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)
>>> a = list(G.generate_dimino())
>>> [perm.cyclic_form for perm in a]
[[[4], [3], [2], [1], [0]], [[0, 1, 2, 3, 4]], [[0, 2, 4, 1, 3]],
[[0, 3, 1, 4, 2]], [[0, 4, 3, 2, 1]], [[2], [1, 3], [0, 4]],
[[2, 3], [1, 4], [0]], [[3], [2, 4], [0, 1]], [[3, 4], [1], [0, 2]],
[[4], [1, 2], [0, 3]]]
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 = range(1, n)
a.append(0)
gen1 = _new_from_array_form(a)
a = range(n)
a.reverse()
gen2 = _new_from_array_form(a)
G = PermutationGroup([gen1, gen2])
G._is_abelian = False
G._degree = n
G._is_transitive = True
G._order = 2 * n
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)
>>> a = list(G.generate_dimino())
>>> len(a)
12
>>> [perm.is_even for perm in a]
[True, True, True, True, True, True, True, True, True, True, True, 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 = range(n)
a[0], a[1], a[2] = a[1], a[2], a[0]
gen1 = _new_from_array_form(a)
if n % 2:
a = range(1, n)
a.append(0)
gen2 = _new_from_array_form(a)
else:
a = range(2, n)
a.append(1)
gen2 = _new_from_array_form([0] + a)
G = PermutationGroup([gen1, gen2])
if n < 4:
G._is_abelian = True
else:
G._is_abelian = False
G._degree = n
G._is_transitive = True
G._is_alt = True
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)
>>> a = list(G.generate_dimino())
>>> len(a)
12
>>> [perm.is_even for perm in a]
[True, True, True, True, True, True, True, True, True, True, True, 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 = range(n)
a[0], a[1], a[2] = a[1], a[2], a[0]
gen1 = _new_from_array_form(a)
if n % 2:
a = range(1, n)
a.append(0)
gen2 = _new_from_array_form(a)
else:
a = range(2, n)
a.append(1)
gen2 = _new_from_array_form([0] + a)
G = PermutationGroup([gen1, gen2])
if n<4:
G._is_abelian = True
else:
G._is_abelian = False
G._degree = n
G._is_transitive = True
G._is_alt = True
return G
def generate_dimino(self, af=False):
"""
yield group elements using Dimino's algorithm
If af == True it yields the array form of the permutations
Reference:
[1] The implementation of various algorithms for Permutation Groups in
the Computer Algebra System: AXIOM, N.J. Doye, M.Sc. Thesis
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([0, 2, 3, 1])
>>> g = PermutationGroup([a, b])
>>> list(g.generate_dimino(af=True))
[[0, 1, 2, 3], [0, 2, 1, 3], [0, 2, 3, 1], [0, 1, 3, 2], [0, 3, 2, 1], [0, 3, 1, 2]]
"""
idn = range(self.degree)
order = 0
element_list = [idn]
set_element_list = set([tuple(idn)])
if af:
yield idn
else:
yield _new_from_array_form(idn)
gens = [p.array_form for p in self.generators]
for i in xrange(len(gens)):
# D elements of the subgroup G_i generated by gens[:i]
D = element_list[:]
N = [idn]
while N:
A = N
N = []
for a in A:
for g in gens[:i + 1]:
ag = perm_af_mul(a, g)
if tuple(ag) not in set_element_list:
# produce G_i*g
for d in D:
order += 1
ap = perm_af_mul(d, ag)
if af:
yield ap
else:
p = _new_from_array_form(ap)
yield p
element_list.append(ap)
set_element_list.add(tuple(ap))
N.append(ap)
self._order = len(element_list)
def generate_dimino(self, af=False):
"""
yield group elements using Dimino's algorithm
If af == True it yields the array form of the permutations
Reference:
[1] The implementation of various algorithms for Permutation Groups in
the Computer Algebra System: AXIOM, N.J. Doye, M.Sc. Thesis
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([0, 2, 3, 1])
>>> g = PermutationGroup([a, b])
>>> list(g.generate_dimino(af=True))
[[0, 1, 2, 3], [0, 2, 1, 3], [0, 2, 3, 1], [0, 1, 3, 2], [0, 3, 2, 1], [0, 3, 1, 2]]
"""
idn = range(self.degree)
order = 0
element_list = [idn]
set_element_list = set([tuple(idn)])
if af:
yield idn
else:
yield _new_from_array_form(idn)
gens = [p.array_form for p in self.generators]
for i in xrange(len(gens)):
# D elements of the subgroup G_i generated by gens[:i]
D = element_list[:]
N = [idn]
while N:
A = N
N = []
for a in A:
for g in gens[:i+1]:
ag = perm_af_mul(a, g)
if tuple(ag) not in set_element_list:
# produce G_i*g
for d in D:
order += 1
ap = perm_af_mul(d, ag)
if af:
yield ap
else:
p = _new_from_array_form(ap)
yield p
element_list.append(ap)
set_element_list.add(tuple(ap))
N.append(ap)
self._order = len(element_list)
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.order()
24
>>> list(G.generate_schreier_sims(af=True))
[[0, 1, 2, 3], [0, 1, 3, 2], [0, 2, 1, 3], [0, 2, 3, 1], [0, 3, 1, 2],
[0, 3, 2, 1], [1, 2, 3, 0], [1, 2, 0, 3], [1, 3, 2, 0],
[1, 3, 0, 2], [1, 0, 2, 3], [1, 0, 3, 2], [2, 3, 0, 1], [2, 3, 1, 0],
[2, 0, 3, 1], [2, 0, 1, 3], [2, 1, 3, 0], [2, 1, 0, 3], [3, 0, 1, 2],
[3, 0, 2, 1], [3, 1, 0, 2], [3, 1, 2, 0], [3, 2, 0, 1], [3, 2, 1, 0]]
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 = range(1,n)
a.append(0)
gen1 = _new_from_array_form(a)
a = range(n)
a[0], a[1] = a[1], a[0]
gen2 = _new_from_array_form(a)
G = PermutationGroup([gen1, gen2])
if n<3:
G._is_abelian = True
else:
G._is_abelian = False
G._degree = n
G._is_transitive = True
G._is_sym = True
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)
>>> a = list(G.generate_dimino())
>>> [perm.cyclic_form for perm in a]
[[[4], [3], [2], [1], [0]], [[0, 1, 2, 3, 4]], [[0, 2, 4, 1, 3]],
[[0, 3, 1, 4, 2]], [[0, 4, 3, 2, 1]], [[2], [1, 3], [0, 4]],
[[2, 3], [1, 4], [0]], [[3], [2, 4], [0, 1]], [[3, 4], [1], [0, 2]],
[[4], [1, 2], [0, 3]]]
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 = range(1, n)
a.append(0)
gen1 = _new_from_array_form(a)
a = range(n)
a.reverse()
gen2 = _new_from_array_form(a)
G = PermutationGroup([gen1, gen2])
G._is_abelian = False
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.order()
24
>>> list(G.generate_schreier_sims(af=True))
[[0, 1, 2, 3], [0, 1, 3, 2], [0, 2, 1, 3], [0, 2, 3, 1], [0, 3, 1, 2],
[0, 3, 2, 1], [1, 2, 3, 0], [1, 2, 0, 3], [1, 3, 2, 0],
[1, 3, 0, 2], [1, 0, 2, 3], [1, 0, 3, 2], [2, 3, 0, 1], [2, 3, 1, 0],
[2, 0, 3, 1], [2, 0, 1, 3], [2, 1, 3, 0], [2, 1, 0, 3], [3, 0, 1, 2],
[3, 0, 2, 1], [3, 1, 0, 2], [3, 1, 2, 0], [3, 2, 0, 1], [3, 2, 1, 0]]
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 = range(1, n)
a.append(0)
gen1 = _new_from_array_form(a)
a = range(n)
a[0], a[1] = a[1], a[0]
gen2 = _new_from_array_form(a)
G = PermutationGroup([gen1, gen2])
if n < 3:
G._is_abelian = True
else:
G._is_abelian = False
G._degree = n
G._is_transitive = True
G._is_sym = True
return G
def coset_unrank(self, rank, af=False):
"""
unrank using Schreier-Sims representation
coset_unrank is the inverse operation of coset_rank
if 0 <= rank < order; otherwise it returns None.
"""
u = self.coset_repr()
if rank < 0 or rank >= self.order():
return None
un = self._coset_repr_n
base = self._base
m = len(u)
nb = len(base)
assert nb == len(un)
v = [0] * m
for i in range(nb - 1, -1, -1):
j = base[i]
rank, c = divmod(rank, un[i])
v[j] = c
a = [u[i][v[i]] for i in range(m)]
h = perm_af_muln(*a)
if af:
return h
else:
return _new_from_array_form(h)
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.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 = range(1, n)
a.append(0)
gen = _new_from_array_form(a)
G = PermutationGroup([gen])
G._is_abelian = True
G._degree = n
G._is_transitive = True
G._order = n
return G
def coset_unrank(self, rank, af=False):
"""
unrank using Schreier-Sims representation
coset_unrank is the inverse operation of coset_rank
if 0 <= rank < order; otherwise it returns None.
"""
u = self.coset_repr()
if rank < 0 or rank >= self.order():
return None
un = self._coset_repr_n
base = self._base
m = len(u)
nb = len(base)
assert nb == len(un)
v = [0]*m
for i in range(nb-1, -1,-1):
j = base[i]
rank, c = divmod(rank, un[i])
v[j] = c
a = [u[i][v[i]] for i in range(m)]
h = perm_af_muln(*a)
if af:
return h
else:
return _new_from_array_form(h)
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.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 = range(1, n)
a.append(0)
gen = _new_from_array_form(a)
G = PermutationGroup([gen])
G._is_abelian = True
G._degree = n
G._is_transitive = True
G._order = n
return G
def generate_schreier_sims(self, af=False):
"""
yield group elements using the Schreier-Sims representation
If af = True it yields the array form of the permutations
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([0, 2, 3, 1])
>>> g = PermutationGroup([a, b])
>>> list(g.generate_schreier_sims(af=True))
[[0, 1, 2, 3], [0, 1, 3, 2], [0, 2, 3, 1], [0, 2, 1, 3], [0, 3, 2, 1], [0, 3, 1, 2]]
"""
def get1(posmax):
n = len(posmax) - 1
for i in range(n,-1,-1):
if posmax[i] != 1:
return i + 1
n = self.degree
u = self.coset_repr()
# stg stack of group elements
stg = [range(n)]
# posmax[i] = len(u[i])
posmax = [len(x) for x in u]
n1 = get1(posmax)
pos = [0]*n1
posmax = posmax[:n1]
h = 0
while 1:
# backtrack when finished iterating over coset
if pos[h] >= posmax[h]:
if h == 0:
raise StopIteration
pos[h] = 0
h -= 1
stg.pop()
continue
p = perm_af_mul(stg[-1], u[h][pos[h]])
pos[h] += 1
stg.append(p)
h += 1
if h == n1:
if af:
yield p
else:
p1 = _new_from_array_form(p)
yield p1
stg.pop()
h -= 1
def generate_schreier_sims(self, af=False):
"""
yield group elements using the Schreier-Sims representation
If af = True it yields the array form of the permutations
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([0, 2, 3, 1])
>>> g = PermutationGroup([a, b])
>>> list(g.generate_schreier_sims(af=True))
[[0, 1, 2, 3], [0, 1, 3, 2], [0, 2, 3, 1], [0, 2, 1, 3], [0, 3, 2, 1], [0, 3, 1, 2]]
"""
def get1(posmax):
n = len(posmax) - 1
for i in range(n, -1, -1):
if posmax[i] != 1:
return i + 1
n = self.degree
u = self.coset_repr()
# stg stack of group elements
stg = [range(n)]
# posmax[i] = len(u[i])
posmax = [len(x) for x in u]
n1 = get1(posmax)
pos = [0] * n1
posmax = posmax[:n1]
h = 0
while 1:
# backtrack when finished iterating over coset
if pos[h] >= posmax[h]:
if h == 0:
raise StopIteration
pos[h] = 0
h -= 1
stg.pop()
continue
p = perm_af_mul(stg[-1], u[h][pos[h]])
pos[h] += 1
stg.append(p)
h += 1
if h == n1:
if af:
yield p
else:
p1 = _new_from_array_form(p)
yield p1
stg.pop()
h -= 1
def DirectProduct(*groups):
"""
Returns the direct product of several groups as a permutation group.
This is implemented much like the __mul__ procedure for taking the direct
product of two permutation groups, but the idea of shifting the
generators is realized in the case of an arbitrary number of groups.
A call to DirectProduct(G1, G2, ..., Gn) is generally expected to be faster
than a call to G1*G2*...*Gn (and thus the need for this algorithm).
Examples
========
>>> from sympy.combinatorics.group_constructs import DirectProduct
>>> from sympy.combinatorics.named_groups import CyclicGroup
>>> C = CyclicGroup(4)
>>> G = DirectProduct(C,C,C)
>>> G.order()
64
See Also
========
__mul__
"""
degrees = []
gens_count = []
total_degree = 0
total_gens = 0
for group in groups:
current_deg = group.degree
current_num_gens = len(group.generators)
degrees.append(current_deg)
total_degree += current_deg
gens_count.append(current_num_gens)
total_gens += current_num_gens
array_gens = []
for i in range(total_gens):
array_gens.append(range(total_degree))
current_gen = 0
current_deg = 0
for i in xrange(len(gens_count)):
for j in xrange(current_gen, current_gen + gens_count[i]):
gen = ((groups[i].generators)[j - current_gen]).array_form
array_gens[j][current_deg:current_deg + degrees[i]] =\
[ x + current_deg for x in gen]
current_gen += gens_count[i]
current_deg += degrees[i]
perm_gens = [_new_from_array_form(array) for array in array_gens]
return PermutationGroup(perm_gens)
def stabilizer(self, alpha):
"""
return the stabilizer subgroup leaving alpha fixed.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([2, 0, 1, 3])
>>> g = PermutationGroup([a, b])
>>> g2 = g.stabilizer(2)
>>> g2
PermutationGroup([Permutation([1, 0, 2, 3])])
"""
if alpha == 0:
self.schreier_sims()
gens = self._stabilizers_gens
gens = [_new_from_array_form(p) for p in gens]
return PermutationGroup(gens)
# h[alpha] = 0; h[0] = alpha
n = self.degree
h = range(n)
h[0] = alpha
h[alpha] = 0
genv = [p.array_form for p in self.generators]
# conjugate the group to breng alpha to 0
gens = [_new_from_array_form(perm_af_muln(h, p, h)) for p in genv]
G = PermutationGroup(gens)
G.schreier_sims()
# stabilizers for 0
gens1 = G._stabilizers_gens
# conjugate the group
gens2 = [_new_from_array_form(perm_af_muln(h, p, h)) for p in gens1]
return PermutationGroup(gens2)
def stabilizer(self, alpha):
"""
return the stabilizer subgroup leaving alpha fixed.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([2, 0, 1, 3])
>>> g = PermutationGroup([a, b])
>>> g2 = g.stabilizer(2)
>>> g2
PermutationGroup([Permutation([1, 0, 2, 3])])
"""
if alpha == 0:
self.schreier_sims()
gens = self._stabilizers_gens
gens = [_new_from_array_form(p) for p in gens]
return PermutationGroup(gens)
# h[alpha] = 0; h[0] = alpha
n = self.degree
h = range(n)
h[0] = alpha
h[alpha] = 0
genv = [p.array_form for p in self.generators]
# conjugate the group to breng alpha to 0
gens = [_new_from_array_form(perm_af_muln(h, p, h)) for p in genv]
G = PermutationGroup(gens)
G.schreier_sims()
# stabilizers for 0
gens1 = G._stabilizers_gens
# conjugate the group
gens2 = [_new_from_array_form(perm_af_muln(h, p, h)) for p in gens1]
return PermutationGroup(gens2)
def _remove_gens(base, strong_gens, basic_orbits=None, strong_gens_distr=None):
"""
Remove redundant generators from a strong generating set.
Parameters
==========
``base`` - a base
``strong_gens`` - a strong generating set relative to ``base``
``basic_orbits`` - basic orbits
``strong_gens_distr`` - strong generators distributed by membership in basic
stabilizers
Returns
=======
A strong generating set with respect to ``base`` which is a subset of
``strong_gens``.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> from sympy.combinatorics.util import _remove_gens
>>> from sympy.combinatorics.testutil import _verify_bsgs
>>> S = SymmetricGroup(15)
>>> base, strong_gens = S.schreier_sims_incremental()
>>> len(strong_gens)
26
>>> new_gens = _remove_gens(base, strong_gens)
>>> len(new_gens)
14
>>> _verify_bsgs(S, base, new_gens)
True
Notes
=====
This procedure is outlined in [1],p.95.
References
==========
[1] Holt, D., Eick, B., O'Brien, E.
"Handbook of computational group theory"
"""
from sympy.combinatorics.perm_groups import PermutationGroup
base_len = len(base)
degree = strong_gens[0].size
identity = _new_from_array_form(range(degree))
if strong_gens_distr is None:
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
temp = strong_gens_distr[:]
if basic_orbits is None:
basic_orbits = []
for i in range(base_len):
stab = PermutationGroup(strong_gens_distr[i])
basic_orbit = stab.orbit(base[i])
basic_orbits.append(basic_orbit)
strong_gens_distr.append([])
res = strong_gens[:]
for i in range(base_len - 1, -1, -1):
gens_copy = strong_gens_distr[i][:]
for gen in strong_gens_distr[i]:
if gen not in strong_gens_distr[i + 1]:
temp_gens = gens_copy[:]
temp_gens.remove(gen)
if temp_gens == []:
continue
temp_group = PermutationGroup(temp_gens)
temp_orbit = temp_group.orbit(base[i])
if temp_orbit == basic_orbits[i]:
gens_copy.remove(gen)
res.remove(gen)
return res
def _distribute_gens_by_base(base, gens):
"""
Distribute the group elements ``gens`` by membership in basic stabilizers.
Notice that for a base `(b_1, b_2, ..., b_k)`, the basic stabilizers
are defined as `G^{(i)} = G_{b_1, ..., b_{i-1}}` for
`i \in\{1, 2, ..., k\}`.
Parameters
==========
``base`` - a sequence of points in `\{0, 1, ..., n-1\}`
``gens`` - a list of elements of a permutation group of degree `n`.
Returns
=======
List of length `k`, where `k` is
the length of ``base``. The `i`-th entry contains those elements in
``gens`` which fix the first `i` elements of ``base`` (so that the
`0`-th entry is equal to ``gens`` itself). If no element fixes the first
`i` elements of ``base``, the `i`-th element is set to a list containing
the identity element.
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> from sympy.combinatorics.util import _distribute_gens_by_base
>>> D = DihedralGroup(3)
>>> D.schreier_sims()
>>> D.strong_gens
[Permutation([1, 2, 0]), Permutation([2, 1, 0]), Permutation([0, 2, 1])]
>>> D.base
[0, 1]
>>> _distribute_gens_by_base(D.base, D.strong_gens)
[[Permutation([1, 2, 0]), Permutation([2, 1, 0]), Permutation([0, 2, 1])],\
[Permutation([0, 2, 1])]]
See Also
========
_strong_gens_from_distr, _orbits_transversals_from_bsgs,
_handle_precomputed_bsgs
"""
base_len = len(base)
stabs = []
degree = gens[0].size
for i in xrange(base_len):
stabs.append([])
num_gens = len(gens)
max_stab_index = 0
for i in xrange(num_gens):
j = 0
while j < base_len - 1 and gens[i](base[j]) == base[j]:
j += 1
if j > max_stab_index:
max_stab_index = j
for k in xrange(j + 1):
stabs[k].append(gens[i])
for i in range(max_stab_index + 1, base_len):
stabs[i].append(_new_from_array_form(range(degree)))
return stabs
def _remove_gens(base, strong_gens, basic_orbits=None, strong_gens_distr=None):
"""
Remove redundant generators from a strong generating set.
Parameters
==========
``base`` - a base
``strong_gens`` - a strong generating set relative to ``base``
``basic_orbits`` - basic orbits
``strong_gens_distr`` - strong generators distributed by membership in basic
stabilizers
Returns
=======
A strong generating set with respect to ``base`` which is a subset of
``strong_gens``.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> from sympy.combinatorics.util import _remove_gens, _verify_bsgs
>>> S = SymmetricGroup(15)
>>> base, strong_gens = S.schreier_sims_incremental()
>>> len(strong_gens)
26
>>> new_gens = _remove_gens(base, strong_gens)
>>> len(new_gens)
14
>>> _verify_bsgs(S, base, new_gens)
True
Notes
=====
This procedure is outlined in [1],p.95.
References
==========
[1] Holt, D., Eick, B., O'Brien, E.
"Handbook of computational group theory"
"""
from sympy.combinatorics.perm_groups import PermutationGroup
base_len = len(base)
degree = strong_gens[0].size
identity = _new_from_array_form(range(degree))
if strong_gens_distr is None:
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
temp = strong_gens_distr[:]
if basic_orbits is None:
basic_orbits = []
for i in range(base_len):
stab = PermutationGroup(strong_gens_distr[i])
basic_orbit = stab.orbit(base[i])
basic_orbits.append(basic_orbit)
strong_gens_distr.append([])
res = strong_gens[:]
for i in range(base_len - 1, -1, -1):
gens_copy = strong_gens_distr[i][:]
for gen in strong_gens_distr[i]:
if gen not in strong_gens_distr[i + 1]:
temp_gens = gens_copy[:]
temp_gens.remove(gen)
if temp_gens == []:
continue
temp_group = PermutationGroup(temp_gens)
temp_orbit = temp_group.orbit(base[i])
if temp_orbit == basic_orbits[i]:
gens_copy.remove(gen)
res.remove(gen)
return res
def _distribute_gens_by_base(base, gens):
"""
Distribute the group elements ``gens`` by membership in basic stabilizers.
Notice that for a base `(b_1, b_2, ..., b_k)`, the basic stabilizers
are defined as `G^{(i)} = G_{b_1, ..., b_{i-1}}` for
`i \in\{1, 2, ..., k\}`.
Parameters
==========
``base`` - a sequence of points in `\{0, 1, ..., n-1\}`
``gens`` - a list of elements of a permutation group of degree `n`.
Returns
=======
List of length `k`, where `k` is
the length of ``base``. The `i`-th entry contains those elements in
``gens`` which fix the first `i` elements of ``base`` (so that the
`0`-th entry is equal to ``gens`` itself). If no element fixes the first
`i` elements of ``base``, the `i`-th element is set to a list containing
the identity element.
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> from sympy.combinatorics.util import _distribute_gens_by_base
>>> D = DihedralGroup(3)
>>> D.schreier_sims()
>>> D.strong_gens
[Permutation([1, 2, 0]), Permutation([2, 1, 0]), Permutation([0, 2, 1])]
>>> D.base
[0, 1]
>>> _distribute_gens_by_base(D.base, D.strong_gens)
[[Permutation([1, 2, 0]), Permutation([2, 1, 0]), Permutation([0, 2, 1])],\
[Permutation([0, 2, 1])]]
See Also
========
_strong_gens_from_distr, _orbits_transversals_from_bsgs,
_handle_precomputed_bsgs
"""
base_len = len(base)
stabs = []
degree = gens[0].size
for i in xrange(base_len):
stabs.append([])
num_gens = len(gens)
max_stab_index = 0
for i in xrange(num_gens):
j = 0
while j < base_len - 1 and gens[i](base[j]) == base[j]:
j += 1
if j > max_stab_index:
max_stab_index = j
for k in xrange(j + 1):
stabs[k].append(gens[i])
for i in range(max_stab_index + 1, base_len):
stabs[i].append(_new_from_array_form(range(degree)))
return stabs
