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from sympy.combinatorics.group_constructs import DirectProduct
from sympy.combinatorics.perm_groups import PermutationGroup
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.
Explanation
===========
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.named_groups import AbelianGroup
>>> AbelianGroup(3, 4)
PermutationGroup([
(6)(0 1 2),
(3 4 5 6)])
>>> _.is_group
True
See Also
========
DirectProduct
References
==========
.. [1] https://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.
Explanation
===========
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)
set_alternating_group_properties(G, n, n)
G._is_alt = True
return G
def set_alternating_group_properties(G, n, degree):
"""Set known properties of an alternating group. """
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 = degree
G._is_transitive = True
G._is_dihedral = False
def CyclicGroup(n):
"""
Generates the cyclic group of order ``n`` as a permutation group.
Explanation
===========
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
G._is_dihedral = (n == 2)
return G
def DihedralGroup(n):
r"""
Generates the dihedral group `D_n` as a permutation group.
Explanation
===========
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] https://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_dihedral = True
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.
Explanation
===========
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] https://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])
set_symmetric_group_properties(G, n, n)
G._is_sym = True
return G
def set_symmetric_group_properties(G, n, degree):
"""Set known properties of a symmetric group. """
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 = degree
G._is_transitive = True
G._is_dihedral = (n in [2, 3]) # cf Landau's func and Stirling's approx
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))
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