def is_normal(self, gr):
"""
test if G=self is a normal subgroup of gr
G is normal in gr if
for each g2 in G, g1 in gr, g = g1*g2*g1**-1 belongs to G
It is sufficient to check this for each g1 in gr.generator and
g2 g2 in G.generator
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([1, 2, 0])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G1 = PermutationGroup([a, Permutation([2, 0, 1])])
>>> G1.is_normal(G)
True
"""
gens2 = [p.array_form for p in self.generators]
gens1 = [p.array_form for p in gr.generators]
for g1 in gens1:
for g2 in gens2:
p = perm_af_muln(g1, g2, perm_af_invert(g1))
if not self.coset_decomposition(p):
return False
return True
def is_normal(self, gr):
"""
test if G=self is a normal subgroup of gr
G is normal in gr if
for each g2 in G, g1 in gr, g = g1*g2*g1**-1 belongs to G
It is sufficient to check this for each g1 in gr.generator and
g2 g2 in G.generator
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([1, 2, 0])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G1 = PermutationGroup([a, Permutation([2, 0, 1])])
>>> G1.is_normal(G)
True
"""
gens2 = [p.array_form for p in self.generators]
gens1 = [p.array_form for p in gr.generators]
for g1 in gens1:
for g2 in gens2:
p = perm_af_muln(g1, g2, perm_af_invert(g1))
if not self.coset_decomposition(p):
return False
return True
def insert(self, g, alpha):
"""
insert permutation `g` in stabilizer chain at point alpha
"""
n = len(g)
if not g == self.idn:
vertex = self.vertex
jg = self.jg
i = _smallest_change(g, alpha)
ig = g[i]
nn = vertex[i].index_neighbor[ig]
if nn >= 0: # if ig is already neighbor of i
jginn = jg[vertex[i].perm[nn]]
if g != jginn:
# cycle consisting of two edges;
# replace jginn by g and insert h = g**-1*jginn
g1 = perm_af_invert(g)
h = perm_af_mul(g1, jginn)
jg[ vertex[i].perm[nn] ] = g
self.insert(h, alpha)
else: # new edge
self.insert_edge(g, i, ig)
self.cycle = [i]
if self.find_cycle(i, i, ig, -1):
cycle = self.cycle
cycle.append(cycle[0])
# find the smallest point (vertex) of the cycle
minn = min(cycle)
cmin = cycle.index(minn)
# now walk around the cycle starting from the smallest
# point, and multiply around the cycle to obtain h
# satisfying h[cmin] = cmin
ap = []
for c in range(cmin, len(cycle)-1) + range(cmin):
i = cycle[c]
j = cycle[c+1]
nn = vertex[i].index_neighbor[j]
p = jg[ vertex[i].perm[nn] ]
if i > j:
p = perm_af_invert(p)
ap.append(p)
ap.reverse()
h = perm_af_muln(*ap)
self.remove_edge(cycle[cmin], cycle[cmin + 1])
self.insert(h, alpha)
def insert(self, g, alpha):
"""
insert permutation `g` in stabilizer chain at point alpha
"""
n = len(g)
if not g == self.idn:
vertex = self.vertex
jg = self.jg
i = _smallest_change(g, alpha)
ig = g[i]
nn = vertex[i].index_neighbor[ig]
if nn >= 0: # if ig is already neighbor of i
jginn = jg[vertex[i].perm[nn]]
if g != jginn:
# cycle consisting of two edges;
# replace jginn by g and insert h = g**-1*jginn
g1 = perm_af_invert(g)
h = perm_af_mul(g1, jginn)
jg[vertex[i].perm[nn]] = g
self.insert(h, alpha)
else: # new edge
self.insert_edge(g, i, ig)
self.cycle = [i]
if self.find_cycle(i, i, ig, -1):
cycle = self.cycle
cycle.append(cycle[0])
# find the smallest point (vertex) of the cycle
minn = min(cycle)
cmin = cycle.index(minn)
# now walk around the cycle starting from the smallest
# point, and multiply around the cycle to obtain h
# satisfying h[cmin] = cmin
ap = []
for c in range(cmin, len(cycle) - 1) + range(cmin):
i = cycle[c]
j = cycle[c + 1]
nn = vertex[i].index_neighbor[j]
p = jg[vertex[i].perm[nn]]
if i > j:
p = perm_af_invert(p)
ap.append(p)
ap.reverse()
h = perm_af_muln(*ap)
self.remove_edge(cycle[cmin], cycle[cmin + 1])
self.insert(h, alpha)
def coset_rank(self, g):
"""
rank using Schreier-Sims representation
The coset rank of `g` is the ordering number in which
it appears in the lexicographic listing according to the
coset decomposition, see coset_decomposition;
the ordering is the same as in G.generate(method='coset').
If `g` does not belong to the group it returns None
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([[0, 1, 3, 7, 6, 4], [2, 5]])
>>> b = Permutation([[0, 1, 3, 2], [4, 5, 7, 6]])
>>> G = PermutationGroup([a, b])
>>> c = Permutation([[0, 1, 2, 3, 4], [5, 6, 7]])
>>> G.coset_rank(c)
>>> c = Permutation([[0, 6], [1, 7], [2, 4], [3, 5]])
>>> G.coset_rank(c)
40
>>> G.coset_unrank(40, af=True)
[6, 7, 4, 5, 2, 3, 0, 1]
"""
u = self.coset_repr()
if isinstance(g, Permutation):
g = g.array_form
g1 = g
m = len(u)
a = []
un = self._coset_repr_n
n = self.degree
rank = 0
base = [1]
for i in un[m:0:-1]:
base.append(base[-1]*i)
base.reverse()
a1 = [0]*m
i1 = -1
for i in self._base:
i1 += 1
x = g1[i]
for j, h in enumerate(u[i]):
if h[i] == x:
a.append(h)
a1[i] = j
rank += j*base[i1]
p2 = perm_af_invert(h)
g1 = perm_af_mul(p2, g1)
break
else:
return None
if perm_af_muln(*a) == g:
return rank
return None
def coset_rank(self, g):
"""
rank using Schreier-Sims representation
The coset rank of `g` is the ordering number in which
it appears in the lexicographic listing according to the
coset decomposition, see coset_decomposition;
the ordering is the same as in G.generate(method='coset').
If `g` does not belong to the group it returns None
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([[0, 1, 3, 7, 6, 4], [2, 5]])
>>> b = Permutation([[0, 1, 3, 2], [4, 5, 7, 6]])
>>> G = PermutationGroup([a, b])
>>> c = Permutation([[0, 1, 2, 3, 4], [5, 6, 7]])
>>> G.coset_rank(c)
>>> c = Permutation([[0, 6], [1, 7], [2, 4], [3, 5]])
>>> G.coset_rank(c)
40
>>> G.coset_unrank(40, af=True)
[6, 7, 4, 5, 2, 3, 0, 1]
"""
u = self.coset_repr()
if isinstance(g, Permutation):
g = g.array_form
g1 = g
m = len(u)
a = []
un = self._coset_repr_n
n = self.degree
rank = 0
base = [1]
for i in un[m:0:-1]:
base.append(base[-1] * i)
base.reverse()
a1 = [0] * m
i1 = -1
for i in self._base:
i1 += 1
x = g1[i]
for j, h in enumerate(u[i]):
if h[i] == x:
a.append(h)
a1[i] = j
rank += j * base[i1]
p2 = perm_af_invert(h)
g1 = perm_af_mul(p2, g1)
break
else:
return None
if perm_af_muln(*a) == g:
return rank
return None
def coset_decomposition(self, g):
"""
Decompose `g` as h_0*...*h_{len(u)}
The Schreier-Sims coset representation u of `G`
gives a univoque decomposition of an element `g`
as h_0*...*h_{len(u)}, where h_i belongs to u[i]
Output: [h_0, .., h_{len(u)}] if `g` belongs to `G`
False otherwise
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([[0, 1, 3, 7, 6, 4], [2, 5]])
>>> b = Permutation([[0, 1, 3, 2], [4, 5, 7, 6]])
>>> G = PermutationGroup([a, b])
>>> c = Permutation([[0, 1, 2, 3, 4], [5, 6, 7]])
>>> G.coset_decomposition(c)
False
>>> c = Permutation([[0, 6], [1, 7], [2, 4], [3, 5]])
>>> G.coset_decomposition(c)
[[6, 4, 2, 0, 7, 5, 3, 1], [0, 4, 1, 5, 2, 6, 3, 7], [0, 1, 2, 3, 4, 5, 6, 7]]
>>> G.has_element(c)
True
"""
u = self.coset_repr()
if isinstance(g, Permutation):
g = g.array_form
g1 = g
n = len(u)
a = []
for i in range(n):
x = g1[i]
for h in u[i]:
if h[i] == x:
a.append(h)
p2 = perm_af_invert(h)
g1 = perm_af_mul(p2, g1)
break
else:
return False
if perm_af_muln(*a) == g:
return a
return False
def coset_decomposition(self, g):
"""
Decompose `g` as h_0*...*h_{len(u)}
The Schreier-Sims coset representation u of `G`
gives a univoque decomposition of an element `g`
as h_0*...*h_{len(u)}, where h_i belongs to u[i]
Output: [h_0, .., h_{len(u)}] if `g` belongs to `G`
False otherwise
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([[0, 1, 3, 7, 6, 4], [2, 5]])
>>> b = Permutation([[0, 1, 3, 2], [4, 5, 7, 6]])
>>> G = PermutationGroup([a, b])
>>> c = Permutation([[0, 1, 2, 3, 4], [5, 6, 7]])
>>> G.coset_decomposition(c)
False
>>> c = Permutation([[0, 6], [1, 7], [2, 4], [3, 5]])
>>> G.coset_decomposition(c)
[[6, 4, 2, 0, 7, 5, 3, 1], [0, 4, 1, 5, 2, 6, 3, 7], [0, 1, 2, 3, 4, 5, 6, 7]]
>>> G.has_element(c)
True
"""
u = self.coset_repr()
if isinstance(g, Permutation):
g = g.array_form
g1 = g
n = len(u)
a = []
for i in range(n):
x = g1[i]
for h in u[i]:
if h[i] == x:
a.append(h)
p2 = perm_af_invert(h)
g1 = perm_af_mul(p2, g1)
break
else:
return False
if perm_af_muln(*a) == g:
return a
return False
def commutator(self):
"""
commutator subgroup
The commutator subgroup is the subgroup generated by all
commutators; it is equal to the normal closure of the set
of commutators of the generators.
see http://groupprops.subwiki.org/wiki/Derived_subgroup
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([1, 0, 2, 4, 3])
>>> b = Permutation([0, 1, 3, 2, 4])
>>> G = PermutationGroup([a, b])
>>> C = G.commutator()
>>> list(C.generate(af=True))
[[0, 1, 2, 3, 4], [0, 1, 3, 4, 2], [0, 1, 4, 2, 3]]
"""
r = self._r
gens = [p.array_form for p in self.generators]
gens_inv = [perm_af_invert(p) for p in gens]
set_commutators = set()
for i in range(r):
for j in range(r):
p1 = gens[i]
p1inv = gens_inv[i]
p2 = gens[j]
p2inv = gens_inv[j]
c = [p1[p2[p1inv[k]]] for k in p2inv]
ct = tuple(c)
if not ct in set_commutators:
set_commutators.add(ct)
cms = [Permutation(p) for p in set_commutators]
G2 = self.normal_closure(cms)
return G2
def normal_closure(self, gens):
"""
normal closure in self of a list gens2 of permutations
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([1, 2, 0])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.order()
6
>>> G1 = G.normal_closure([a])
>>> list(G1.generate(af=True))
[[0, 1, 2], [1, 2, 0], [2, 0, 1]]
"""
G2 = PermutationGroup(gens)
if G2.is_normal(self):
return G2
gens1 = [p.array_form for p in self.generators]
b = 0
while not b:
gens2 = [p.array_form for p in G2.generators]
b = 1
for g1 in gens1:
if not b:
break
for g2 in gens2:
p = perm_af_muln(g1, g2, perm_af_invert(g1))
p = Permutation(p)
if not G2.has_element(p):
gens2 = G2.generators + [p]
G2 = PermutationGroup(gens2)
b = 0
break
return G2
def commutator(self):
"""
commutator subgroup
The commutator subgroup is the subgroup generated by all
commutators; it is equal to the normal closure of the set
of commutators of the generators.
see http://groupprops.subwiki.org/wiki/Derived_subgroup
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([1, 0, 2, 4, 3])
>>> b = Permutation([0, 1, 3, 2, 4])
>>> G = PermutationGroup([a, b])
>>> C = G.commutator()
>>> list(C.generate(af=True))
[[0, 1, 2, 3, 4], [0, 1, 3, 4, 2], [0, 1, 4, 2, 3]]
"""
r = self._r
gens = [p.array_form for p in self.generators]
gens_inv = [perm_af_invert(p) for p in gens]
set_commutators = set()
for i in range(r):
for j in range(r):
p1 = gens[i]
p1inv = gens_inv[i]
p2 = gens[j]
p2inv = gens_inv[j]
c = [p1[p2[p1inv[k]]] for k in p2inv]
ct = tuple(c)
if not ct in set_commutators:
set_commutators.add(ct)
cms = [Permutation(p) for p in set_commutators]
G2 = self.normal_closure(cms)
return G2
def normal_closure(self, gens):
"""
normal closure in self of a list gens2 of permutations
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([1, 2, 0])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.order()
6
>>> G1 = G.normal_closure([a])
>>> list(G1.generate(af=True))
[[0, 1, 2], [1, 2, 0], [2, 0, 1]]
"""
G2 = PermutationGroup(gens)
if G2.is_normal(self):
return G2
gens1 = [p.array_form for p in self.generators]
b = 0
while not b:
gens2 = [p.array_form for p in G2.generators]
b = 1
for g1 in gens1:
if not b:
break
for g2 in gens2:
p = perm_af_muln(g1, g2, perm_af_invert(g1))
p = Permutation(p)
if not G2.has_element(p):
gens2 = G2.generators + [p]
G2 = PermutationGroup(gens2)
b = 0
break
return G2
def schreier_sims(self):
"""
Schreier-Sims algorithm.
It computes the generators of the stabilizers chain
G > G_{b_1} > .. > G_{b1,..,b_r} > 1
in which G_{b_1,..,b_i} stabilizes b_1,..,b_i,
and the corresponding `s` cosets.
An element of the group can be written univoquely
as the product h_1*..*h_s.
We use Jerrum's filter in our implementation of the
Schreier-Sims algorithm. It runs in polynomial time.
This implementation is a translation of the C++ implementation in
http://www.m8j.net
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.schreier_sims()
>>> G.stabilizers_gens()
[[0, 2, 1]]
>>> G.coset_repr()
[[[0, 1, 2], [1, 0, 2], [2, 0, 1]], [[0, 1, 2], [0, 2, 1]]]
"""
if self._coset_repr:
return
JGr = _JGraph(self)
alpha = 0
n = JGr.n
self._order = 1
coset_repr = []
num_generators = []
generators = []
gen = range(n)
base = {}
JGr.gens += [None]*(n - len(JGr.gens))
while 1:
self._coset_repr_n = 0
self._coset_repr = [None]*n
JGr.schreier_tree(alpha, gen)
cri = []
for p in self._coset_repr:
if not p:
cri.append(p)
else:
cri.append(perm_af_invert(p))
JGr.jerrum_filter(alpha, cri)
if self._coset_repr_n > 1:
base[alpha] = self._coset_repr_n
self._order *= self._coset_repr_n
coset_repr.append([p for p in self._coset_repr if p])
d = {}
for p in self._coset_repr:
if p:
d[p[alpha]] = p
num_generators.append(JGr.r)
if JGr.r:
generators.extend(JGr.gens[:JGr.r])
if JGr.r <= 0:
break
alpha += 1
self._coset_repr = coset_repr
a = []
for p in generators:
if p not in a:
a.append(p)
self._stabilizers_gens = a
i = len(JGr.gens) - 1
while not JGr.gens[i]:
i -= 1
JGr.gens = JGr.gens[:i+1]
self._base = base.keys()
self._coset_repr_n = base.values()
def schreier_sims(self):
"""
Schreier-Sims algorithm.
It computes the generators of the stabilizers chain
G > G_{b_1} > .. > G_{b1,..,b_r} > 1
in which G_{b_1,..,b_i} stabilizes b_1,..,b_i,
and the corresponding `s` cosets.
An element of the group can be written univoquely
as the product h_1*..*h_s.
We use Jerrum's filter in our implementation of the
Schreier-Sims algorithm. It runs in polynomial time.
This implementation is a translation of the C++ implementation in
http://www.m8j.net
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.schreier_sims()
>>> G.stabilizers_gens()
[[0, 2, 1]]
>>> G.coset_repr()
[[[0, 1, 2], [1, 0, 2], [2, 0, 1]], [[0, 1, 2], [0, 2, 1]]]
"""
if self._coset_repr:
return
JGr = _JGraph(self)
alpha = 0
n = JGr.n
self._order = 1
coset_repr = []
num_generators = []
generators = []
gen = range(n)
base = {}
JGr.gens += [None] * (n - len(JGr.gens))
while 1:
self._coset_repr_n = 0
self._coset_repr = [None] * n
JGr.schreier_tree(alpha, gen)
cri = []
for p in self._coset_repr:
if not p:
cri.append(p)
else:
cri.append(perm_af_invert(p))
JGr.jerrum_filter(alpha, cri)
if self._coset_repr_n > 1:
base[alpha] = self._coset_repr_n
self._order *= self._coset_repr_n
coset_repr.append([p for p in self._coset_repr if p])
d = {}
for p in self._coset_repr:
if p:
d[p[alpha]] = p
num_generators.append(JGr.r)
if JGr.r:
generators.extend(JGr.gens[:JGr.r])
if JGr.r <= 0:
break
alpha += 1
self._coset_repr = coset_repr
a = []
for p in generators:
if p not in a:
a.append(p)
self._stabilizers_gens = a
i = len(JGr.gens) - 1
while not JGr.gens[i]:
i -= 1
JGr.gens = JGr.gens[:i + 1]
self._base = base.keys()
self._coset_repr_n = base.values()