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from sympy.ntheory.primetest import isprime
from sympy.combinatorics.perm_groups import PermutationGroup
from sympy.printing.defaults import DefaultPrinting
from sympy.combinatorics.free_groups import free_group
class PolycyclicGroup(DefaultPrinting):
is_group = True
is_solvable = True
def __init__(self, pc_sequence, pc_series, relative_order, collector=None):
"""
Parameters
==========
pc_sequence : list
A sequence of elements whose classes generate the cyclic factor
groups of pc_series.
pc_series : list
A subnormal sequence of subgroups where each factor group is cyclic.
relative_order : list
The orders of factor groups of pc_series.
collector : Collector
By default, it is None. Collector class provides the
polycyclic presentation with various other functionalities.
"""
self.pcgs = pc_sequence
self.pc_series = pc_series
self.relative_order = relative_order
self.collector = Collector(self.pcgs, pc_series, relative_order) if not collector else collector
def is_prime_order(self):
return all(isprime(order) for order in self.relative_order)
def length(self):
return len(self.pcgs)
class Collector(DefaultPrinting):
"""
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of Computational Group Theory"
Section 8.1.3
"""
def __init__(self, pcgs, pc_series, relative_order, free_group_=None, pc_presentation=None):
"""
Most of the parameters for the Collector class are the same as for PolycyclicGroup.
Others are described below.
Parameters
==========
free_group_ : tuple
free_group_ provides the mapping of polycyclic generating
sequence with the free group elements.
pc_presentation : dict
Provides the presentation of polycyclic groups with the
help of power and conjugate relators.
See Also
========
PolycyclicGroup
"""
self.pcgs = pcgs
self.pc_series = pc_series
self.relative_order = relative_order
self.free_group = free_group('x:{}'.format(len(pcgs)))[0] if not free_group_ else free_group_
self.index = {s: i for i, s in enumerate(self.free_group.symbols)}
self.pc_presentation = self.pc_relators()
def minimal_uncollected_subword(self, word):
r"""
Returns the minimal uncollected subwords.
Explanation
===========
A word ``v`` defined on generators in ``X`` is a minimal
uncollected subword of the word ``w`` if ``v`` is a subword
of ``w`` and it has one of the following form
* `v = {x_{i+1}}^{a_j}x_i`
* `v = {x_{i+1}}^{a_j}{x_i}^{-1}`
* `v = {x_i}^{a_j}`
for `a_j` not in `\{1, \ldots, s-1\}`. Where, ``s`` is the power
exponent of the corresponding generator.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics import free_group
>>> G = SymmetricGroup(4)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> F, x1, x2 = free_group("x1, x2")
>>> word = x2**2*x1**7
>>> collector.minimal_uncollected_subword(word)
((x2, 2),)
"""
# To handle the case word = <identity>
if not word:
return None
array = word.array_form
re = self.relative_order
index = self.index
for i in range(len(array)):
s1, e1 = array[i]
if re[index[s1]] and (e1 < 0 or e1 > re[index[s1]]-1):
return ((s1, e1), )
for i in range(len(array)-1):
s1, e1 = array[i]
s2, e2 = array[i+1]
if index[s1] > index[s2]:
e = 1 if e2 > 0 else -1
return ((s1, e1), (s2, e))
return None
def relations(self):
"""
Separates the given relators of pc presentation in power and
conjugate relations.
Returns
=======
(power_rel, conj_rel)
Separates pc presentation into power and conjugate relations.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> G = SymmetricGroup(3)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> power_rel, conj_rel = collector.relations()
>>> power_rel
{x0**2: (), x1**3: ()}
>>> conj_rel
{x0**-1*x1*x0: x1**2}
See Also
========
pc_relators
"""
power_relators = {}
conjugate_relators = {}
for key, value in self.pc_presentation.items():
if len(key.array_form) == 1:
power_relators[key] = value
else:
conjugate_relators[key] = value
return power_relators, conjugate_relators
def subword_index(self, word, w):
"""
Returns the start and ending index of a given
subword in a word.
Parameters
==========
word : FreeGroupElement
word defined on free group elements for a
polycyclic group.
w : FreeGroupElement
subword of a given word, whose starting and
ending index to be computed.
Returns
=======
(i, j)
A tuple containing starting and ending index of ``w``
in the given word.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics import free_group
>>> G = SymmetricGroup(4)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> F, x1, x2 = free_group("x1, x2")
>>> word = x2**2*x1**7
>>> w = x2**2*x1
>>> collector.subword_index(word, w)
(0, 3)
>>> w = x1**7
>>> collector.subword_index(word, w)
(2, 9)
"""
low = -1
high = -1
for i in range(len(word)-len(w)+1):
if word.subword(i, i+len(w)) == w:
low = i
high = i+len(w)
break
if low == high == -1:
return -1, -1
return low, high
def map_relation(self, w):
"""
Return a conjugate relation.
Explanation
===========
Given a word formed by two free group elements, the
corresponding conjugate relation with those free
group elements is formed and mapped with the collected
word in the polycyclic presentation.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics import free_group
>>> G = SymmetricGroup(3)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> F, x0, x1 = free_group("x0, x1")
>>> w = x1*x0
>>> collector.map_relation(w)
x1**2
See Also
========
pc_presentation
"""
array = w.array_form
s1 = array[0][0]
s2 = array[1][0]
key = ((s2, -1), (s1, 1), (s2, 1))
key = self.free_group.dtype(key)
return self.pc_presentation[key]
def collected_word(self, word):
r"""
Return the collected form of a word.
Explanation
===========
A word ``w`` is called collected, if `w = {x_{i_1}}^{a_1} * \ldots *
{x_{i_r}}^{a_r}` with `i_1 < i_2< \ldots < i_r` and `a_j` is in
`\{1, \ldots, {s_j}-1\}`.
Otherwise w is uncollected.
Parameters
==========
word : FreeGroupElement
An uncollected word.
Returns
=======
word
A collected word of form `w = {x_{i_1}}^{a_1}, \ldots,
{x_{i_r}}^{a_r}` with `i_1, i_2, \ldots, i_r` and `a_j \in
\{1, \ldots, {s_j}-1\}`.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> from sympy.combinatorics import free_group
>>> G = SymmetricGroup(4)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> F, x0, x1, x2, x3 = free_group("x0, x1, x2, x3")
>>> word = x3*x2*x1*x0
>>> collected_word = collector.collected_word(word)
>>> free_to_perm = {}
>>> free_group = collector.free_group
>>> for sym, gen in zip(free_group.symbols, collector.pcgs):
... free_to_perm[sym] = gen
>>> G1 = PermutationGroup()
>>> for w in word:
... sym = w[0]
... perm = free_to_perm[sym]
... G1 = PermutationGroup([perm] + G1.generators)
>>> G2 = PermutationGroup()
>>> for w in collected_word:
... sym = w[0]
... perm = free_to_perm[sym]
... G2 = PermutationGroup([perm] + G2.generators)
The two are not identical, but they are equivalent:
>>> G1.equals(G2), G1 == G2
(True, False)
See Also
========
minimal_uncollected_subword
"""
free_group = self.free_group
while True:
w = self.minimal_uncollected_subword(word)
if not w:
break
low, high = self.subword_index(word, free_group.dtype(w))
if low == -1:
continue
s1, e1 = w[0]
if len(w) == 1:
re = self.relative_order[self.index[s1]]
q = e1 // re
r = e1-q*re
key = ((w[0][0], re), )
key = free_group.dtype(key)
if self.pc_presentation[key]:
presentation = self.pc_presentation[key].array_form
sym, exp = presentation[0]
word_ = ((w[0][0], r), (sym, q*exp))
word_ = free_group.dtype(word_)
else:
if r != 0:
word_ = ((w[0][0], r), )
word_ = free_group.dtype(word_)
else:
word_ = None
word = word.eliminate_word(free_group.dtype(w), word_)
if len(w) == 2 and w[1][1] > 0:
s2, e2 = w[1]
s2 = ((s2, 1), )
s2 = free_group.dtype(s2)
word_ = self.map_relation(free_group.dtype(w))
word_ = s2*word_**e1
word_ = free_group.dtype(word_)
word = word.substituted_word(low, high, word_)
elif len(w) == 2 and w[1][1] < 0:
s2, e2 = w[1]
s2 = ((s2, 1), )
s2 = free_group.dtype(s2)
word_ = self.map_relation(free_group.dtype(w))
word_ = s2**-1*word_**e1
word_ = free_group.dtype(word_)
word = word.substituted_word(low, high, word_)
return word
def pc_relators(self):
r"""
Return the polycyclic presentation.
Explanation
===========
There are two types of relations used in polycyclic
presentation.
* Power relations : Power relators are of the form `x_i^{re_i}`,
where `i \in \{0, \ldots, \mathrm{len(pcgs)}\}`, ``x`` represents polycyclic
generator and ``re`` is the corresponding relative order.
* Conjugate relations : Conjugate relators are of the form `x_j^-1x_ix_j`,
where `j < i \in \{0, \ldots, \mathrm{len(pcgs)}\}`.
Returns
=======
A dictionary with power and conjugate relations as key and
their collected form as corresponding values.
Notes
=====
Identity Permutation is mapped with empty ``()``.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.permutations import Permutation
>>> S = SymmetricGroup(49).sylow_subgroup(7)
>>> der = S.derived_series()
>>> G = der[len(der)-2]
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> pcgs = PcGroup.pcgs
>>> len(pcgs)
6
>>> free_group = collector.free_group
>>> pc_resentation = collector.pc_presentation
>>> free_to_perm = {}
>>> for s, g in zip(free_group.symbols, pcgs):
... free_to_perm[s] = g
>>> for k, v in pc_resentation.items():
... k_array = k.array_form
... if v != ():
... v_array = v.array_form
... lhs = Permutation()
... for gen in k_array:
... s = gen[0]
... e = gen[1]
... lhs = lhs*free_to_perm[s]**e
... if v == ():
... assert lhs.is_identity
... continue
... rhs = Permutation()
... for gen in v_array:
... s = gen[0]
... e = gen[1]
... rhs = rhs*free_to_perm[s]**e
... assert lhs == rhs
"""
free_group = self.free_group
rel_order = self.relative_order
pc_relators = {}
perm_to_free = {}
pcgs = self.pcgs
for gen, s in zip(pcgs, free_group.generators):
perm_to_free[gen**-1] = s**-1
perm_to_free[gen] = s
pcgs = pcgs[::-1]
series = self.pc_series[::-1]
rel_order = rel_order[::-1]
collected_gens = []
for i, gen in enumerate(pcgs):
re = rel_order[i]
relation = perm_to_free[gen]**re
G = series[i]
l = G.generator_product(gen**re, original = True)
l.reverse()
word = free_group.identity
for g in l:
word = word*perm_to_free[g]
word = self.collected_word(word)
pc_relators[relation] = word if word else ()
self.pc_presentation = pc_relators
collected_gens.append(gen)
if len(collected_gens) > 1:
conj = collected_gens[len(collected_gens)-1]
conjugator = perm_to_free[conj]
for j in range(len(collected_gens)-1):
conjugated = perm_to_free[collected_gens[j]]
relation = conjugator**-1*conjugated*conjugator
gens = conj**-1*collected_gens[j]*conj
l = G.generator_product(gens, original = True)
l.reverse()
word = free_group.identity
for g in l:
word = word*perm_to_free[g]
word = self.collected_word(word)
pc_relators[relation] = word if word else ()
self.pc_presentation = pc_relators
return pc_relators
def exponent_vector(self, element):
r"""
Return the exponent vector of length equal to the
length of polycyclic generating sequence.
Explanation
===========
For a given generator/element ``g`` of the polycyclic group,
it can be represented as `g = {x_1}^{e_1}, \ldots, {x_n}^{e_n}`,
where `x_i` represents polycyclic generators and ``n`` is
the number of generators in the free_group equal to the length
of pcgs.
Parameters
==========
element : Permutation
Generator of a polycyclic group.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.permutations import Permutation
>>> G = SymmetricGroup(4)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> pcgs = PcGroup.pcgs
>>> collector.exponent_vector(G[0])
[1, 0, 0, 0]
>>> exp = collector.exponent_vector(G[1])
>>> g = Permutation()
>>> for i in range(len(exp)):
... g = g*pcgs[i]**exp[i] if exp[i] else g
>>> assert g == G[1]
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of Computational Group Theory"
Section 8.1.1, Definition 8.4
"""
free_group = self.free_group
G = PermutationGroup()
for g in self.pcgs:
G = PermutationGroup([g] + G.generators)
gens = G.generator_product(element, original = True)
gens.reverse()
perm_to_free = {}
for sym, g in zip(free_group.generators, self.pcgs):
perm_to_free[g**-1] = sym**-1
perm_to_free[g] = sym
w = free_group.identity
for g in gens:
w = w*perm_to_free[g]
word = self.collected_word(w)
index = self.index
exp_vector = [0]*len(free_group)
word = word.array_form
for t in word:
exp_vector[index[t[0]]] = t[1]
return exp_vector
def depth(self, element):
r"""
Return the depth of a given element.
Explanation
===========
The depth of a given element ``g`` is defined by
`\mathrm{dep}[g] = i` if `e_1 = e_2 = \ldots = e_{i-1} = 0`
and `e_i != 0`, where ``e`` represents the exponent-vector.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> G = SymmetricGroup(3)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> collector.depth(G[0])
2
>>> collector.depth(G[1])
1
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of Computational Group Theory"
Section 8.1.1, Definition 8.5
"""
exp_vector = self.exponent_vector(element)
return next((i+1 for i, x in enumerate(exp_vector) if x), len(self.pcgs)+1)
def leading_exponent(self, element):
r"""
Return the leading non-zero exponent.
Explanation
===========
The leading exponent for a given element `g` is defined
by `\mathrm{leading\_exponent}[g]` `= e_i`, if `\mathrm{depth}[g] = i`.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> G = SymmetricGroup(3)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> collector.leading_exponent(G[1])
1
"""
exp_vector = self.exponent_vector(element)
depth = self.depth(element)
if depth != len(self.pcgs)+1:
return exp_vector[depth-1]
return None
def _sift(self, z, g):
h = g
d = self.depth(h)
while d < len(self.pcgs) and z[d-1] != 1:
k = z[d-1]
e = self.leading_exponent(h)*(self.leading_exponent(k))**-1
e = e % self.relative_order[d-1]
h = k**-e*h
d = self.depth(h)
return h
def induced_pcgs(self, gens):
"""
Parameters
==========
gens : list
A list of generators on which polycyclic subgroup
is to be defined.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> S = SymmetricGroup(8)
>>> G = S.sylow_subgroup(2)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> gens = [G[0], G[1]]
>>> ipcgs = collector.induced_pcgs(gens)
>>> [gen.order() for gen in ipcgs]
[2, 2, 2]
>>> G = S.sylow_subgroup(3)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> gens = [G[0], G[1]]
>>> ipcgs = collector.induced_pcgs(gens)
>>> [gen.order() for gen in ipcgs]
[3]
"""
z = [1]*len(self.pcgs)
G = gens
while G:
g = G.pop(0)
h = self._sift(z, g)
d = self.depth(h)
if d < len(self.pcgs):
for gen in z:
if gen != 1:
G.append(h**-1*gen**-1*h*gen)
z[d-1] = h;
z = [gen for gen in z if gen != 1]
return z
def constructive_membership_test(self, ipcgs, g):
"""
Return the exponent vector for induced pcgs.
"""
e = [0]*len(ipcgs)
h = g
d = self.depth(h)
for i, gen in enumerate(ipcgs):
while self.depth(gen) == d:
f = self.leading_exponent(h)*self.leading_exponent(gen)
f = f % self.relative_order[d-1]
h = gen**(-f)*h
e[i] = f
d = self.depth(h)
if h == 1:
return e
return False
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