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Source code for sympy.combinatorics.named_groups

from __future__ import print_function, division

from sympy.combinatorics.perm_groups import PermutationGroup
from sympy.combinatorics.group_constructs import DirectProduct
from sympy.combinatorics.permutations import Permutation

_af_new = Permutation._af_new

[docs]def AbelianGroup(*cyclic_orders): """ Returns the direct product of cyclic groups with the given orders. According to the structure theorem for finite abelian groups ([1]), every finite abelian group can be written as the direct product of finitely many cyclic groups. [1] http://groupprops.subwiki.org/wiki/Structure_theorem_for_finitely_generated_abelian_groups Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.named_groups import AbelianGroup >>> AbelianGroup(3, 4) PermutationGroup([ Permutation(6)(0, 1, 2), Permutation(3, 4, 5, 6)]) >>> _.is_group() False See Also ======== DirectProduct """ groups = [] degree = 0 order = 1 for size in cyclic_orders: degree += size order *= size groups.append(CyclicGroup(size)) G = DirectProduct(*groups) G._is_abelian = True G._degree = degree G._order = order return G
[docs]def AlternatingGroup(n): """ Generates the alternating group on n elements as a permutation group. For n > 2, the generators taken are (0 1 2), (0 1 2 ... n-1) for n odd and (0 1 2), (1 2 ... n-1) for n even (See [1], p.31, ex.6.9.). After the group is generated, some of its basic properties are set. The cases n = 1, 2 are handled separately. Examples ======== >>> from sympy.combinatorics.named_groups import AlternatingGroup >>> G = AlternatingGroup(4) >>> G.is_group() False >>> a = list(G.generate_dimino()) >>> len(a) 12 >>> all(perm.is_even for perm in a) True See Also ======== SymmetricGroup, CyclicGroup, DihedralGroup References ========== [1] Armstrong, M. "Groups and Symmetry" """ # small cases are special if n in (1, 2): return PermutationGroup([Permutation([0])]) a = list(range(n)) a[0], a[1], a[2] = a[1], a[2], a[0] gen1 = a if n % 2: a = list(range(1, n)) a.append(0) gen2 = a else: a = list(range(2, n)) a.append(1) a.insert(0, 0) gen2 = a gens = [gen1, gen2] if gen1 == gen2: gens = gens[:1] G = PermutationGroup([_af_new(a) for a in gens], dups=False) if n < 4: G._is_abelian = True G._is_nilpotent = True else: G._is_abelian = False G._is_nilpotent = False if n < 5: G._is_solvable = True else: G._is_solvable = False G._degree = n G._is_transitive = True G._is_alt = True return G
[docs]def CyclicGroup(n): """ Generates the cyclic group of order n as a permutation group. The generator taken is the n-cycle (0 1 2 ... n-1) (in cycle notation). After the group is generated, some of its basic properties are set. Examples ======== >>> from sympy.combinatorics.named_groups import CyclicGroup >>> G = CyclicGroup(6) >>> G.is_group() False >>> G.order() 6 >>> list(G.generate_schreier_sims(af=True)) [[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]] See Also ======== SymmetricGroup, DihedralGroup, AlternatingGroup """ a = list(range(1, n)) a.append(0) gen = _af_new(a) G = PermutationGroup([gen]) G._is_abelian = True G._is_nilpotent = True G._is_solvable = True G._degree = n G._is_transitive = True G._order = n return G
[docs]def DihedralGroup(n): r""" Generates the dihedral group D_n as a permutation group. The dihedral group D_n is the group of symmetries of the regular n-gon. The generators taken are the n-cycle a = (0 1 2 ... n-1) (a rotation of the n-gon) and b = (0 n-1)(1 n-2)... (a reflection of the n-gon) in cycle rotation. It is easy to see that these satisfy a**n = b**2 = 1 and bab = ~a so they indeed generate D_n (See [1]). After the group is generated, some of its basic properties are set. Examples ======== >>> from sympy.combinatorics.named_groups import DihedralGroup >>> G = DihedralGroup(5) >>> G.is_group() False >>> a = list(G.generate_dimino()) >>> [perm.cyclic_form for perm in a] [[], [[0, 1, 2, 3, 4]], [[0, 2, 4, 1, 3]], [[0, 3, 1, 4, 2]], [[0, 4, 3, 2, 1]], [[0, 4], [1, 3]], [[1, 4], [2, 3]], [[0, 1], [2, 4]], [[0, 2], [3, 4]], [[0, 3], [1, 2]]] See Also ======== SymmetricGroup, CyclicGroup, AlternatingGroup References ========== [1] http://en.wikipedia.org/wiki/Dihedral_group """ # small cases are special if n == 1: return PermutationGroup([Permutation([1, 0])]) if n == 2: return PermutationGroup([Permutation([1, 0, 3, 2]), Permutation([2, 3, 0, 1]), Permutation([3, 2, 1, 0])]) a = list(range(1, n)) a.append(0) gen1 = _af_new(a) a = list(range(n)) a.reverse() gen2 = _af_new(a) G = PermutationGroup([gen1, gen2]) # if n is a power of 2, group is nilpotent if n & (n-1) == 0: G._is_nilpotent = True else: G._is_nilpotent = False G._is_abelian = False G._is_solvable = True G._degree = n G._is_transitive = True G._order = 2*n return G
[docs]def SymmetricGroup(n): """ Generates the symmetric group on n elements as a permutation group. The generators taken are the n-cycle (0 1 2 ... n-1) and the transposition (0 1) (in cycle notation). (See [1]). After the group is generated, some of its basic properties are set. Examples ======== >>> from sympy.combinatorics.named_groups import SymmetricGroup >>> G = SymmetricGroup(4) >>> G.is_group() False >>> G.order() 24 >>> list(G.generate_schreier_sims(af=True)) [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 1, 2, 0], [0, 2, 3, 1], [1, 3, 0, 2], [2, 0, 1, 3], [3, 2, 0, 1], [0, 3, 1, 2], [1, 0, 2, 3], [2, 1, 3, 0], [3, 0, 1, 2], [0, 1, 3, 2], [1, 2, 0, 3], [2, 3, 1, 0], [3, 1, 0, 2], [0, 2, 1, 3], [1, 3, 2, 0], [2, 0, 3, 1], [3, 2, 1, 0], [0, 3, 2, 1], [1, 0, 3, 2], [2, 1, 0, 3], [3, 0, 2, 1]] See Also ======== CyclicGroup, DihedralGroup, AlternatingGroup References ========== [1] http://en.wikipedia.org/wiki/Symmetric_group#Generators_and_relations """ if n == 1: G = PermutationGroup([Permutation([0])]) elif n == 2: G = PermutationGroup([Permutation([1, 0])]) else: a = list(range(1, n)) a.append(0) gen1 = _af_new(a) a = list(range(n)) a[0], a[1] = a[1], a[0] gen2 = _af_new(a) G = PermutationGroup([gen1, gen2]) if n < 3: G._is_abelian = True G._is_nilpotent = True else: G._is_abelian = False G._is_nilpotent = False if n < 5: G._is_solvable = True else: G._is_solvable = False G._degree = n G._is_transitive = True G._is_sym = True return G
def RubikGroup(n): """Return a group of Rubik's cube generators. >>> from sympy.combinatorics.named_groups import RubikGroup >>> RubikGroup(2).is_group() False """ from sympy.combinatorics.generators import rubik if n <= 1: raise ValueError("Invalid cube . n has to be greater than 1") return PermutationGroup(rubik(n))