.subgroup(gap_group=group) for group in LCS] def molien_series(self): r""" @@ -2736,7 +3059,7 @@ True """ NS = self._gap_().NormalSubgroups() - return [PermutationGroup(gap_group=group) for group in NS] + return [self.subgroup(gap_group=group) for group in NS] def poincare_series(self, p=2, n=10): """ @@ -2788,11 +3111,11 @@ sage: G = PermutationGroup(['(1,2,3)', '(2,3)']) sage: G.sylow_subgroup(2) - Permutation Group with generators [(2,3)] + Subgroup of (Permutation Group with generators [(2,3), (1,2,3)]) generated by [(2,3)] sage: G.sylow_subgroup(5) - Permutation Group with generators [()] + Subgroup of (Permutation Group with generators [(2,3), (1,2,3)]) generated by [()] """ - return PermutationGroup(gap_group=self._gap_().SylowSubgroup(p)) + return self.subgroup(gap_group=self._gap_().SylowSubgroup(p)) def upper_central_series(self): """ @@ -2806,11 +3129,11 @@ sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]]) sage: G.upper_central_series() - [Permutation Group with generators [()]] + [Subgroup of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]) generated by [()]] """ current_randstate().set_seed_gap() UCS = self._gap_().UpperCentralSeriesOfGroup() - return [PermutationGroup(gap_group=group) for group in UCS] + return [self.subgroup(gap_group=group) for group in UCS] class PermutationGroup_subgroup(PermutationGroup_generic): """ @@ -2822,9 +3145,9 @@ sage: G = CyclicPermutationGroup(4) sage: gens = G.gens() sage: H = DihedralGroup(4) - sage: PermutationGroup_subgroup(H,list(gens)) - Subgroup of Dihedral group of order 8 as a permutation group generated by [(1,2,3,4)] - sage: K=PermutationGroup_subgroup(H,list(gens)) + sage: H.subgroup(gens) + Subgroup of (Dihedral group of order 8 as a permutation group) generated by [(1,2,3,4)] + sage: K = H.subgroup(gens) sage: K.list() [(), (1,2,3,4), (1,3)(2,4), (1,4,3,2)] sage: K.ambient_group() @@ -2832,7 +3155,8 @@ sage: K.gens() [(1,2,3,4)] """ - def __init__(self, ambient, gens, from_group = False, check=True, canonicalize=True): + def __init__(self, ambient, gens=None, gap_group=None, domain=None, + category=None, canonicalize=True, check=True): r""" Initialization method for the ``PermutationGroup_subgroup`` class. @@ -2862,9 +3186,9 @@ sage: H = CyclicPermutationGroup(4) sage: gens = H.gens(); gens [(1,2,3,4)] - sage: S = PermutationGroup_subgroup(G,gens) + sage: S = G.subgroup(gens) sage: S - Subgroup of Dihedral group of order 8 as a permutation group generated by [(1,2,3,4)] + Subgroup of (Dihedral group of order 8 as a permutation group) generated by [(1,2,3,4)] sage: S.list() [(), (1,2,3,4), (1,3)(2,4), (1,4,3,2)] sage: S.ambient_group() @@ -2883,31 +3207,28 @@ """ if not isinstance(ambient, PermutationGroup_generic): raise TypeError, "ambient (=%s) must be perm group."%ambient - if not isinstance(gens, list): - raise TypeError, "gens (=%s) must be a list"%gens - - self.__ambient_group = ambient - self._gens = gens - cmd = 'Group(%s)'%gens - cmd = cmd.replace("'","") # get rid of quotes - self._gap_string = cmd - - G = ambient + if domain is None: + domain = ambient.domain() + if category is None: + category = ambient.category() + PermutationGroup_generic.__init__(self, gens=gens, gap_group=gap_group, domain=domain, + category=category, canonicalize=canonicalize) + + self._ambient_group = ambient if check: - for g in gens: - if g not in G: + for g in self.gens(): + if g not in ambient: raise TypeError, "each generator must be in the ambient group" - self.__ambient_group = G - - PermutationGroup_generic.__init__(self, gens, canonicalize=canonicalize) def __cmp__(self, other): r""" Compare ``self`` and ``other``. - First, ``self`` and ``other`` are compared as permutation groups, see :method:`sage.groups.perm_gps.permgroup.PermutationGroup_generic.__cmp__`. - Second, if both are equal, the ambient groups are compared, where (if necessary) - ``other`` is considered a subgroup of itself. + First, ``self`` and ``other`` are compared as permutation + groups, see :method:`PermutationGroup_generic.__cmp__`. + Second, if both are equal, the ambient groups are compared, + where (if necessary) ``other`` is considered a subgroup of + itself. EXAMPLES:: @@ -2941,7 +3262,7 @@ sage: G Symmetric group of order 6! as a permutation group sage: G3 - Subgroup of SymmetricGroup(6) generated by [(1,2), (1,2,3,4,5,6)] + Subgroup of (Symmetric group of order 6! as a permutation group) generated by [(1,2), (1,2,3,4,5,6)] sage: G is G3 False sage: G == G3 # as permutation groups @@ -2986,11 +3307,11 @@ sage: gens = H.gens() sage: S = PermutationGroup_subgroup(G, list(gens)) sage: S - Subgroup of Dihedral group of order 8 as a permutation group generated by [(1,2,3,4)] + Subgroup of (Dihedral group of order 8 as a permutation group) generated by [(1,2,3,4)] sage: S._repr_() - 'Subgroup of Dihedral group of order 8 as a permutation group generated by [(1,2,3,4)]' + 'Subgroup of (Dihedral group of order 8 as a permutation group) generated by [(1,2,3,4)]' """ - s = "Subgroup of %s generated by %s"%(self.ambient_group(), self.gens()) + s = "Subgroup of (%s) generated by %s"%(self.ambient_group(), self.gens()) return s def _latex_(self): @@ -3007,9 +3328,9 @@ sage: gens = H.gens() sage: S = PermutationGroup_subgroup(G, list(gens)) sage: latex(S) - Subgroup of Dihedral group of order 8 as a permutation group generated by [(1,2,3,4)] + Subgroup of (Dihedral group of order 8 as a permutation group) generated by [(1,2,3,4)] sage: S._latex_() - 'Subgroup of Dihedral group of order 8 as a permutation group generated by [(1,2,3,4)]' + 'Subgroup of (Dihedral group of order 8 as a permutation group) generated by [(1,2,3,4)]' """ return self._repr_() @@ -3032,25 +3353,25 @@ sage: S.ambient_group() == G True """ - return self.__ambient_group - - def gens(self): + return self._ambient_group + + + def is_normal(self, other=None): """ - Return the generators for this subgroup. + Return ``True`` if this group is a normal subgroup of + ``other``. If ``other`` is not specified, then it is assumed + to be the ambient group. - EXAMPLES: - - An example involving the dihedral group on four elements, - `D_8`:: + EXAMPLES:: - sage: G = DihedralGroup(4) - sage: H = CyclicPermutationGroup(4) - sage: gens = H.gens() - sage: S = PermutationGroup_subgroup(G, list(gens)) - sage: S.gens() - [(1,2,3,4)] - sage: S.gens() == list(H.gens()) - True + sage: S = SymmetricGroup(['a','b','c']) + sage: H = S.subgroup([('a', 'b', 'c')]); H + Subgroup of (Symmetric group of order 3! as a permutation group) generated by [('a','b','c')] + sage: H.is_normal() + True + """ - return self._gens - + if other is None: + other = self.ambient_group() + return PermutationGroup_generic.is_normal(self, other) + diff --git a/sage/groups/perm_gps/permgroup_element.pxd b/sage/groups/perm_gps/permgroup_element.pxd --- a/sage/groups/perm_gps/permgroup_element.pxd +++ b/sage/groups/perm_gps/permgroup_element.pxd @@ -9,5 +9,6 @@ cdef Element _gap_element cdef __tuple cdef PermutationGroupElement _new_c(self) + cpdef _gap_list(self) cpdef list(self) cdef public __custom_name diff --git a/sage/groups/perm_gps/permgroup_element.pyx b/sage/groups/perm_gps/permgroup_element.pyx --- a/sage/groups/perm_gps/permgroup_element.pyx +++ b/sage/groups/perm_gps/permgroup_element.pyx @@ -65,13 +65,11 @@ from sage.rings.all import ZZ, Integer, is_MPolynomial, is_Polynomial from sage.matrix.all import MatrixSpace from sage.interfaces.all import gap, is_GapElement, is_ExpectElement - +from sage.sets.all import FiniteEnumeratedSet import sage.structure.coerce as coerce import operator -from sage.rings.integer import Integer - from sage.rings.fast_arith cimport arith_llong cdef arith_llong arith = arith_llong() cdef extern from *: @@ -80,22 +78,66 @@ #import permgroup_named def make_permgroup_element(G, x): - G._deg = len(x) + """ + Returns a PermutationGroupElement given the permutation group + ``G`` and the permutation ``x`` in list notation. + + This is function is used when unpickling old (pre-domain) versions + of permutation groups and their elements. This now does a bit of + processing and calls :func:`make_permgroup_element_v2` which is + used in unpickling the current PermutationGroupElements. + + EXAMPLES:: + + sage: from sage.groups.perm_gps.permgroup_element import make_permgroup_element + sage: S = SymmetricGroup(3) + sage: make_permgroup_element(S, [1,3,2]) + (2,3) + """ + domain = FiniteEnumeratedSet(range(1, len(x)+1)) + return make_permgroup_element_v2(G, x, domain) + +def make_permgroup_element_v2(G, x, domain): + """ + Returns a PermutationGroupElement given the permutation group + ``G``, the permutation ``x`` in list notation, and the domain + ``domain`` of the permutation group. + + This is function is used when unpickling permutation groups and + their elements. + + EXAMPLES:: + + sage: from sage.groups.perm_gps.permgroup_element import make_permgroup_element_v2 + sage: S = SymmetricGroup(3) + sage: make_permgroup_element_v2(S, [1,3,2], S.domain()) + (2,3) + """ + # Note that it has to be in-sync with the __init__ method of + # PermutationGroup_generic since the elements have to be created + # before the PermutationGroup_generic is initialized. The + # constructor for PermutationGroupElement requires that + # G._domain_to_gap be set. + G._domain = domain + G._deg = len(domain) + G._domain_to_gap = dict([(key, i+1) for i, key in enumerate(domain)]) + G._domain_from_gap = dict([(i+1, key) for i, key in enumerate(domain)]) return G(x, check=False) + def is_PermutationGroupElement(x): + """ + Returns True if ``x`` is a PermutationGroupElement. + + EXAMPLES:: + + sage: p = PermutationGroupElement([(1,2),(3,4,5)]) + sage: from sage.groups.perm_gps.permgroup_element import is_PermutationGroupElement + sage: is_PermutationGroupElement(p) + True + """ return isinstance(x, PermutationGroupElement) - -def gap_format(x): - """ - Put a permutation in Gap format, as a string. - """ - if isinstance(x, list) and not isinstance(x[0], tuple): - return gap.eval('PermList(%s)' % x) - x = str(x).replace(' ','').replace('\n','') - return x.replace('),(',')(').replace('[','').replace(']','').replace(',)',')') - def string_to_tuples(g): """ EXAMPLES:: @@ -120,6 +162,93 @@ g = '[' + g + ']' return sage_eval(g, preparse=False) +def standardize_generator(g, convert_dict=None): + """ + Standardizes the input for permutation group elements to a list of + tuples. This was factored out of the + PermutationGroupElement.__init__ since + PermutationGroup_generic.__init__ needs to do the same computation + in order to compute the domain of a group when it's not explicitly + specified. + + INPUT: + + - ``g`` - a list, tuple, string, GapElement, + PermuationGroupElement + + - ``convert_dict`` - (optional) a dictionary used to convert the + points to a number compatible with GAP. + + OUTPUT: + + The permutation in as a list of cycles. + + EXAMPLES:: + + sage: from sage.groups.perm_gps.permgroup_element import standardize_generator + sage: standardize_generator('(1,2)') + [(1, 2)] + + sage: p = PermutationGroupElement([(1,2)]) + sage: standardize_generator(p) + [(1, 2)] + sage: standardize_generator(p._gap_()) + [(1, 2)] + sage: standardize_generator((1,2)) + [(1, 2)] + sage: standardize_generator([(1,2)]) + [(1, 2)] + + :: + + sage: d = {'a': 1, 'b': 2} + sage: p = SymmetricGroup(['a', 'b']).gen(0); p + ('a','b') + sage: standardize_generator(p, convert_dict=d) + [(1, 2)] + sage: standardize_generator(p._gap_(), convert_dict=d) + [(1, 2)] + sage: standardize_generator(('a','b'), convert_dict=d) + [(1, 2)] + sage: standardize_generator([('a','b')], convert_dict=d) + [(1, 2)] + """ + from sage.interfaces.gap import GapElement + from sage.combinat.permutation import Permutation + from sage.libs.pari.gen import gen + + if isinstance(g, gen): + g = list(g) + + needs_conversion = True + if isinstance(g, GapElement): + g = str(g) + needs_conversion = False + if isinstance(g, PermutationGroupElement): + g = g.cycle_tuples() + if isinstance(g, str): + g = string_to_tuples(g) + if isinstance(g, tuple) and (len(g) == 0 or not isinstance(g[0], tuple)): + g = [g] + + #Get the permutation in list notation + if PyList_CheckExact(g) and (len(g) == 0 or not PY_TYPE_CHECK(g[0], tuple)): + if convert_dict is not None and needs_conversion: + g = [convert_dict[x] for x in g] + + + #FIXME: Currently, we need to verify that g defines an actual + #permutation since the constructor Permutation does not + #perform this check. When it does, we should remove this code. + #See #8392 + if set(g) != set(range(1, len(g)+1)): + raise ValueError, "Invalid permutation vector: %s"%g + return Permutation(g).cycle_tuples() + else: + if convert_dict is not None and needs_conversion: + g = [tuple([convert_dict[x] for x in cycle])for cycle in g] + + return g cdef class PermutationGroupElement(MultiplicativeGroupElement): """ @@ -292,42 +421,32 @@ sage: PermutationGroupElement([()]) () """ - from sage.interfaces.gap import GapElement from sage.groups.perm_gps.permgroup_named import SymmetricGroup from sage.groups.perm_gps.permgroup import PermutationGroup_generic + from sage.combinat.permutation import from_cycles - #Convert GAP elements to strings - if isinstance(g, GapElement): - g = str(g) - elif isinstance(g, PermutationGroupElement): - g = g.list() - - #Convert all the string to tuples - if isinstance(g, str): - g = string_to_tuples(g) + convert_dict = parent._domain_to_gap if parent is not None else None + try: + v = standardize_generator(g, convert_dict) + except KeyError: + raise ValueError, "Invalid permutation vector: %s" % g - if isinstance(g, tuple) and (len(g) == 0 or not isinstance(g[0], tuple)): - g = [g] - - #Get the permutation in list notation - if PyList_CheckExact(g) and (len(g) == 0 or not PY_TYPE_CHECK(g[0], tuple)): - v = g if len(g) != 0 else [1] - self.__gap = 'PermList(%s)'%v - else: - from sage.combinat.permutation import Permutation - v = Permutation(g)._list - self.__gap = str(g)[1:-1].replace('), (',')(').replace(',)',')').strip(',') + + degree = max([1] + [max(cycle+(1,)) for cycle in v]) + v = from_cycles(degree, v) + + self.__gap = 'PermList(%s)'%v if parent is None: - parent = SymmetricGroup(max(len(v),1)) - + parent = SymmetricGroup(len(v)) + if check and parent.__class__ != SymmetricGroup: if not (parent is None or isinstance(parent, PermutationGroup_generic)): raise TypeError, 'parent must be a permutation group' if parent is not None: P = parent._gap_() if not P.parent()(self.__gap) in P: - raise TypeError, 'permutation %s not in %s'%(self.__gap, parent) + raise TypeError, 'permutation %s not in %s'%(g, parent) Element.__init__(self, parent) @@ -361,7 +480,19 @@ sage_free(self.perm) def __reduce__(self): - return make_permgroup_element, (self._parent, self.list()) + """ + Returns a function and its arguments needed to create this + permutation group element. This is used in pickling. + + EXAMPLES:: + + sage: g = PermutationGroupElement([(1,2,3),(4,5)]); g + (1,2,3)(4,5) + sage: func, args = g.__reduce__() + sage: func(*args) + (1,2,3)(4,5) + """ + return make_permgroup_element_v2, (self._parent, self.list(), self._parent.domain()) cdef PermutationGroupElement _new_c(self): cdef PermutationGroupElement other = PY_NEW_SAME_TYPE(self) @@ -375,24 +506,56 @@ other.perm = sage_malloc(sizeof(int) * other.n) return other - def _gap_(self, G=None): - if self._gap_element is None or \ - (G is not None and self._gap_element._parent is not G): - if G is None: - import sage.interfaces.gap - G = sage.interfaces.gap.gap - self._gap_element = G("PermList(%s)" % self.list()) + def _gap_(self, gap=None): + """ + Returns + + EXAMPLES:: + + sage: g = PermutationGroupElement([(1,2,3),(4,5)]); g + (1,2,3)(4,5) + sage: a = g._gap_(); a + (1,2,3)(4,5) + sage: g._gap_() is g._gap_() + True + + Note that only one GapElement is cached: + + sage: gap2 = Gap() + sage: b = g._gap_(gap2) + sage: c = g._gap_() + sage: a is c + False + """ + if (self._gap_element is None or + (gap is not None and self._gap_element._parent is not gap)): + if gap is None: + from sage.interfaces.gap import gap + self._gap_element = gap(self._gap_init_()) return self._gap_element + def _gap_init_(self): + """ + Returns a GAP string representation for this + PermutationGroupElement. + + EXAMPLES:: + + sage: g = PermutationGroupElement([(1,2,3),(4,5)]) + sage: g._gap_init_() + 'PermList([2, 3, 1, 5, 4])' + """ + return 'PermList(%s)'%self._gap_list() + def _repr_(self): """ Return string representation of this permutation. - EXAMPLES: We create the permutation `(1,2,3)(4,5)` and - print it. - - :: - + EXAMPLES: + + We create the permutation `(1,2,3)(4,5)` and + print it. :: + sage: g = PermutationGroupElement([(1,2,3),(4,5)]) sage: g._repr_() '(1,2,3)(4,5)' @@ -407,13 +570,24 @@ sage: g (1,2,3)(4,5) """ - cycles = self.cycle_tuples() - if len(cycles) == 0: - return '()' - return ''.join([str(c) for c in cycles]).replace(', ',',') + return self.cycle_string() def _latex_(self): - return str(self) + """ + Returns a latex representation of this permutation. + + EXAMPLES:: + + sage: g = PermutationGroupElement([(1,2,3),(4,5)]) + sage: latex(g) + (1,2,3)(4,5) + + sage: S = SymmetricGroup(['a', 'b']) + sage: latex(S.gens()) + \left[(\texttt{a},\texttt{b})\right] + """ + from sage.misc.latex import latex + return "".join(["(" + ",".join([latex(x) for x in cycle])+")" for cycle in self.cycle_tuples()]) def __getitem__(self, i): """ @@ -519,14 +693,30 @@ sage: g(x) Traceback (most recent call last): ... - ValueError: Must be an integer, list, tuple or string. + ValueError: Must be in the domain or a list, tuple or string. sage: g(3/2) Traceback (most recent call last): ... - ValueError: Must be an integer, list, tuple or string. + ValueError: Must be in the domain or a list, tuple or string. """ + to_gap = self._parent._domain_to_gap + from_gap = self._parent._domain_from_gap cdef int j - if isinstance(i,(list,tuple,str)): + + try: + i = to_gap[i] + except (KeyError, TypeError): + # We currently have to include this to maintain the + # current behavior where if you pass in an integer which + # is not in the domain of the permutation group, then that + # integer itself will be returned. + if isinstance(i, (long, int, Integer)): + return i + + + if not isinstance(i,(list,tuple,str)): + raise ValueError, "Must be in the domain or a list, tuple or string." + permuted = [i[self.perm[j]] for j from 0 <= j < self.n] if PY_TYPE_CHECK(i, tuple): permuted = tuple(permuted) @@ -535,13 +725,11 @@ permuted += i[self.n:] return permuted else: - if not isinstance(i, (int, Integer)): - raise ValueError("Must be an integer, list, tuple or string.") j = i if 1 <= j <= self.n: - return self.perm[j-1]+1 + return from_gap[self.perm[j-1]+1] else: - return i + return from_gap[i] cpdef _act_on_(self, x, bint self_on_left): """ @@ -590,6 +778,15 @@ return left(tuple(sigma_x)) cpdef MonoidElement _mul_(left, MonoidElement _right): + """ + EXAMPLES:: + + sage: S = SymmetricGroup(['a', 'b']) + sage: s = S([('a', 'b')]); s + ('a','b') + sage: s*s + () + """ cdef PermutationGroupElement prod = left._new_c() cdef PermutationGroupElement right = _right cdef int i @@ -614,7 +811,55 @@ for i from 0 <= i < self.n: inv.perm[self.perm[i]] = i return inv + + cpdef _gap_list(self): + """ + Returns this permutation in list notation compatible with the + GAP numbering. + + EXAMPLES:: + + sage: S = SymmetricGroup(3) + sage: s = S.gen(0); s + (1,2,3) + sage: s._gap_list() + [2, 3, 1] + + :: + sage: S = SymmetricGroup(['a', 'b', 'c']) + sage: s = S.gen(0); s + ('a','b','c') + sage: s._gap_list() + [2, 3, 1] + """ + cdef int i + return [self.perm[i]+1 for i from 0 <= i < self.n] + + def _gap_cycle_string(self): + """ + Returns a cycle string for this permutation compatible with + the GAP numbering. + + EXAMPLES:: + + sage: S = SymmetricGroup(3) + sage: s = S.gen(0); s + (1,2,3) + sage: s._gap_cycle_string() + '(1,2,3)' + + :: + + sage: S = SymmetricGroup(['a', 'b', 'c']) + sage: s = S.gen(0); s + ('a','b','c') + sage: s._gap_cycle_string() + '(1,2,3)' + """ + from sage.combinat.permutation import Permutation + return Permutation(self._gap_list()).cycle_string() + cpdef list(self): """ Returns list of the images of the integers from 1 to n under this @@ -633,9 +878,26 @@ (1,2) sage: x.list() [2, 1, 3, 4] + + TESTS:: + + sage: S = SymmetricGroup(0) + sage: x = S.one(); x + () + sage: x.list() + [] """ cdef int i - return [self.perm[i]+1 for i from 0 <= i < self.n] + + #We need to do this to handle the case of SymmetricGroup(0) + #where the domain is (), but the permutation group element has + #an underlying representation of [1]. The 1 doesn't + #correspond to anything in the domain + if len(self._parent._domain) == 0: + return [] + else: + from_gap = self._parent._domain_from_gap + return [from_gap[self.perm[i]+1] for i from 0 <= i < self.n] def __hash__(self): """ @@ -654,11 +916,11 @@ 1592966088 # 32-bit 2865702456085625800 # 64-bit """ - return hash(self.tuple()) - + return hash(tuple(self._gap_list())) + def tuple(self): """ - Return tuple of images of integers under self. + Return tuple of images of the domain under self. EXAMPLES:: @@ -666,6 +928,10 @@ sage: s = G([2,1,5,3,4]) sage: s.tuple() (2, 1, 5, 3, 4) + + sage: S = SymmetricGroup(['a', 'b']) + sage: S.gen().tuple() + ('b', 'a') """ if self.__tuple is None: self.__tuple = tuple(self.list()) @@ -696,10 +962,11 @@ sage: x.dict() {1: 2, 2: 1, 3: 3, 4: 4} """ + from_gap = self._parent._domain_from_gap cdef int i u = {} for i from 0 <= i < self.n: - u[i+1] = self.perm[i]+1 + u[i+1] = from_gap[self.perm[i]+1] return u def order(self): @@ -816,7 +1083,7 @@ def orbit(self, n, bint sorted=True): """ Returns the orbit of the integer `n` under this group - element, as a sorted list of integers. + element, as a sorted list. EXAMPLES:: @@ -828,20 +1095,34 @@ [1, 2, 3] sage: g.orbit(10) [10] + + :: + + sage: s = SymmetricGroup(['a', 'b']).gen(0); s + ('a','b') + sage: s.orbit('a') + ['a', 'b'] """ + to_gap = self._parent._domain_to_gap + from_gap = self._parent._domain_from_gap + try: + n = to_gap[n] + except KeyError: + return [n] + cdef int i = n cdef int start = i if 1 <= i <= self.n: - L = [i] + L = [from_gap[i]] i = self.perm[i-1]+1 while i != start: - PyList_Append(L,i) + PyList_Append(L,from_gap[i]) i = self.perm[i-1]+1 if sorted: L.sort() return L else: - return [n] + return from_gap[n] def cycles(self): """ @@ -876,28 +1157,83 @@ sage_free(seen) return L - def cycle_tuples(self): + def cycle_tuples(self, singletons=False): """ Return self as a list of disjoint cycles, represented as tuples rather than permutation group elements. + + INPUT: + + - ``singletons`` - boolean (default: False) whether or not consider the + cycle that correspond to fixed point + + EXAMPLES:: + + sage: p = PermutationGroupElement('(2,6)(4,5,1)') + sage: p.cycle_tuples() + [(1, 4, 5), (2, 6)] + sage: p.cycle_tuples(singletons=True) + [(1, 4, 5), (2, 6), (3,)] + + EXAMPLES:: + + sage: S = SymmetricGroup(4) + sage: S.gen(0).cycle_tuples() + [(1, 2, 3, 4)] + + :: + + sage: S = SymmetricGroup(['a','b','c','d']) + sage: S.gen(0).cycle_tuples() + [('a', 'b', 'c', 'd')] + sage: S([('a', 'b'), ('c', 'd')]).cycle_tuples() + [('a', 'b'), ('c', 'd')] """ + from_gap = self._parent._domain_from_gap L = [] cdef int i, k cdef bint* seen = sage_malloc(sizeof(bint) * self.n) for i from 0 <= i < self.n: seen[i] = 0 for i from 0 <= i < self.n: - if seen[i] or self.perm[i] == i: + if seen[i]: continue - cycle = [i+1] - k = self.perm[i] - while k != i: - PyList_Append(cycle, k+1) - seen[k] = 1 - k = self.perm[k] - PyList_Append(L, tuple(cycle)) + if self.perm[i] == i: + if singletons: + PyList_Append(L, (from_gap[i+1],)) + # it is not necessary to put seen[i] to 1 as we will never + # see i again + else: + continue + else: + cycle = [from_gap[i+1]] + k = self.perm[i] + while k != i: + PyList_Append(cycle, from_gap[k+1]) + seen[k] = 1 + k = self.perm[k] + PyList_Append(L, tuple(cycle)) sage_free(seen) return L + def cycle_string(self, singletons=False): + """ + Return string representation of this permutation. + + EXAMPLES:: + + sage: g = PermutationGroupElement([(1,2,3),(4,5)]) + sage: g.cycle_string() + '(1,2,3)(4,5)' + + sage: g = PermutationGroupElement([3,2,1]) + sage: g.cycle_string(singletons=True) + '(1,3)(2)' + """ + cycles = self.cycle_tuples(singletons) + if len(cycles) == 0: + return '()' + return ''.join([repr(c) for c in cycles]).replace(', ',',').replace(',)',')') + def has_descent(self, i, side = "right", positive = False): """ INPUT: @@ -944,18 +1280,17 @@ True sage: S._test_has_descent() """ + to_gap = self._parent._domain_to_gap + from_gap = self._parent._domain_from_gap if side == "right": self = ~self + try: - if not self.parent()._has_natural_set(): - set = self.parent()._set - pos = set.index(i) - i1 = set[pos+1] - res = set.index(self(i)) > set.index(self(i1)) - return res is not positive - except AttributeError: - pass - return (self(i) > self(i+1)) is not positive + i1 = from_gap[to_gap[i]+1] + except KeyError: + return False + + return (to_gap[self(i)] > to_gap[self(i1)]) is not positive def matrix(self): """ @@ -980,7 +1315,7 @@ entries[i, self.perm[i]] = 1 return M(entries) - def word_problem(g, words, display=True): + def word_problem(self, words, display=True): """ G and H are permutation groups, g in G, H is a subgroup of G generated by a list (words) of elements of G. If g is in H, return @@ -995,8 +1330,7 @@ EXAMPLE:: sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]], canonicalize=False) - sage: g1 = G.gens()[0] - sage: g2 = G.gens()[1] + sage: g1, g2 = G.gens() sage: h = g1^2*g2*g1 sage: h.word_problem([g1,g2], False) ('x1^2*x2^-1*x1', '(1,2,3)(4,5)^2*(3,4)^-1*(1,2,3)(4,5)') @@ -1005,16 +1339,17 @@ [['(1,2,3)(4,5)', 2], ['(3,4)', -1], ['(1,2,3)(4,5)', 1]] ('x1^2*x2^-1*x1', '(1,2,3)(4,5)^2*(3,4)^-1*(1,2,3)(4,5)') """ + if not self._parent._has_natural_domain(): + raise NotImplementedError + import copy from sage.groups.perm_gps.permgroup import PermutationGroup from sage.interfaces.all import gap - + G = gap(words[0].parent()) - g = words[0].parent()(g) - gensH = gap(words) - H = gensH.Group() - hom = G.EpimorphismFromFreeGroup() - ans = hom.PreImagesRepresentative(gap(g)) + g = words[0].parent()(self) + H = gap.Group(words) + ans = G.EpimorphismFromFreeGroup().PreImagesRepresentative(g) l1 = str(ans) l2 = copy.copy(l1) diff --git a/sage/groups/perm_gps/permgroup_morphism.py b/sage/groups/perm_gps/permgroup_morphism.py --- a/sage/groups/perm_gps/permgroup_morphism.py +++ b/sage/groups/perm_gps/permgroup_morphism.py @@ -15,9 +15,9 @@ sage: g = G([(1,2,3,4)]) sage: phi = PermutationGroupMorphism_im_gens(G, H, map(H, G.gens())) sage: phi.image(G) - Permutation Group with generators [(1,2,3,4)] + Subgroup of (Dihedral group of order 8 as a permutation group) generated by [(1,2,3,4)] sage: phi.kernel() - Permutation Group with generators [()] + Subgroup of (Cyclic group of order 4 as a permutation group) generated by [()] sage: phi.image(g) (1,2,3,4) sage: phi(g) @@ -87,7 +87,7 @@ sage: g = G([(1,2,3,4)]) sage: phi = PermutationGroupMorphism_im_gens(G, H, [1]) sage: phi.kernel() - Permutation Group with generators [(1,2,3,4)] + Subgroup of (Cyclic group of order 4 as a permutation group) generated by [(1,2,3,4)] :: @@ -98,7 +98,7 @@ sage: G.is_isomorphic(pr1.kernel()) True """ - return PermutationGroup(gap_group=self._gap_().Kernel()) + return self.domain().subgroup(gap_group=self._gap_().Kernel()) def image(self, J): """ @@ -112,7 +112,7 @@ sage: g = G([(1,2,3,4)]) sage: phi = PermutationGroupMorphism_im_gens(G, H, map(H, G.gens())) sage: phi.image(G) - Permutation Group with generators [(1,2,3,4)] + Subgroup of (Dihedral group of order 8 as a permutation group) generated by [(1,2,3,4)] sage: phi.image(g) (1,2,3,4) @@ -123,17 +123,18 @@ sage: H = D[0] sage: pr1 = D[3] sage: pr1.image(G) - Permutation Group with generators [(3,7,5)(4,8,6), (1,2,6)(3,4,8)] + Subgroup of (The projective special linear group of degree 2 over Finite Field of size 7) generated by [(3,7,5)(4,8,6), (1,2,6)(3,4,8)] sage: G.is_isomorphic(pr1.image(G)) True """ + H = self.codomain() if J in self.domain(): J = PermutationGroup([J]) G = self._gap_().Image(J) - return PermutationGroup(gap_group=G).gens()[0] + return H.subgroup(gap_group=G).gens()[0] else: G = self._gap_().Image(J) - return PermutationGroup(gap_group=G) + return H.subgroup(gap_group=G) def __call__(self, g): """ @@ -177,7 +178,7 @@ sage: H = G.subgroup([G([(1,2,3,4)])]) sage: PermutationGroupMorphism_from_gap(H, G, gap.Identity) Permutation group morphism: - From: Subgroup of Permutation Group with generators [(1,2)(3,4), (1,2,3,4)] generated by [(1,2,3,4)] + From: Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(1,2,3,4)] To: Permutation Group with generators [(1,2)(3,4), (1,2,3,4)] Defn: Identity """ diff --git a/sage/groups/perm_gps/permgroup_named.py b/sage/groups/perm_gps/permgroup_named.py --- a/sage/groups/perm_gps/permgroup_named.py +++ b/sage/groups/perm_gps/permgroup_named.py @@ -37,7 +37,7 @@ -- PSp(2n,q), projective symplectic linear group of $2n\times 2n$ matrices over the finite field GF(q) --- PSU(n,q), projective special unitary group of $n\times n$ matrices having +-- PSU(n,q), projective special unitary group of $n \times n$ matrices having coefficients in the finite field $GF(q^2)$ that respect a fixed nondegenerate sesquilinear form, of determinant 1. @@ -78,32 +78,50 @@ from sage.groups.abelian_gps.abelian_group import AbelianGroup from sage.misc.functional import is_even from sage.misc.cachefunc import cached_method +from sage.misc.misc import deprecated_function_alias from sage.groups.perm_gps.permgroup import PermutationGroup_generic from sage.groups.perm_gps.permgroup_element import PermutationGroupElement from sage.structure.unique_representation import UniqueRepresentation from sage.structure.parent import Parent from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets +from sage.sets.finite_enumerated_set import FiniteEnumeratedSet from sage.sets.disjoint_union_enumerated_sets import DisjointUnionEnumeratedSets -from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets from sage.categories.enumerated_sets import EnumeratedSets from sage.sets.non_negative_integers import NonNegativeIntegers from sage.sets.family import Family class PermutationGroup_unique(UniqueRepresentation, PermutationGroup_generic): + @staticmethod + def __classcall__(cls, *args, **kwds): + """ + This makes sure that domain is a FiniteEnumeratedSet before it gets passed + on to the __init__ method. + EXAMPLES:: + + sage: SymmetricGroup(['a','b']).domain() #indirect doctest + {'a', 'b'} + """ + domain = kwds.pop('domain', None) + if domain is not None: + if domain not in FiniteEnumeratedSets(): + domain = FiniteEnumeratedSet(domain) + kwds['domain'] = domain + return super(PermutationGroup_unique, cls).__classcall__(cls, *args, **kwds) + def __eq__(self, other): - """ - Overrides the default equality testing provided by - UniqueRepresentation by forcing a call to :meth:.`__cmp__`. + """ + Overrides the default equality testing provided by + UniqueRepresentation by forcing a call to :meth:.`__cmp__`. - EXAMPLES:: - - sage: G = SymmetricGroup(6) - sage: G3 = G.subgroup([G((1,2,3,4,5,6)),G((1,2))]) - sage: G == G3 - True - """ - return self.__cmp__(other) == 0 + EXAMPLES:: + + sage: G = SymmetricGroup(6) + sage: G3 = G.subgroup([G((1,2,3,4,5,6)),G((1,2))]) + sage: G == G3 + True + """ + return self.__cmp__(other) == 0 class PermutationGroup_symalt(PermutationGroup_unique): @@ -113,7 +131,7 @@ """ @staticmethod - def __classcall__(cls, n): + def __classcall__(cls, domain): """ Normalizes the input of the constructor into a set @@ -142,86 +160,29 @@ sage: SymmetricGroup(-1) Traceback (most recent call last): ... - ValueError: n (=-1) must be an integer >= 0 or a list + ValueError: domain (=-1) must be an integer >= 0 or a list """ - if not isinstance(n, (tuple,list)): - try: - n = Integer(n) - except TypeError: - raise ValueError, "n (=%s) must be an integer >= 0 or a list (but n has type %s)"%(n,type(n)) - else: - if n < 0: - raise ValueError, "n (=%s) must be an integer >= 0 or a list"%n - n = range(1, n+1) + if domain not in FiniteEnumeratedSets(): + if not isinstance(domain, (tuple, list)): + try: + domain = Integer(domain) + except TypeError: + raise ValueError, "domain (=%s) must be an integer >= 0 or a finite set (but domain has type %s)"%(domain,type(domain)) + + if domain < 0: + raise ValueError, "domain (=%s) must be an integer >= 0 or a list"%domain + else: + domain = range(1, domain+1) + v = FiniteEnumeratedSet(domain) + else: + v = domain - try: - v = tuple(Integer(z) for z in n) - except TypeError: - raise ValueError, "each entry of list must be an integer" + return super(PermutationGroup_symalt, cls).__classcall__(cls, domain=v) - if v and min(v) < 1: # for SymmetricGroup(0), v is empty - raise ValueError, "each element of list must be positive" - - return super(PermutationGroup_symalt, cls).__classcall__(cls, v) - - def set(self): - """ - Returns the list of positive integers on which this group acts. - - EXAMPLES: - sage: SymmetricGroup(3).set() - (1, 2, 3) - sage: SymmetricGroup([2,3,4]).set() - (2, 3, 4) - sage: AlternatingGroup(3).set() - (1, 2, 3) - sage: AlternatingGroup([2,3,4]).set() - (2, 3, 4) - """ - return self._set - - @cached_method - def _has_natural_set(self): - """ - Returns true if the underlying set is of the form (1,...,n) - - EXAMPLES:: - - sage: SymmetricGroup(3)._has_natural_set() - True - sage: SymmetricGroup((1,2,3))._has_natural_set() - True - sage: SymmetricGroup((1,3))._has_natural_set() - False - sage: SymmetricGroup((3,2,1))._has_natural_set() - False - """ - set = self.set() - return set == tuple(range(1,len(set)+1)) - - def __str__(self): - """ - EXAMPLES: - sage: S = SymmetricGroup([2,3,7]); S - Symmetric group of order 3! as a permutation group - sage: str(S) - 'SymmetricGroup((2, 3, 7))' - sage: S = SymmetricGroup(5); S - Symmetric group of order 5! as a permutation group - sage: str(S) - 'SymmetricGroup(5)' - sage: A = AlternatingGroup([2,3,7]); A - Alternating group of order 3!/2 as a permutation group - sage: str(A) - 'AlternatingGroup((2, 3, 7))' - """ - set = self._set - if self._has_natural_set(): - set = len(set) - return "%s(%s)"%(self._gap_name, set) + set = deprecated_function_alias(PermutationGroup_generic.domain, 'Sage Version 4.7.1') class SymmetricGroup(PermutationGroup_symalt): - def __init__(self, _set): + def __init__(self, domain=None): """ The full symmetric group of order $n!$, as a permutation group. If n is a list or tuple of positive integers then it returns the @@ -244,13 +205,13 @@ sage: G = SymmetricGroup([1,2,4,5]) sage: G Symmetric group of order 4! as a permutation group - sage: G.set() - (1, 2, 4, 5) + sage: G.domain() + {1, 2, 4, 5} sage: G = SymmetricGroup(4) sage: G Symmetric group of order 4! as a permutation group - sage: G.set() - (1, 2, 3, 4) + sage: G.domain() + {1, 2, 3, 4} sage: G.category() Join of Category of finite permutation groups and Category of finite weyl groups sage: TestSuite(G).run() @@ -262,22 +223,38 @@ from sage.categories.finite_weyl_groups import FiniteWeylGroups from sage.categories.finite_permutation_groups import FinitePermutationGroups from sage.categories.category import Category + #Note that we skip the call to the superclass initializer in order to #avoid infinite recursion since SymmetricGroup is called by #PermutationGroupElement - super(PermutationGroup_generic,self).__init__(category = Category.join([FinitePermutationGroups(), FiniteWeylGroups()])) - - self._set = _set - self._deg = max(self._set+(0,)) # _set cat be empty - n = len(self._set) + super(PermutationGroup_generic, self).__init__(category = Category.join([FinitePermutationGroups(), FiniteWeylGroups()])) + + self._domain = domain + self._deg = len(self._domain) + self._domain_to_gap = dict((key, i+1) for i, key in enumerate(self._domain)) + self._domain_from_gap = dict((i+1, key) for i, key in enumerate(self._domain)) #Create the generators for the symmetric group - gens = [ tuple(self._set) ] - if n > 2: - gens.append( tuple(self._set[:2]) ) + gens = [ tuple(self._domain) ] + if len(self._domain) > 2: + gens.append( tuple(self._domain[:2]) ) self._gens = [PermutationGroupElement(g, self, check=False) for g in gens] - self._gap_string = '%s(%s)'%(self._gap_name, n) + + def _gap_init_(self, gap=None): + """ + Returns the string used to create this group in GAP. + + EXAMPLES:: + + sage: S = SymmetricGroup(3) + sage: S._gap_init_() + 'SymmetricGroup(3)' + sage: S = SymmetricGroup(['a', 'b', 'c']) + sage: S._gap_init_() + 'SymmetricGroup(3)' + """ + return 'SymmetricGroup(%s)'%self.degree() @cached_method def index_set(self): @@ -288,13 +265,13 @@ sage: S8 = SymmetricGroup(8) sage: S8.index_set() - (1, 2, 3, 4, 5, 6, 7) + [1, 2, 3, 4, 5, 6, 7] sage: S = SymmetricGroup([3,1,4,5]) sage: S.index_set() - (3, 1, 4) + [3, 1, 4] """ - return self.set()[:-1] + return self.domain()[:-1] def __cmp__(self, x): """ @@ -307,7 +284,7 @@ True """ if isinstance(x, SymmetricGroup): - return cmp((self._deg, self._set), (x._deg, x._set)) + return cmp((self._deg, self._domain), (x._deg, x._domain)) else: return PermutationGroup_generic.__cmp__(self, x) @@ -317,9 +294,7 @@ sage: A = SymmetricGroup([2,3,7]); A Symmetric group of order 3! as a permutation group """ - return "Symmetric group of order %s! as a permutation group"%len(self._set) - - _gap_name = 'SymmetricGroup' + return "Symmetric group of order %s! as a permutation group"%self.degree() def simple_reflection(self, i): """ @@ -336,7 +311,7 @@ sage: A.simple_reflections() Finite family {2: (2,3), 3: (3,7)} """ - return self([(i, self._set[self._set.index(i)+1])], check=False) + return self([(i, self._domain[self._domain.index(i)+1])], check=False) def major_index(self, parameter=None): r""" @@ -368,7 +343,7 @@ return q_factorial(self.degree(), parameter) class AlternatingGroup(PermutationGroup_symalt): - def __init__(self, _set): + def __init__(self, domain=None): """ The alternating group of order $n!/2$, as a permutation group. @@ -390,25 +365,13 @@ sage: G = AlternatingGroup([1,2,4,5]) sage: G Alternating group of order 4!/2 as a permutation group - sage: G.set() - (1, 2, 4, 5) + sage: G.domain() + {1, 2, 4, 5} sage: G.category() Category of finite permutation groups sage: TestSuite(G).run() """ - n = len(_set) - #Create the generators for the symmetric group - if n == 1: - gens = [ [] ] - else: - gens = [ tuple(_set) ] - if n > 2: - gens.append( tuple(_set[:2]) ) - - PermutationGroup_symalt.__init__(self, gap_group='%s(%s)'%(self._gap_name,n)) - - self._set = _set - self._deg = max(_set) + PermutationGroup_symalt.__init__(self, gap_group='AlternatingGroup(%s)'%len(domain), domain=domain) def _repr_(self): """ @@ -416,10 +379,23 @@ sage: A = AlternatingGroup([2,3,7]); A Alternating group of order 3!/2 as a permutation group """ - return "Alternating group of order %s!/2 as a permutation group" % len(self._set) + return "Alternating group of order %s!/2 as a permutation group"%self.degree() - _gap_name = 'AlternatingGroup' + def _gap_init_(self, gap=None): + """ + Returns the string used to create this group in GAP. + EXAMPLES:: + + sage: A = AlternatingGroup(3) + sage: A._gap_init_() + 'AlternatingGroup(3)' + sage: A = AlternatingGroup(['a', 'b', 'c']) + sage: A._gap_init_() + 'AlternatingGroup(3)' + """ + return 'AlternatingGroup(%s)'%(self.degree()) + class CyclicPermutationGroup(PermutationGroup_unique): def __init__(self, n): """ @@ -494,7 +470,7 @@ n = self.order() a = list(factor(n)) invs = [x[0]**x[1] for x in a] - G = AbelianGroup(len(a),invs) + G = AbelianGroup(len(a), invs) return G class DiCyclicGroup(PermutationGroup_unique): @@ -636,7 +612,7 @@ a = [tuple(range(1, halfr+1)), tuple(range(halfr+1, r+1))] # With an odd part, a cycle of length m will give the right order for a if m > 1: - a.append( tuple(range(r+1,r+m+1)) ) + a.append( tuple(range(r+1, r+m+1)) ) # Representation of x # Four-cycles that will conjugate the generator a properly @@ -644,9 +620,9 @@ for i in range(0, fourthr)] # With an odd part, transpositions will conjugate the m-cycle to create inverse if m > 1: - x += [(r+i+1,r+m-i) for i in range(0, (m-1)//2)] + x += [(r+i+1, r+m-i) for i in range(0, (m-1)//2)] - PermutationGroup_generic.__init__(self, gens=[a,x]) + PermutationGroup_generic.__init__(self, gens=[a, x]) def _repr_(self): r""" @@ -962,7 +938,7 @@ self._d = d self._n = n - self._set = range(1, d+1) + self._domain = range(1, d+1) def _repr_(self): """ @@ -1198,13 +1174,15 @@ sage: [TransitiveGroups(i).cardinality() for i in range(11)] # requires optional database_gap [1, 1, 1, 2, 5, 5, 16, 7, 50, 34, 45] - .. warning:: The database_gap contains all transitive groups - up to degree 30:: + .. warning:: - sage: TransitiveGroups(31).cardinality() # requires optional database_gap - Traceback (most recent call last): - ... - NotImplementedError: Only the transitive groups of order less than 30 are available in GAP's database + The database_gap contains all transitive groups + up to degree 30:: + + sage: TransitiveGroups(31).cardinality() # requires optional database_gap + Traceback (most recent call last): + ... + NotImplementedError: Only the transitive groups of order less than 30 are available in GAP's database TESTS:: @@ -1283,7 +1261,6 @@ Category of finite permutation groups sage: TestSuite(G).run() """ - from sage.groups.perm_gps.permgroup import PermutationGroup_generic id = 'Group([()])' if n == 1 else 'PGL(%s,%s)'%(n,q) PermutationGroup_generic.__init__(self, gap_group=id) self._q = q @@ -1585,7 +1562,10 @@ class SuzukiGroup(PermutationGroup_unique): def __init__(self, q, name='a'): r""" - The Suzuki group over GF(q), $^2 B_2(2^{2k+1}) = Sz(2^{2k+1})$. A wrapper for the GAP function SuzukiGroup. + The Suzuki group over GF(q), + $^2 B_2(2^{2k+1}) = Sz(2^{2k+1})$. + + A wrapper for the GAP function SuzukiGroup. INPUT: q -- 2^n, an odd power of 2; the size of the ground @@ -1595,11 +1575,13 @@ finite field GF(q) OUTPUT: - A Suzuki group. + + - A Suzuki group. - EXAMPLES: + EXAMPLES:: + sage: SuzukiGroup(8) - Permutation Group with generators [(1,28,10,44)(3,50,11,42)(4,43,53,64)(5,9,39,52)(6,36,63,13)(7,51,60,57)(8,33,37,16)(12,24,55,29)(14,30,48,47)(15,19,61,54)(17,59,22,62)(18,23,34,31)(20,38,49,25)(21,26,45,58)(27,32,41,65)(35,46,40,56), (1,2)(3,10)(4,42)(5,18)(6,50)(7,26)(8,58)(9,34)(12,28)(13,45)(14,44)(15,23)(16,31)(17,21)(19,39)(20,38)(22,25)(24,61)(27,60)(29,65)(30,55)(32,33)(35,52)(36,49)(37,59)(40,54)(41,62)(43,53)(46,48)(47,56)(51,63)(57,64)] + Permutation Group with generators [(1,2)(3,10)(4,42)(5,18)(6,50)(7,26)(8,58)(9,34)(12,28)(13,45)(14,44)(15,23)(16,31)(17,21)(19,39)(20,38)(22,25)(24,61)(27,60)(29,65)(30,55)(32,33)(35,52)(36,49)(37,59)(40,54)(41,62)(43,53)(46,48)(47,56)(51,63)(57,64), (1,28,10,44)(3,50,11,42)(4,43,53,64)(5,9,39,52)(6,36,63,13)(7,51,60,57)(8,33,37,16)(12,24,55,29)(14,30,48,47)(15,19,61,54)(17,59,22,62)(18,23,34,31)(20,38,49,25)(21,26,45,58)(27,32,41,65)(35,46,40,56)] sage: print SuzukiGroup(8) The Suzuki group over Finite Field in a of size 2^3 @@ -1612,7 +1594,8 @@ Finite Field in alpha of size 2^5 REFERENCES: - http://en.wikipedia.org/wiki/Group_of_Lie_type\#Suzuki-Ree_groups + + - http://en.wikipedia.org/wiki/Group_of_Lie_type\#Suzuki-Ree_groups """ q = Integer(q) from sage.rings.arith import valuation diff --git a/sage/rings/number_field/galois_group.py b/sage/rings/number_field/galois_group.py --- a/sage/rings/number_field/galois_group.py +++ b/sage/rings/number_field/galois_group.py @@ -343,8 +343,10 @@ sage: G.subgroup([ G(1), G([(1,5,2),(3,4,6)]), G([(1,2,5),(3,6,4)])]) Subgroup [(), (1,5,2)(3,4,6), (1,2,5)(3,6,4)] of Galois group of Number Field in b with defining polynomial x^6 - 14*x^4 + 20*x^3 + 49*x^2 - 140*x + 307 """ - if len(elts) == self.order(): return self - else: return GaloisGroup_subgroup(self, elts) + if len(elts) == self.order(): + return self + else: + return GaloisGroup_subgroup(self, elts) # Proper number theory starts here. All the functions below make no sense # unless the field is Galois. @@ -560,11 +562,14 @@ sage: GaloisGroup_subgroup( G, [ G(1), G([(1,5,2),(3,4,6)]), G([(1,2,5),(3,6,4)])]) Subgroup [(), (1,5,2)(3,4,6), (1,2,5)(3,6,4)] of Galois group of Number Field in b with defining polynomial x^6 - 14*x^4 + 20*x^3 + 49*x^2 - 140*x + 307 """ - + #XXX: This should be fixed so that this can use GaloisGroup_v2.__init__ PermutationGroup_generic.__init__(self, elts, canonicalize = True) self._ambient = ambient self._number_field = ambient.number_field() - self._galois_closure=ambient._galois_closure + self._galois_closure = ambient._galois_closure + self._pari_data = ambient._pari_data + self._pari_gc = ambient._pari_gc + self._gc_map = ambient._gc_map self._elts = elts def fixed_field(self): @@ -683,7 +688,8 @@ return min(w) def __cmp__(self, other): - r"""Compare self to other. For some bizarre reason, if you just let it + r""" + Compare self to other. For some bizarre reason, if you just let it inherit the cmp routine from PermutationGroupElement, cmp(x, y) works but sorting lists doesn't. @@ -691,9 +697,9 @@ sage: K. = NumberField(x^6 + 40*x^3 + 1372);G = K.galois_group() sage: sorted([G.artin_symbol(Q) for Q in K.primes_above(5)]) - [(1,2)(3,4)(5,6), (1,3)(2,6)(4,5), (1,5)(2,4)(3,6)] + [(1,3)(2,6)(4,5), (1,2)(3,4)(5,6), (1,5)(2,4)(3,6)] """ - return cmp(self.list(), other.list()) + return PermutationGroupElement.__cmp__(self, other) diff --git a/sage/rings/number_field/number_field.py b/sage/rings/number_field/number_field.py --- a/sage/rings/number_field/number_field.py +++ b/sage/rings/number_field/number_field.py @@ -6785,7 +6785,7 @@ sage: z = CyclotomicField(3).an_element(); z zeta3 sage: c = K.character([1,z,z**2]); c - Character of Subgroup of AlternatingGroup(4) generated by [(2,3,4)] + Character of Subgroup of (Alternating group of order 4!/2 as a permutation group) generated by [(2,3,4)] sage: c(g^2); z^2 -zeta3 - 1 -zeta3 - 1 diff --git a/sage/sets/finite_enumerated_set.py b/sage/sets/finite_enumerated_set.py --- a/sage/sets/finite_enumerated_set.py +++ b/sage/sets/finite_enumerated_set.py @@ -181,6 +181,18 @@ """ return Integer(len(self._elements)) + def index(self, x): + """ + Returns the index of ``x`` in this finite enumerated set. + + EXAMPLES:: + + sage: S = FiniteEnumeratedSet(['a','b','c']) + sage: S.index('b') + 1 + """ + return self._elements.index(x) + def _element_constructor_(self, el): """ TESTS:: diff --git a/sage/structure/parent.pyx b/sage/structure/parent.pyx --- a/sage/structure/parent.pyx +++ b/sage/structure/parent.pyx @@ -1668,7 +1668,7 @@ sage: S3.register_embedding(phi) sage: S3.coerce_embedding() Generic morphism: - From: AlternatingGroup(3) + From: Alternating group of order 3!/2 as a permutation group To: Special Linear Group of degree 3 over Rational Field sage: S3.coerce_embedding()(p) [0 0 1] diff --git a/sage/structure/parent_gens.pyx b/sage/structure/parent_gens.pyx --- a/sage/structure/parent_gens.pyx +++ b/sage/structure/parent_gens.pyx @@ -368,7 +368,7 @@ if d.has_key('_element_constructor'): return parent.Parent.__setstate__(self, d) try: - self.__dict__ = d + self.__dict__.update(d) self._generator_orders = d['_generator_orders'] except (AttributeError,KeyError): pass