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from __future__ import print_function, division 

 

from sympy.core.compatibility import range 

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 

 

 

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. 

 

Examples 

======== 

 

>>> from sympy.combinatorics import Permutation 

>>> Permutation.print_cyclic = True 

>>> from sympy.combinatorics.named_groups import AbelianGroup 

>>> AbelianGroup(3, 4) 

PermutationGroup([ 

(6)(0 1 2), 

(3 4 5 6)]) 

>>> _.is_group 

True 

 

See Also 

======== 

 

DirectProduct 

 

References 

========== 

 

[1] http://groupprops.subwiki.org/wiki/Structure_theorem_for_finitely_generated_abelian_groups 

 

""" 

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 

 

 

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 

True 

>>> 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 

 

 

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 

True 

>>> 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 

 

 

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 

True 

>>> 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 

 

 

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 

True 

>>> 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 

True 

""" 

from sympy.combinatorics.generators import rubik 

if n <= 1: 

raise ValueError("Invalid cube . n has to be greater than 1") 

return PermutationGroup(rubik(n))