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Traktor/myenv/Lib/site-packages/sympy/combinatorics/perm_groups.py

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losowanie zdjec
2024-05-26 05:12:46 +02:00
from math import factorial as _factorial, log, prod
from itertools import chain, islice, product
from sympy.combinatorics import Permutation
from sympy.combinatorics.permutations import (_af_commutes_with, _af_invert,
_af_rmul, _af_rmuln, _af_pow, Cycle)
from sympy.combinatorics.util import (_check_cycles_alt_sym,
_distribute_gens_by_base, _orbits_transversals_from_bsgs,
_handle_precomputed_bsgs, _base_ordering, _strong_gens_from_distr,
_strip, _strip_af)
from sympy.core import Basic
from sympy.core.random import _randrange, randrange, choice
from sympy.core.symbol import Symbol
from sympy.core.sympify import _sympify
from sympy.functions.combinatorial.factorials import factorial
from sympy.ntheory import primefactors, sieve
from sympy.ntheory.factor_ import (factorint, multiplicity)
from sympy.ntheory.primetest import isprime
from sympy.utilities.iterables import has_variety, is_sequence, uniq
rmul = Permutation.rmul_with_af
_af_new = Permutation._af_new
class PermutationGroup(Basic):
r"""The class defining a Permutation group.
Explanation
===========
``PermutationGroup([p1, p2, ..., pn])`` returns the permutation group
generated by the list of permutations. This group can be supplied
to Polyhedron if one desires to decorate the elements to which the
indices of the permutation refer.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> from sympy.combinatorics import Polyhedron
The permutations corresponding to motion of the front, right and
bottom face of a $2 \times 2$ Rubik's cube are defined:
>>> F = Permutation(2, 19, 21, 8)(3, 17, 20, 10)(4, 6, 7, 5)
>>> R = Permutation(1, 5, 21, 14)(3, 7, 23, 12)(8, 10, 11, 9)
>>> D = Permutation(6, 18, 14, 10)(7, 19, 15, 11)(20, 22, 23, 21)
These are passed as permutations to PermutationGroup:
>>> G = PermutationGroup(F, R, D)
>>> G.order()
3674160
The group can be supplied to a Polyhedron in order to track the
objects being moved. An example involving the $2 \times 2$ Rubik's cube is
given there, but here is a simple demonstration:
>>> a = Permutation(2, 1)
>>> b = Permutation(1, 0)
>>> G = PermutationGroup(a, b)
>>> P = Polyhedron(list('ABC'), pgroup=G)
>>> P.corners
(A, B, C)
>>> P.rotate(0) # apply permutation 0
>>> P.corners
(A, C, B)
>>> P.reset()
>>> P.corners
(A, B, C)
Or one can make a permutation as a product of selected permutations
and apply them to an iterable directly:
>>> P10 = G.make_perm([0, 1])
>>> P10('ABC')
['C', 'A', 'B']
See Also
========
sympy.combinatorics.polyhedron.Polyhedron,
sympy.combinatorics.permutations.Permutation
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of Computational Group Theory"
.. [2] Seress, A.
"Permutation Group Algorithms"
.. [3] https://en.wikipedia.org/wiki/Schreier_vector
.. [4] https://en.wikipedia.org/wiki/Nielsen_transformation#Product_replacement_algorithm
.. [5] Frank Celler, Charles R.Leedham-Green, Scott H.Murray,
Alice C.Niemeyer, and E.A.O'Brien. "Generating Random
Elements of a Finite Group"
.. [6] https://en.wikipedia.org/wiki/Block_%28permutation_group_theory%29
.. [7] https://algorithmist.com/wiki/Union_find
.. [8] https://en.wikipedia.org/wiki/Multiply_transitive_group#Multiply_transitive_groups
.. [9] https://en.wikipedia.org/wiki/Center_%28group_theory%29
.. [10] https://en.wikipedia.org/wiki/Centralizer_and_normalizer
.. [11] https://groupprops.subwiki.org/wiki/Derived_subgroup
.. [12] https://en.wikipedia.org/wiki/Nilpotent_group
.. [13] https://www.math.colostate.edu/~hulpke/CGT/cgtnotes.pdf
.. [14] https://docs.gap-system.org/doc/ref/manual.pdf
"""
is_group = True
def __new__(cls, *args, dups=True, **kwargs):
"""The default constructor. Accepts Cycle and Permutation forms.
Removes duplicates unless ``dups`` keyword is ``False``.
"""
if not args:
args = [Permutation()]
else:
args = list(args[0] if is_sequence(args[0]) else args)
if not args:
args = [Permutation()]
if any(isinstance(a, Cycle) for a in args):
args = [Permutation(a) for a in args]
if has_variety(a.size for a in args):
degree = kwargs.pop('degree', None)
if degree is None:
degree = max(a.size for a in args)
for i in range(len(args)):
if args[i].size != degree:
args[i] = Permutation(args[i], size=degree)
if dups:
args = list(uniq([_af_new(list(a)) for a in args]))
if len(args) > 1:
args = [g for g in args if not g.is_identity]
return Basic.__new__(cls, *args, **kwargs)
def __init__(self, *args, **kwargs):
self._generators = list(self.args)
self._order = None
self._center = []
self._is_abelian = None
self._is_transitive = None
self._is_sym = None
self._is_alt = None
self._is_primitive = None
self._is_nilpotent = None
self._is_solvable = None
self._is_trivial = None
self._transitivity_degree = None
self._max_div = None
self._is_perfect = None
self._is_cyclic = None
self._is_dihedral = None
self._r = len(self._generators)
self._degree = self._generators[0].size
# these attributes are assigned after running schreier_sims
self._base = []
self._strong_gens = []
self._strong_gens_slp = []
self._basic_orbits = []
self._transversals = []
self._transversal_slp = []
# these attributes are assigned after running _random_pr_init
self._random_gens = []
# finite presentation of the group as an instance of `FpGroup`
self._fp_presentation = None
def __getitem__(self, i):
return self._generators[i]
def __contains__(self, i):
"""Return ``True`` if *i* is contained in PermutationGroup.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> p = Permutation(1, 2, 3)
>>> Permutation(3) in PermutationGroup(p)
True
"""
if not isinstance(i, Permutation):
raise TypeError("A PermutationGroup contains only Permutations as "
"elements, not elements of type %s" % type(i))
return self.contains(i)
def __len__(self):
return len(self._generators)
def equals(self, other):
"""Return ``True`` if PermutationGroup generated by elements in the
group are same i.e they represent the same PermutationGroup.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> p = Permutation(0, 1, 2, 3, 4, 5)
>>> G = PermutationGroup([p, p**2])
>>> H = PermutationGroup([p**2, p])
>>> G.generators == H.generators
False
>>> G.equals(H)
True
"""
if not isinstance(other, PermutationGroup):
return False
set_self_gens = set(self.generators)
set_other_gens = set(other.generators)
# before reaching the general case there are also certain
# optimisation and obvious cases requiring less or no actual
# computation.
if set_self_gens == set_other_gens:
return True
# in the most general case it will check that each generator of
# one group belongs to the other PermutationGroup and vice-versa
for gen1 in set_self_gens:
if not other.contains(gen1):
return False
for gen2 in set_other_gens:
if not self.contains(gen2):
return False
return True
def __mul__(self, other):
"""
Return the direct product of two permutation groups as a permutation
group.
Explanation
===========
This implementation realizes the direct product by shifting the index
set for the generators of the second group: so if we have ``G`` acting
on ``n1`` points and ``H`` acting on ``n2`` points, ``G*H`` acts on
``n1 + n2`` points.
Examples
========
>>> from sympy.combinatorics.named_groups import CyclicGroup
>>> G = CyclicGroup(5)
>>> H = G*G
>>> H
PermutationGroup([
(9)(0 1 2 3 4),
(5 6 7 8 9)])
>>> H.order()
25
"""
if isinstance(other, Permutation):
return Coset(other, self, dir='+')
gens1 = [perm._array_form for perm in self.generators]
gens2 = [perm._array_form for perm in other.generators]
n1 = self._degree
n2 = other._degree
start = list(range(n1))
end = list(range(n1, n1 + n2))
for i in range(len(gens2)):
gens2[i] = [x + n1 for x in gens2[i]]
gens2 = [start + gen for gen in gens2]
gens1 = [gen + end for gen in gens1]
together = gens1 + gens2
gens = [_af_new(x) for x in together]
return PermutationGroup(gens)
def _random_pr_init(self, r, n, _random_prec_n=None):
r"""Initialize random generators for the product replacement algorithm.
Explanation
===========
The implementation uses a modification of the original product
replacement algorithm due to Leedham-Green, as described in [1],
pp. 69-71; also, see [2], pp. 27-29 for a detailed theoretical
analysis of the original product replacement algorithm, and [4].
The product replacement algorithm is used for producing random,
uniformly distributed elements of a group `G` with a set of generators
`S`. For the initialization ``_random_pr_init``, a list ``R`` of
`\max\{r, |S|\}` group generators is created as the attribute
``G._random_gens``, repeating elements of `S` if necessary, and the
identity element of `G` is appended to ``R`` - we shall refer to this
last element as the accumulator. Then the function ``random_pr()``
is called ``n`` times, randomizing the list ``R`` while preserving
the generation of `G` by ``R``. The function ``random_pr()`` itself
takes two random elements ``g, h`` among all elements of ``R`` but
the accumulator and replaces ``g`` with a randomly chosen element
from `\{gh, g(~h), hg, (~h)g\}`. Then the accumulator is multiplied
by whatever ``g`` was replaced by. The new value of the accumulator is
then returned by ``random_pr()``.
The elements returned will eventually (for ``n`` large enough) become
uniformly distributed across `G` ([5]). For practical purposes however,
the values ``n = 50, r = 11`` are suggested in [1].
Notes
=====
THIS FUNCTION HAS SIDE EFFECTS: it changes the attribute
self._random_gens
See Also
========
random_pr
"""
deg = self.degree
random_gens = [x._array_form for x in self.generators]
k = len(random_gens)
if k < r:
for i in range(k, r):
random_gens.append(random_gens[i - k])
acc = list(range(deg))
random_gens.append(acc)
self._random_gens = random_gens
# handle randomized input for testing purposes
if _random_prec_n is None:
for i in range(n):
self.random_pr()
else:
for i in range(n):
self.random_pr(_random_prec=_random_prec_n[i])
def _union_find_merge(self, first, second, ranks, parents, not_rep):
"""Merges two classes in a union-find data structure.
Explanation
===========
Used in the implementation of Atkinson's algorithm as suggested in [1],
pp. 83-87. The class merging process uses union by rank as an
optimization. ([7])
Notes
=====
THIS FUNCTION HAS SIDE EFFECTS: the list of class representatives,
``parents``, the list of class sizes, ``ranks``, and the list of
elements that are not representatives, ``not_rep``, are changed due to
class merging.
See Also
========
minimal_block, _union_find_rep
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of computational group theory"
.. [7] https://algorithmist.com/wiki/Union_find
"""
rep_first = self._union_find_rep(first, parents)
rep_second = self._union_find_rep(second, parents)
if rep_first != rep_second:
# union by rank
if ranks[rep_first] >= ranks[rep_second]:
new_1, new_2 = rep_first, rep_second
else:
new_1, new_2 = rep_second, rep_first
total_rank = ranks[new_1] + ranks[new_2]
if total_rank > self.max_div:
return -1
parents[new_2] = new_1
ranks[new_1] = total_rank
not_rep.append(new_2)
return 1
return 0
def _union_find_rep(self, num, parents):
"""Find representative of a class in a union-find data structure.
Explanation
===========
Used in the implementation of Atkinson's algorithm as suggested in [1],
pp. 83-87. After the representative of the class to which ``num``
belongs is found, path compression is performed as an optimization
([7]).
Notes
=====
THIS FUNCTION HAS SIDE EFFECTS: the list of class representatives,
``parents``, is altered due to path compression.
See Also
========
minimal_block, _union_find_merge
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of computational group theory"
.. [7] https://algorithmist.com/wiki/Union_find
"""
rep, parent = num, parents[num]
while parent != rep:
rep = parent
parent = parents[rep]
# path compression
temp, parent = num, parents[num]
while parent != rep:
parents[temp] = rep
temp = parent
parent = parents[temp]
return rep
@property
def base(self):
r"""Return a base from the Schreier-Sims algorithm.
Explanation
===========
For a permutation group `G`, a base is a sequence of points
`B = (b_1, b_2, \dots, b_k)` such that no element of `G` apart
from the identity fixes all the points in `B`. The concepts of
a base and strong generating set and their applications are
discussed in depth in [1], pp. 87-89 and [2], pp. 55-57.
An alternative way to think of `B` is that it gives the
indices of the stabilizer cosets that contain more than the
identity permutation.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> G = PermutationGroup([Permutation(0, 1, 3)(2, 4)])
>>> G.base
[0, 2]
See Also
========
strong_gens, basic_transversals, basic_orbits, basic_stabilizers
"""
if self._base == []:
self.schreier_sims()
return self._base
def baseswap(self, base, strong_gens, pos, randomized=False,
transversals=None, basic_orbits=None, strong_gens_distr=None):
r"""Swap two consecutive base points in base and strong generating set.
Explanation
===========
If a base for a group `G` is given by `(b_1, b_2, \dots, b_k)`, this
function returns a base `(b_1, b_2, \dots, b_{i+1}, b_i, \dots, b_k)`,
where `i` is given by ``pos``, and a strong generating set relative
to that base. The original base and strong generating set are not
modified.
The randomized version (default) is of Las Vegas type.
Parameters
==========
base, strong_gens
The base and strong generating set.
pos
The position at which swapping is performed.
randomized
A switch between randomized and deterministic version.
transversals
The transversals for the basic orbits, if known.
basic_orbits
The basic orbits, if known.
strong_gens_distr
The strong generators distributed by basic stabilizers, if known.
Returns
=======
(base, strong_gens)
``base`` is the new base, and ``strong_gens`` is a generating set
relative to it.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.testutil import _verify_bsgs
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> S = SymmetricGroup(4)
>>> S.schreier_sims()
>>> S.base
[0, 1, 2]
>>> base, gens = S.baseswap(S.base, S.strong_gens, 1, randomized=False)
>>> base, gens
([0, 2, 1],
[(0 1 2 3), (3)(0 1), (1 3 2),
(2 3), (1 3)])
check that base, gens is a BSGS
>>> S1 = PermutationGroup(gens)
>>> _verify_bsgs(S1, base, gens)
True
See Also
========
schreier_sims
Notes
=====
The deterministic version of the algorithm is discussed in
[1], pp. 102-103; the randomized version is discussed in [1], p.103, and
[2], p.98. It is of Las Vegas type.
Notice that [1] contains a mistake in the pseudocode and
discussion of BASESWAP: on line 3 of the pseudocode,
`|\beta_{i+1}^{\left\langle T\right\rangle}|` should be replaced by
`|\beta_{i}^{\left\langle T\right\rangle}|`, and the same for the
discussion of the algorithm.
"""
# construct the basic orbits, generators for the stabilizer chain
# and transversal elements from whatever was provided
transversals, basic_orbits, strong_gens_distr = \
_handle_precomputed_bsgs(base, strong_gens, transversals,
basic_orbits, strong_gens_distr)
base_len = len(base)
degree = self.degree
# size of orbit of base[pos] under the stabilizer we seek to insert
# in the stabilizer chain at position pos + 1
size = len(basic_orbits[pos])*len(basic_orbits[pos + 1]) \
//len(_orbit(degree, strong_gens_distr[pos], base[pos + 1]))
# initialize the wanted stabilizer by a subgroup
if pos + 2 > base_len - 1:
T = []
else:
T = strong_gens_distr[pos + 2][:]
# randomized version
if randomized is True:
stab_pos = PermutationGroup(strong_gens_distr[pos])
schreier_vector = stab_pos.schreier_vector(base[pos + 1])
# add random elements of the stabilizer until they generate it
while len(_orbit(degree, T, base[pos])) != size:
new = stab_pos.random_stab(base[pos + 1],
schreier_vector=schreier_vector)
T.append(new)
# deterministic version
else:
Gamma = set(basic_orbits[pos])
Gamma.remove(base[pos])
if base[pos + 1] in Gamma:
Gamma.remove(base[pos + 1])
# add elements of the stabilizer until they generate it by
# ruling out member of the basic orbit of base[pos] along the way
while len(_orbit(degree, T, base[pos])) != size:
gamma = next(iter(Gamma))
x = transversals[pos][gamma]
temp = x._array_form.index(base[pos + 1]) # (~x)(base[pos + 1])
if temp not in basic_orbits[pos + 1]:
Gamma = Gamma - _orbit(degree, T, gamma)
else:
y = transversals[pos + 1][temp]
el = rmul(x, y)
if el(base[pos]) not in _orbit(degree, T, base[pos]):
T.append(el)
Gamma = Gamma - _orbit(degree, T, base[pos])
# build the new base and strong generating set
strong_gens_new_distr = strong_gens_distr[:]
strong_gens_new_distr[pos + 1] = T
base_new = base[:]
base_new[pos], base_new[pos + 1] = base_new[pos + 1], base_new[pos]
strong_gens_new = _strong_gens_from_distr(strong_gens_new_distr)
for gen in T:
if gen not in strong_gens_new:
strong_gens_new.append(gen)
return base_new, strong_gens_new
@property
def basic_orbits(self):
r"""
Return the basic orbits relative to a base and strong generating set.
Explanation
===========
If `(b_1, b_2, \dots, b_k)` is a base for a group `G`, and
`G^{(i)} = G_{b_1, b_2, \dots, b_{i-1}}` is the ``i``-th basic stabilizer
(so that `G^{(1)} = G`), the ``i``-th basic orbit relative to this base
is the orbit of `b_i` under `G^{(i)}`. See [1], pp. 87-89 for more
information.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> S = SymmetricGroup(4)
>>> S.basic_orbits
[[0, 1, 2, 3], [1, 2, 3], [2, 3]]
See Also
========
base, strong_gens, basic_transversals, basic_stabilizers
"""
if self._basic_orbits == []:
self.schreier_sims()
return self._basic_orbits
@property
def basic_stabilizers(self):
r"""
Return a chain of stabilizers relative to a base and strong generating
set.
Explanation
===========
The ``i``-th basic stabilizer `G^{(i)}` relative to a base
`(b_1, b_2, \dots, b_k)` is `G_{b_1, b_2, \dots, b_{i-1}}`. For more
information, see [1], pp. 87-89.
Examples
========
>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> A = AlternatingGroup(4)
>>> A.schreier_sims()
>>> A.base
[0, 1]
>>> for g in A.basic_stabilizers:
... print(g)
...
PermutationGroup([
(3)(0 1 2),
(1 2 3)])
PermutationGroup([
(1 2 3)])
See Also
========
base, strong_gens, basic_orbits, basic_transversals
"""
if self._transversals == []:
self.schreier_sims()
strong_gens = self._strong_gens
base = self._base
if not base: # e.g. if self is trivial
return []
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
basic_stabilizers = []
for gens in strong_gens_distr:
basic_stabilizers.append(PermutationGroup(gens))
return basic_stabilizers
@property
def basic_transversals(self):
"""
Return basic transversals relative to a base and strong generating set.
Explanation
===========
The basic transversals are transversals of the basic orbits. They
are provided as a list of dictionaries, each dictionary having
keys - the elements of one of the basic orbits, and values - the
corresponding transversal elements. See [1], pp. 87-89 for more
information.
Examples
========
>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> A = AlternatingGroup(4)
>>> A.basic_transversals
[{0: (3), 1: (3)(0 1 2), 2: (3)(0 2 1), 3: (0 3 1)}, {1: (3), 2: (1 2 3), 3: (1 3 2)}]
See Also
========
strong_gens, base, basic_orbits, basic_stabilizers
"""
if self._transversals == []:
self.schreier_sims()
return self._transversals
def composition_series(self):
r"""
Return the composition series for a group as a list
of permutation groups.
Explanation
===========
The composition series for a group `G` is defined as a
subnormal series `G = H_0 > H_1 > H_2 \ldots` A composition
series is a subnormal series such that each factor group
`H(i+1) / H(i)` is simple.
A subnormal series is a composition series only if it is of
maximum length.
The algorithm works as follows:
Starting with the derived series the idea is to fill
the gap between `G = der[i]` and `H = der[i+1]` for each
`i` independently. Since, all subgroups of the abelian group
`G/H` are normal so, first step is to take the generators
`g` of `G` and add them to generators of `H` one by one.
The factor groups formed are not simple in general. Each
group is obtained from the previous one by adding one
generator `g`, if the previous group is denoted by `H`
then the next group `K` is generated by `g` and `H`.
The factor group `K/H` is cyclic and it's order is
`K.order()//G.order()`. The series is then extended between
`K` and `H` by groups generated by powers of `g` and `H`.
The series formed is then prepended to the already existing
series.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.named_groups import CyclicGroup
>>> S = SymmetricGroup(12)
>>> G = S.sylow_subgroup(2)
>>> C = G.composition_series()
>>> [H.order() for H in C]
[1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1]
>>> G = S.sylow_subgroup(3)
>>> C = G.composition_series()
>>> [H.order() for H in C]
[243, 81, 27, 9, 3, 1]
>>> G = CyclicGroup(12)
>>> C = G.composition_series()
>>> [H.order() for H in C]
[12, 6, 3, 1]
"""
der = self.derived_series()
if not all(g.is_identity for g in der[-1].generators):
raise NotImplementedError('Group should be solvable')
series = []
for i in range(len(der)-1):
H = der[i+1]
up_seg = []
for g in der[i].generators:
K = PermutationGroup([g] + H.generators)
order = K.order() // H.order()
down_seg = []
for p, e in factorint(order).items():
for _ in range(e):
down_seg.append(PermutationGroup([g] + H.generators))
g = g**p
up_seg = down_seg + up_seg
H = K
up_seg[0] = der[i]
series.extend(up_seg)
series.append(der[-1])
return series
def coset_transversal(self, H):
"""Return a transversal of the right cosets of self by its subgroup H
using the second method described in [1], Subsection 4.6.7
"""
if not H.is_subgroup(self):
raise ValueError("The argument must be a subgroup")
if H.order() == 1:
return self._elements
self._schreier_sims(base=H.base) # make G.base an extension of H.base
base = self.base
base_ordering = _base_ordering(base, self.degree)
identity = Permutation(self.degree - 1)
transversals = self.basic_transversals[:]
# transversals is a list of dictionaries. Get rid of the keys
# so that it is a list of lists and sort each list in
# the increasing order of base[l]^x
for l, t in enumerate(transversals):
transversals[l] = sorted(t.values(),
key = lambda x: base_ordering[base[l]^x])
orbits = H.basic_orbits
h_stabs = H.basic_stabilizers
g_stabs = self.basic_stabilizers
indices = [x.order()//y.order() for x, y in zip(g_stabs, h_stabs)]
# T^(l) should be a right transversal of H^(l) in G^(l) for
# 1<=l<=len(base). While H^(l) is the trivial group, T^(l)
# contains all the elements of G^(l) so we might just as well
# start with l = len(h_stabs)-1
if len(g_stabs) > len(h_stabs):
T = g_stabs[len(h_stabs)]._elements
else:
T = [identity]
l = len(h_stabs)-1
t_len = len(T)
while l > -1:
T_next = []
for u in transversals[l]:
if u == identity:
continue
b = base_ordering[base[l]^u]
for t in T:
p = t*u
if all(base_ordering[h^p] >= b for h in orbits[l]):
T_next.append(p)
if t_len + len(T_next) == indices[l]:
break
if t_len + len(T_next) == indices[l]:
break
T += T_next
t_len += len(T_next)
l -= 1
T.remove(identity)
T = [identity] + T
return T
def _coset_representative(self, g, H):
"""Return the representative of Hg from the transversal that
would be computed by ``self.coset_transversal(H)``.
"""
if H.order() == 1:
return g
# The base of self must be an extension of H.base.
if not(self.base[:len(H.base)] == H.base):
self._schreier_sims(base=H.base)
orbits = H.basic_orbits[:]
h_transversals = [list(_.values()) for _ in H.basic_transversals]
transversals = [list(_.values()) for _ in self.basic_transversals]
base = self.base
base_ordering = _base_ordering(base, self.degree)
def step(l, x):
gamma = sorted(orbits[l], key = lambda y: base_ordering[y^x])[0]
i = [base[l]^h for h in h_transversals[l]].index(gamma)
x = h_transversals[l][i]*x
if l < len(orbits)-1:
for u in transversals[l]:
if base[l]^u == base[l]^x:
break
x = step(l+1, x*u**-1)*u
return x
return step(0, g)
def coset_table(self, H):
"""Return the standardised (right) coset table of self in H as
a list of lists.
"""
# Maybe this should be made to return an instance of CosetTable
# from fp_groups.py but the class would need to be changed first
# to be compatible with PermutationGroups
if not H.is_subgroup(self):
raise ValueError("The argument must be a subgroup")
T = self.coset_transversal(H)
n = len(T)
A = list(chain.from_iterable((gen, gen**-1)
for gen in self.generators))
table = []
for i in range(n):
row = [self._coset_representative(T[i]*x, H) for x in A]
row = [T.index(r) for r in row]
table.append(row)
# standardize (this is the same as the algorithm used in coset_table)
# If CosetTable is made compatible with PermutationGroups, this
# should be replaced by table.standardize()
A = range(len(A))
gamma = 1
for alpha, a in product(range(n), A):
beta = table[alpha][a]
if beta >= gamma:
if beta > gamma:
for x in A:
z = table[gamma][x]
table[gamma][x] = table[beta][x]
table[beta][x] = z
for i in range(n):
if table[i][x] == beta:
table[i][x] = gamma
elif table[i][x] == gamma:
table[i][x] = beta
gamma += 1
if gamma >= n-1:
return table
def center(self):
r"""
Return the center of a permutation group.
Explanation
===========
The center for a group `G` is defined as
`Z(G) = \{z\in G | \forall g\in G, zg = gz \}`,
the set of elements of `G` that commute with all elements of `G`.
It is equal to the centralizer of `G` inside `G`, and is naturally a
subgroup of `G` ([9]).
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> D = DihedralGroup(4)
>>> G = D.center()
>>> G.order()
2
See Also
========
centralizer
Notes
=====
This is a naive implementation that is a straightforward application
of ``.centralizer()``
"""
return self.centralizer(self)
def centralizer(self, other):
r"""
Return the centralizer of a group/set/element.
Explanation
===========
The centralizer of a set of permutations ``S`` inside
a group ``G`` is the set of elements of ``G`` that commute with all
elements of ``S``::
`C_G(S) = \{ g \in G | gs = sg \forall s \in S\}` ([10])
Usually, ``S`` is a subset of ``G``, but if ``G`` is a proper subgroup of
the full symmetric group, we allow for ``S`` to have elements outside
``G``.
It is naturally a subgroup of ``G``; the centralizer of a permutation
group is equal to the centralizer of any set of generators for that
group, since any element commuting with the generators commutes with
any product of the generators.
Parameters
==========
other
a permutation group/list of permutations/single permutation
Examples
========
>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... CyclicGroup)
>>> S = SymmetricGroup(6)
>>> C = CyclicGroup(6)
>>> H = S.centralizer(C)
>>> H.is_subgroup(C)
True
See Also
========
subgroup_search
Notes
=====
The implementation is an application of ``.subgroup_search()`` with
tests using a specific base for the group ``G``.
"""
if hasattr(other, 'generators'):
if other.is_trivial or self.is_trivial:
return self
degree = self.degree
identity = _af_new(list(range(degree)))
orbits = other.orbits()
num_orbits = len(orbits)
orbits.sort(key=lambda x: -len(x))
long_base = []
orbit_reps = [None]*num_orbits
orbit_reps_indices = [None]*num_orbits
orbit_descr = [None]*degree
for i in range(num_orbits):
orbit = list(orbits[i])
orbit_reps[i] = orbit[0]
orbit_reps_indices[i] = len(long_base)
for point in orbit:
orbit_descr[point] = i
long_base = long_base + orbit
base, strong_gens = self.schreier_sims_incremental(base=long_base)
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
i = 0
for i in range(len(base)):
if strong_gens_distr[i] == [identity]:
break
base = base[:i]
base_len = i
for j in range(num_orbits):
if base[base_len - 1] in orbits[j]:
break
rel_orbits = orbits[: j + 1]
num_rel_orbits = len(rel_orbits)
transversals = [None]*num_rel_orbits
for j in range(num_rel_orbits):
rep = orbit_reps[j]
transversals[j] = dict(
other.orbit_transversal(rep, pairs=True))
trivial_test = lambda x: True
tests = [None]*base_len
for l in range(base_len):
if base[l] in orbit_reps:
tests[l] = trivial_test
else:
def test(computed_words, l=l):
g = computed_words[l]
rep_orb_index = orbit_descr[base[l]]
rep = orbit_reps[rep_orb_index]
im = g._array_form[base[l]]
im_rep = g._array_form[rep]
tr_el = transversals[rep_orb_index][base[l]]
# using the definition of transversal,
# base[l]^g = rep^(tr_el*g);
# if g belongs to the centralizer, then
# base[l]^g = (rep^g)^tr_el
return im == tr_el._array_form[im_rep]
tests[l] = test
def prop(g):
return [rmul(g, gen) for gen in other.generators] == \
[rmul(gen, g) for gen in other.generators]
return self.subgroup_search(prop, base=base,
strong_gens=strong_gens, tests=tests)
elif hasattr(other, '__getitem__'):
gens = list(other)
return self.centralizer(PermutationGroup(gens))
elif hasattr(other, 'array_form'):
return self.centralizer(PermutationGroup([other]))
def commutator(self, G, H):
"""
Return the commutator of two subgroups.
Explanation
===========
For a permutation group ``K`` and subgroups ``G``, ``H``, the
commutator of ``G`` and ``H`` is defined as the group generated
by all the commutators `[g, h] = hgh^{-1}g^{-1}` for ``g`` in ``G`` and
``h`` in ``H``. It is naturally a subgroup of ``K`` ([1], p.27).
Examples
========
>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... AlternatingGroup)
>>> S = SymmetricGroup(5)
>>> A = AlternatingGroup(5)
>>> G = S.commutator(S, A)
>>> G.is_subgroup(A)
True
See Also
========
derived_subgroup
Notes
=====
The commutator of two subgroups `H, G` is equal to the normal closure
of the commutators of all the generators, i.e. `hgh^{-1}g^{-1}` for `h`
a generator of `H` and `g` a generator of `G` ([1], p.28)
"""
ggens = G.generators
hgens = H.generators
commutators = []
for ggen in ggens:
for hgen in hgens:
commutator = rmul(hgen, ggen, ~hgen, ~ggen)
if commutator not in commutators:
commutators.append(commutator)
res = self.normal_closure(commutators)
return res
def coset_factor(self, g, factor_index=False):
"""Return ``G``'s (self's) coset factorization of ``g``
Explanation
===========
If ``g`` is an element of ``G`` then it can be written as the product
of permutations drawn from the Schreier-Sims coset decomposition,
The permutations returned in ``f`` are those for which
the product gives ``g``: ``g = f[n]*...f[1]*f[0]`` where ``n = len(B)``
and ``B = G.base``. f[i] is one of the permutations in
``self._basic_orbits[i]``.
If factor_index==True,
returns a tuple ``[b[0],..,b[n]]``, where ``b[i]``
belongs to ``self._basic_orbits[i]``
Examples
========
>>> from sympy.combinatorics import Permutation, 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])
Define g:
>>> g = Permutation(7)(1, 2, 4)(3, 6, 5)
Confirm that it is an element of G:
>>> G.contains(g)
True
Thus, it can be written as a product of factors (up to
3) drawn from u. See below that a factor from u1 and u2
and the Identity permutation have been used:
>>> f = G.coset_factor(g)
>>> f[2]*f[1]*f[0] == g
True
>>> f1 = G.coset_factor(g, True); f1
[0, 4, 4]
>>> tr = G.basic_transversals
>>> f[0] == tr[0][f1[0]]
True
If g is not an element of G then [] is returned:
>>> c = Permutation(5, 6, 7)
>>> G.coset_factor(c)
[]
See Also
========
sympy.combinatorics.util._strip
"""
if isinstance(g, (Cycle, Permutation)):
g = g.list()
if len(g) != self._degree:
# this could either adjust the size or return [] immediately
# but we don't choose between the two and just signal a possible
# error
raise ValueError('g should be the same size as permutations of G')
I = list(range(self._degree))
basic_orbits = self.basic_orbits
transversals = self._transversals
factors = []
base = self.base
h = g
for i in range(len(base)):
beta = h[base[i]]
if beta == base[i]:
factors.append(beta)
continue
if beta not in basic_orbits[i]:
return []
u = transversals[i][beta]._array_form
h = _af_rmul(_af_invert(u), h)
factors.append(beta)
if h != I:
return []
if factor_index:
return factors
tr = self.basic_transversals
factors = [tr[i][factors[i]] for i in range(len(base))]
return factors
def generator_product(self, g, original=False):
r'''
Return a list of strong generators `[s1, \dots, sn]`
s.t `g = sn \times \dots \times s1`. If ``original=True``, make the
list contain only the original group generators
'''
product = []
if g.is_identity:
return []
if g in self.strong_gens:
if not original or g in self.generators:
return [g]
else:
slp = self._strong_gens_slp[g]
for s in slp:
product.extend(self.generator_product(s, original=True))
return product
elif g**-1 in self.strong_gens:
g = g**-1
if not original or g in self.generators:
return [g**-1]
else:
slp = self._strong_gens_slp[g]
for s in slp:
product.extend(self.generator_product(s, original=True))
l = len(product)
product = [product[l-i-1]**-1 for i in range(l)]
return product
f = self.coset_factor(g, True)
for i, j in enumerate(f):
slp = self._transversal_slp[i][j]
for s in slp:
if not original:
product.append(self.strong_gens[s])
else:
s = self.strong_gens[s]
product.extend(self.generator_product(s, original=True))
return product
def coset_rank(self, g):
"""rank using Schreier-Sims representation.
Explanation
===========
The coset rank of ``g`` is the ordering number in which
it appears in the lexicographic listing according to the
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 import Permutation, 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(7)(2, 4)(3, 5)
>>> G.coset_rank(c)
16
>>> G.coset_unrank(16)
(7)(2 4)(3 5)
See Also
========
coset_factor
"""
factors = self.coset_factor(g, True)
if not factors:
return None
rank = 0
b = 1
transversals = self._transversals
base = self._base
basic_orbits = self._basic_orbits
for i in range(len(base)):
k = factors[i]
j = basic_orbits[i].index(k)
rank += b*j
b = b*len(transversals[i])
return rank
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.
"""
if rank < 0 or rank >= self.order():
return None
base = self.base
transversals = self.basic_transversals
basic_orbits = self.basic_orbits
m = len(base)
v = [0]*m
for i in range(m):
rank, c = divmod(rank, len(transversals[i]))
v[i] = basic_orbits[i][c]
a = [transversals[i][v[i]]._array_form for i in range(m)]
h = _af_rmuln(*a)
if af:
return h
else:
return _af_new(h)
@property
def degree(self):
"""Returns the size of the permutations in the group.
Explanation
===========
The number of permutations comprising the group is given by
``len(group)``; the number of permutations that can be generated
by the group is given by ``group.order()``.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([1, 0, 2])
>>> G = PermutationGroup([a])
>>> G.degree
3
>>> len(G)
1
>>> G.order()
2
>>> list(G.generate())
[(2), (2)(0 1)]
See Also
========
order
"""
return self._degree
@property
def identity(self):
'''
Return the identity element of the permutation group.
'''
return _af_new(list(range(self.degree)))
@property
def elements(self):
"""Returns all the elements of the permutation group as a set
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> p = PermutationGroup(Permutation(1, 3), Permutation(1, 2))
>>> p.elements
{(1 2 3), (1 3 2), (1 3), (2 3), (3), (3)(1 2)}
"""
return set(self._elements)
@property
def _elements(self):
"""Returns all the elements of the permutation group as a list
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> p = PermutationGroup(Permutation(1, 3), Permutation(1, 2))
>>> p._elements
[(3), (3)(1 2), (1 3), (2 3), (1 2 3), (1 3 2)]
"""
return list(islice(self.generate(), None))
def derived_series(self):
r"""Return the derived series for the group.
Explanation
===========
The derived series for a group `G` is defined as
`G = G_0 > G_1 > G_2 > \ldots` where `G_i = [G_{i-1}, G_{i-1}]`,
i.e. `G_i` is the derived subgroup of `G_{i-1}`, for
`i\in\mathbb{N}`. When we have `G_k = G_{k-1}` for some
`k\in\mathbb{N}`, the series terminates.
Returns
=======
A list of permutation groups containing the members of the derived
series in the order `G = G_0, G_1, G_2, \ldots`.
Examples
========
>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... AlternatingGroup, DihedralGroup)
>>> A = AlternatingGroup(5)
>>> len(A.derived_series())
1
>>> S = SymmetricGroup(4)
>>> len(S.derived_series())
4
>>> S.derived_series()[1].is_subgroup(AlternatingGroup(4))
True
>>> S.derived_series()[2].is_subgroup(DihedralGroup(2))
True
See Also
========
derived_subgroup
"""
res = [self]
current = self
nxt = self.derived_subgroup()
while not current.is_subgroup(nxt):
res.append(nxt)
current = nxt
nxt = nxt.derived_subgroup()
return res
def derived_subgroup(self):
r"""Compute the derived subgroup.
Explanation
===========
The derived subgroup, or commutator subgroup is the subgroup generated
by all commutators `[g, h] = hgh^{-1}g^{-1}` for `g, h\in G` ; it is
equal to the normal closure of the set of commutators of the generators
([1], p.28, [11]).
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([1, 0, 2, 4, 3])
>>> b = Permutation([0, 1, 3, 2, 4])
>>> G = PermutationGroup([a, b])
>>> C = G.derived_subgroup()
>>> list(C.generate(af=True))
[[0, 1, 2, 3, 4], [0, 1, 3, 4, 2], [0, 1, 4, 2, 3]]
See Also
========
derived_series
"""
r = self._r
gens = [p._array_form for p in self.generators]
set_commutators = set()
degree = self._degree
rng = list(range(degree))
for i in range(r):
for j in range(r):
p1 = gens[i]
p2 = gens[j]
c = list(range(degree))
for k in rng:
c[p2[p1[k]]] = p1[p2[k]]
ct = tuple(c)
if ct not in set_commutators:
set_commutators.add(ct)
cms = [_af_new(p) for p in set_commutators]
G2 = self.normal_closure(cms)
return G2
def generate(self, method="coset", af=False):
"""Return iterator to generate the elements of the group.
Explanation
===========
Iteration is done with one of these methods::
method='coset' using the Schreier-Sims coset representation
method='dimino' using the Dimino method
If ``af = True`` it yields the array form of the permutations
Examples
========
>>> from sympy.combinatorics import PermutationGroup
>>> from sympy.combinatorics.polyhedron import tetrahedron
The permutation group given in the tetrahedron object is also
true groups:
>>> G = tetrahedron.pgroup
>>> G.is_group
True
Also the group generated by the permutations in the tetrahedron
pgroup -- even the first two -- is a proper group:
>>> H = PermutationGroup(G[0], G[1])
>>> J = PermutationGroup(list(H.generate())); J
PermutationGroup([
(0 1)(2 3),
(1 2 3),
(1 3 2),
(0 3 1),
(0 2 3),
(0 3)(1 2),
(0 1 3),
(3)(0 2 1),
(0 3 2),
(3)(0 1 2),
(0 2)(1 3)])
>>> _.is_group
True
"""
if method == "coset":
return self.generate_schreier_sims(af)
elif method == "dimino":
return self.generate_dimino(af)
else:
raise NotImplementedError('No generation defined for %s' % method)
def generate_dimino(self, af=False):
"""Yield group elements using Dimino's algorithm.
If ``af == True`` it yields the array form of the permutations.
Examples
========
>>> from sympy.combinatorics import Permutation, 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]]
References
==========
.. [1] The Implementation of Various Algorithms for Permutation Groups in
the Computer Algebra System: AXIOM, N.J. Doye, M.Sc. Thesis
"""
idn = list(range(self.degree))
order = 0
element_list = [idn]
set_element_list = {tuple(idn)}
if af:
yield idn
else:
yield _af_new(idn)
gens = [p._array_form for p in self.generators]
for i in range(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 = _af_rmul(a, g)
if tuple(ag) not in set_element_list:
# produce G_i*g
for d in D:
order += 1
ap = _af_rmul(d, ag)
if af:
yield ap
else:
p = _af_new(ap)
yield p
element_list.append(ap)
set_element_list.add(tuple(ap))
N.append(ap)
self._order = len(element_list)
def generate_schreier_sims(self, af=False):
"""Yield group elements using the Schreier-Sims representation
in coset_rank order
If ``af = True`` it yields the array form of the permutations
Examples
========
>>> from sympy.combinatorics import Permutation, 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, 2, 1, 3], [0, 3, 2, 1],
[0, 1, 3, 2], [0, 2, 3, 1], [0, 3, 1, 2]]
"""
n = self._degree
u = self.basic_transversals
basic_orbits = self._basic_orbits
if len(u) == 0:
for x in self.generators:
if af:
yield x._array_form
else:
yield x
return
if len(u) == 1:
for i in basic_orbits[0]:
if af:
yield u[0][i]._array_form
else:
yield u[0][i]
return
u = list(reversed(u))
basic_orbits = basic_orbits[::-1]
# stg stack of group elements
stg = [list(range(n))]
posmax = [len(x) for x in u]
n1 = len(posmax) - 1
pos = [0]*n1
h = 0
while 1:
# backtrack when finished iterating over coset
if pos[h] >= posmax[h]:
if h == 0:
return
pos[h] = 0
h -= 1
stg.pop()
continue
p = _af_rmul(u[h][basic_orbits[h][pos[h]]]._array_form, stg[-1])
pos[h] += 1
stg.append(p)
h += 1
if h == n1:
if af:
for i in basic_orbits[-1]:
p = _af_rmul(u[-1][i]._array_form, stg[-1])
yield p
else:
for i in basic_orbits[-1]:
p = _af_rmul(u[-1][i]._array_form, stg[-1])
p1 = _af_new(p)
yield p1
stg.pop()
h -= 1
@property
def generators(self):
"""Returns the generators of the group.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.generators
[(1 2), (2)(0 1)]
"""
return self._generators
def contains(self, g, strict=True):
"""Test if permutation ``g`` belong to self, ``G``.
Explanation
===========
If ``g`` is an element of ``G`` it can be written as a product
of factors drawn from the cosets of ``G``'s stabilizers. To see
if ``g`` is one of the actual generators defining the group use
``G.has(g)``.
If ``strict`` is not ``True``, ``g`` will be resized, if necessary,
to match the size of permutations in ``self``.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation(1, 2)
>>> b = Permutation(2, 3, 1)
>>> G = PermutationGroup(a, b, degree=5)
>>> G.contains(G[0]) # trivial check
True
>>> elem = Permutation([[2, 3]], size=5)
>>> G.contains(elem)
True
>>> G.contains(Permutation(4)(0, 1, 2, 3))
False
If strict is False, a permutation will be resized, if
necessary:
>>> H = PermutationGroup(Permutation(5))
>>> H.contains(Permutation(3))
False
>>> H.contains(Permutation(3), strict=False)
True
To test if a given permutation is present in the group:
>>> elem in G.generators
False
>>> G.has(elem)
False
See Also
========
coset_factor, sympy.core.basic.Basic.has, __contains__
"""
if not isinstance(g, Permutation):
return False
if g.size != self.degree:
if strict:
return False
g = Permutation(g, size=self.degree)
if g in self.generators:
return True
return bool(self.coset_factor(g.array_form, True))
@property
def is_perfect(self):
"""Return ``True`` if the group is perfect.
A group is perfect if it equals to its derived subgroup.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation(1,2,3)(4,5)
>>> b = Permutation(1,2,3,4,5)
>>> G = PermutationGroup([a, b])
>>> G.is_perfect
False
"""
if self._is_perfect is None:
self._is_perfect = self.equals(self.derived_subgroup())
return self._is_perfect
@property
def is_abelian(self):
"""Test if the group is Abelian.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.is_abelian
False
>>> a = Permutation([0, 2, 1])
>>> G = PermutationGroup([a])
>>> G.is_abelian
True
"""
if self._is_abelian is not None:
return self._is_abelian
self._is_abelian = True
gens = [p._array_form for p in self.generators]
for x in gens:
for y in gens:
if y <= x:
continue
if not _af_commutes_with(x, y):
self._is_abelian = False
return False
return True
def abelian_invariants(self):
"""
Returns the abelian invariants for the given group.
Let ``G`` be a nontrivial finite abelian group. Then G is isomorphic to
the direct product of finitely many nontrivial cyclic groups of
prime-power order.
Explanation
===========
The prime-powers that occur as the orders of the factors are uniquely
determined by G. More precisely, the primes that occur in the orders of the
factors in any such decomposition of ``G`` are exactly the primes that divide
``|G|`` and for any such prime ``p``, if the orders of the factors that are
p-groups in one such decomposition of ``G`` are ``p^{t_1} >= p^{t_2} >= ... p^{t_r}``,
then the orders of the factors that are p-groups in any such decomposition of ``G``
are ``p^{t_1} >= p^{t_2} >= ... p^{t_r}``.
The uniquely determined integers ``p^{t_1} >= p^{t_2} >= ... p^{t_r}``, taken
for all primes that divide ``|G|`` are called the invariants of the nontrivial
group ``G`` as suggested in ([14], p. 542).
Notes
=====
We adopt the convention that the invariants of a trivial group are [].
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.abelian_invariants()
[2]
>>> from sympy.combinatorics import CyclicGroup
>>> G = CyclicGroup(7)
>>> G.abelian_invariants()
[7]
"""
if self.is_trivial:
return []
gns = self.generators
inv = []
G = self
H = G.derived_subgroup()
Hgens = H.generators
for p in primefactors(G.order()):
ranks = []
while True:
pows = []
for g in gns:
elm = g**p
if not H.contains(elm):
pows.append(elm)
K = PermutationGroup(Hgens + pows) if pows else H
r = G.order()//K.order()
G = K
gns = pows
if r == 1:
break
ranks.append(multiplicity(p, r))
if ranks:
pows = [1]*ranks[0]
for i in ranks:
for j in range(i):
pows[j] = pows[j]*p
inv.extend(pows)
inv.sort()
return inv
def is_elementary(self, p):
"""Return ``True`` if the group is elementary abelian. An elementary
abelian group is a finite abelian group, where every nontrivial
element has order `p`, where `p` is a prime.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> G = PermutationGroup([a])
>>> G.is_elementary(2)
True
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([3, 1, 2, 0])
>>> G = PermutationGroup([a, b])
>>> G.is_elementary(2)
True
>>> G.is_elementary(3)
False
"""
return self.is_abelian and all(g.order() == p for g in self.generators)
def _eval_is_alt_sym_naive(self, only_sym=False, only_alt=False):
"""A naive test using the group order."""
if only_sym and only_alt:
raise ValueError(
"Both {} and {} cannot be set to True"
.format(only_sym, only_alt))
n = self.degree
sym_order = _factorial(n)
order = self.order()
if order == sym_order:
self._is_sym = True
self._is_alt = False
if only_alt:
return False
return True
elif 2*order == sym_order:
self._is_sym = False
self._is_alt = True
if only_sym:
return False
return True
return False
def _eval_is_alt_sym_monte_carlo(self, eps=0.05, perms=None):
"""A test using monte-carlo algorithm.
Parameters
==========
eps : float, optional
The criterion for the incorrect ``False`` return.
perms : list[Permutation], optional
If explicitly given, it tests over the given candidates
for testing.
If ``None``, it randomly computes ``N_eps`` and chooses
``N_eps`` sample of the permutation from the group.
See Also
========
_check_cycles_alt_sym
"""
if perms is None:
n = self.degree
if n < 17:
c_n = 0.34
else:
c_n = 0.57
d_n = (c_n*log(2))/log(n)
N_eps = int(-log(eps)/d_n)
perms = (self.random_pr() for i in range(N_eps))
return self._eval_is_alt_sym_monte_carlo(perms=perms)
for perm in perms:
if _check_cycles_alt_sym(perm):
return True
return False
def is_alt_sym(self, eps=0.05, _random_prec=None):
r"""Monte Carlo test for the symmetric/alternating group for degrees
>= 8.
Explanation
===========
More specifically, it is one-sided Monte Carlo with the
answer True (i.e., G is symmetric/alternating) guaranteed to be
correct, and the answer False being incorrect with probability eps.
For degree < 8, the order of the group is checked so the test
is deterministic.
Notes
=====
The algorithm itself uses some nontrivial results from group theory and
number theory:
1) If a transitive group ``G`` of degree ``n`` contains an element
with a cycle of length ``n/2 < p < n-2`` for ``p`` a prime, ``G`` is the
symmetric or alternating group ([1], pp. 81-82)
2) The proportion of elements in the symmetric/alternating group having
the property described in 1) is approximately `\log(2)/\log(n)`
([1], p.82; [2], pp. 226-227).
The helper function ``_check_cycles_alt_sym`` is used to
go over the cycles in a permutation and look for ones satisfying 1).
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> D = DihedralGroup(10)
>>> D.is_alt_sym()
False
See Also
========
_check_cycles_alt_sym
"""
if _random_prec is not None:
N_eps = _random_prec['N_eps']
perms= (_random_prec[i] for i in range(N_eps))
return self._eval_is_alt_sym_monte_carlo(perms=perms)
if self._is_sym or self._is_alt:
return True
if self._is_sym is False and self._is_alt is False:
return False
n = self.degree
if n < 8:
return self._eval_is_alt_sym_naive()
elif self.is_transitive():
return self._eval_is_alt_sym_monte_carlo(eps=eps)
self._is_sym, self._is_alt = False, False
return False
@property
def is_nilpotent(self):
"""Test if the group is nilpotent.
Explanation
===========
A group `G` is nilpotent if it has a central series of finite length.
Alternatively, `G` is nilpotent if its lower central series terminates
with the trivial group. Every nilpotent group is also solvable
([1], p.29, [12]).
Examples
========
>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... CyclicGroup)
>>> C = CyclicGroup(6)
>>> C.is_nilpotent
True
>>> S = SymmetricGroup(5)
>>> S.is_nilpotent
False
See Also
========
lower_central_series, is_solvable
"""
if self._is_nilpotent is None:
lcs = self.lower_central_series()
terminator = lcs[len(lcs) - 1]
gens = terminator.generators
degree = self.degree
identity = _af_new(list(range(degree)))
if all(g == identity for g in gens):
self._is_solvable = True
self._is_nilpotent = True
return True
else:
self._is_nilpotent = False
return False
else:
return self._is_nilpotent
def is_normal(self, gr, strict=True):
"""Test if ``G=self`` is a normal subgroup of ``gr``.
Explanation
===========
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.generators and
g2 in G.generators.
Examples
========
>>> from sympy.combinatorics import Permutation, 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
"""
if not self.is_subgroup(gr, strict=strict):
return False
d_self = self.degree
d_gr = gr.degree
if self.is_trivial and (d_self == d_gr or not strict):
return True
if self._is_abelian:
return True
new_self = self.copy()
if not strict and d_self != d_gr:
if d_self < d_gr:
new_self = PermGroup(new_self.generators + [Permutation(d_gr - 1)])
else:
gr = PermGroup(gr.generators + [Permutation(d_self - 1)])
gens2 = [p._array_form for p in new_self.generators]
gens1 = [p._array_form for p in gr.generators]
for g1 in gens1:
for g2 in gens2:
p = _af_rmuln(g1, g2, _af_invert(g1))
if not new_self.coset_factor(p, True):
return False
return True
def is_primitive(self, randomized=True):
r"""Test if a group is primitive.
Explanation
===========
A permutation group ``G`` acting on a set ``S`` is called primitive if
``S`` contains no nontrivial block under the action of ``G``
(a block is nontrivial if its cardinality is more than ``1``).
Notes
=====
The algorithm is described in [1], p.83, and uses the function
minimal_block to search for blocks of the form `\{0, k\}` for ``k``
ranging over representatives for the orbits of `G_0`, the stabilizer of
``0``. This algorithm has complexity `O(n^2)` where ``n`` is the degree
of the group, and will perform badly if `G_0` is small.
There are two implementations offered: one finds `G_0`
deterministically using the function ``stabilizer``, and the other
(default) produces random elements of `G_0` using ``random_stab``,
hoping that they generate a subgroup of `G_0` with not too many more
orbits than `G_0` (this is suggested in [1], p.83). Behavior is changed
by the ``randomized`` flag.
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> D = DihedralGroup(10)
>>> D.is_primitive()
False
See Also
========
minimal_block, random_stab
"""
if self._is_primitive is not None:
return self._is_primitive
if self.is_transitive() is False:
return False
if randomized:
random_stab_gens = []
v = self.schreier_vector(0)
for _ in range(len(self)):
random_stab_gens.append(self.random_stab(0, v))
stab = PermutationGroup(random_stab_gens)
else:
stab = self.stabilizer(0)
orbits = stab.orbits()
for orb in orbits:
x = orb.pop()
if x != 0 and any(e != 0 for e in self.minimal_block([0, x])):
self._is_primitive = False
return False
self._is_primitive = True
return True
def minimal_blocks(self, randomized=True):
'''
For a transitive group, return the list of all minimal
block systems. If a group is intransitive, return `False`.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> DihedralGroup(6).minimal_blocks()
[[0, 1, 0, 1, 0, 1], [0, 1, 2, 0, 1, 2]]
>>> G = PermutationGroup(Permutation(1,2,5))
>>> G.minimal_blocks()
False
See Also
========
minimal_block, is_transitive, is_primitive
'''
def _number_blocks(blocks):
# number the blocks of a block system
# in order and return the number of
# blocks and the tuple with the
# reordering
n = len(blocks)
appeared = {}
m = 0
b = [None]*n
for i in range(n):
if blocks[i] not in appeared:
appeared[blocks[i]] = m
b[i] = m
m += 1
else:
b[i] = appeared[blocks[i]]
return tuple(b), m
if not self.is_transitive():
return False
blocks = []
num_blocks = []
rep_blocks = []
if randomized:
random_stab_gens = []
v = self.schreier_vector(0)
for i in range(len(self)):
random_stab_gens.append(self.random_stab(0, v))
stab = PermutationGroup(random_stab_gens)
else:
stab = self.stabilizer(0)
orbits = stab.orbits()
for orb in orbits:
x = orb.pop()
if x != 0:
block = self.minimal_block([0, x])
num_block, _ = _number_blocks(block)
# a representative block (containing 0)
rep = {j for j in range(self.degree) if num_block[j] == 0}
# check if the system is minimal with
# respect to the already discovere ones
minimal = True
blocks_remove_mask = [False] * len(blocks)
for i, r in enumerate(rep_blocks):
if len(r) > len(rep) and rep.issubset(r):
# i-th block system is not minimal
blocks_remove_mask[i] = True
elif len(r) < len(rep) and r.issubset(rep):
# the system being checked is not minimal
minimal = False
break
# remove non-minimal representative blocks
blocks = [b for i, b in enumerate(blocks) if not blocks_remove_mask[i]]
num_blocks = [n for i, n in enumerate(num_blocks) if not blocks_remove_mask[i]]
rep_blocks = [r for i, r in enumerate(rep_blocks) if not blocks_remove_mask[i]]
if minimal and num_block not in num_blocks:
blocks.append(block)
num_blocks.append(num_block)
rep_blocks.append(rep)
return blocks
@property
def is_solvable(self):
"""Test if the group is solvable.
``G`` is solvable if its derived series terminates with the trivial
group ([1], p.29).
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> S = SymmetricGroup(3)
>>> S.is_solvable
True
See Also
========
is_nilpotent, derived_series
"""
if self._is_solvable is None:
if self.order() % 2 != 0:
return True
ds = self.derived_series()
terminator = ds[len(ds) - 1]
gens = terminator.generators
degree = self.degree
identity = _af_new(list(range(degree)))
if all(g == identity for g in gens):
self._is_solvable = True
return True
else:
self._is_solvable = False
return False
else:
return self._is_solvable
def is_subgroup(self, G, strict=True):
"""Return ``True`` if all elements of ``self`` belong to ``G``.
If ``strict`` is ``False`` then if ``self``'s degree is smaller
than ``G``'s, the elements will be resized to have the same degree.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> from sympy.combinatorics import SymmetricGroup, CyclicGroup
Testing is strict by default: the degree of each group must be the
same:
>>> p = Permutation(0, 1, 2, 3, 4, 5)
>>> G1 = PermutationGroup([Permutation(0, 1, 2), Permutation(0, 1)])
>>> G2 = PermutationGroup([Permutation(0, 2), Permutation(0, 1, 2)])
>>> G3 = PermutationGroup([p, p**2])
>>> assert G1.order() == G2.order() == G3.order() == 6
>>> G1.is_subgroup(G2)
True
>>> G1.is_subgroup(G3)
False
>>> G3.is_subgroup(PermutationGroup(G3[1]))
False
>>> G3.is_subgroup(PermutationGroup(G3[0]))
True
To ignore the size, set ``strict`` to ``False``:
>>> S3 = SymmetricGroup(3)
>>> S5 = SymmetricGroup(5)
>>> S3.is_subgroup(S5, strict=False)
True
>>> C7 = CyclicGroup(7)
>>> G = S5*C7
>>> S5.is_subgroup(G, False)
True
>>> C7.is_subgroup(G, 0)
False
"""
if isinstance(G, SymmetricPermutationGroup):
if self.degree != G.degree:
return False
return True
if not isinstance(G, PermutationGroup):
return False
if self == G or self.generators[0]==Permutation():
return True
if G.order() % self.order() != 0:
return False
if self.degree == G.degree or \
(self.degree < G.degree and not strict):
gens = self.generators
else:
return False
return all(G.contains(g, strict=strict) for g in gens)
@property
def is_polycyclic(self):
"""Return ``True`` if a group is polycyclic. A group is polycyclic if
it has a subnormal series with cyclic factors. For finite groups,
this is the same as if the group is solvable.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([2, 0, 1, 3])
>>> G = PermutationGroup([a, b])
>>> G.is_polycyclic
True
"""
return self.is_solvable
def is_transitive(self, strict=True):
"""Test if the group is transitive.
Explanation
===========
A group is transitive if it has a single orbit.
If ``strict`` is ``False`` the group is transitive if it has
a single orbit of length different from 1.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([2, 0, 1, 3])
>>> G1 = PermutationGroup([a, b])
>>> G1.is_transitive()
False
>>> G1.is_transitive(strict=False)
True
>>> c = Permutation([2, 3, 0, 1])
>>> G2 = PermutationGroup([a, c])
>>> G2.is_transitive()
True
>>> d = Permutation([1, 0, 2, 3])
>>> e = Permutation([0, 1, 3, 2])
>>> G3 = PermutationGroup([d, e])
>>> G3.is_transitive() or G3.is_transitive(strict=False)
False
"""
if self._is_transitive: # strict or not, if True then True
return self._is_transitive
if strict:
if self._is_transitive is not None: # we only store strict=True
return self._is_transitive
ans = len(self.orbit(0)) == self.degree
self._is_transitive = ans
return ans
got_orb = False
for x in self.orbits():
if len(x) > 1:
if got_orb:
return False
got_orb = True
return got_orb
@property
def is_trivial(self):
"""Test if the group is the trivial group.
This is true if the group contains only the identity permutation.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> G = PermutationGroup([Permutation([0, 1, 2])])
>>> G.is_trivial
True
"""
if self._is_trivial is None:
self._is_trivial = len(self) == 1 and self[0].is_Identity
return self._is_trivial
def lower_central_series(self):
r"""Return the lower central series for the group.
The lower central series for a group `G` is the series
`G = G_0 > G_1 > G_2 > \ldots` where
`G_k = [G, G_{k-1}]`, i.e. every term after the first is equal to the
commutator of `G` and the previous term in `G1` ([1], p.29).
Returns
=======
A list of permutation groups in the order `G = G_0, G_1, G_2, \ldots`
Examples
========
>>> from sympy.combinatorics.named_groups import (AlternatingGroup,
... DihedralGroup)
>>> A = AlternatingGroup(4)
>>> len(A.lower_central_series())
2
>>> A.lower_central_series()[1].is_subgroup(DihedralGroup(2))
True
See Also
========
commutator, derived_series
"""
res = [self]
current = self
nxt = self.commutator(self, current)
while not current.is_subgroup(nxt):
res.append(nxt)
current = nxt
nxt = self.commutator(self, current)
return res
@property
def max_div(self):
"""Maximum proper divisor of the degree of a permutation group.
Explanation
===========
Obviously, this is the degree divided by its minimal proper divisor
(larger than ``1``, if one exists). As it is guaranteed to be prime,
the ``sieve`` from ``sympy.ntheory`` is used.
This function is also used as an optimization tool for the functions
``minimal_block`` and ``_union_find_merge``.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> G = PermutationGroup([Permutation([0, 2, 1, 3])])
>>> G.max_div
2
See Also
========
minimal_block, _union_find_merge
"""
if self._max_div is not None:
return self._max_div
n = self.degree
if n == 1:
return 1
for x in sieve:
if n % x == 0:
d = n//x
self._max_div = d
return d
def minimal_block(self, points):
r"""For a transitive group, finds the block system generated by
``points``.
Explanation
===========
If a group ``G`` acts on a set ``S``, a nonempty subset ``B`` of ``S``
is called a block under the action of ``G`` if for all ``g`` in ``G``
we have ``gB = B`` (``g`` fixes ``B``) or ``gB`` and ``B`` have no
common points (``g`` moves ``B`` entirely). ([1], p.23; [6]).
The distinct translates ``gB`` of a block ``B`` for ``g`` in ``G``
partition the set ``S`` and this set of translates is known as a block
system. Moreover, we obviously have that all blocks in the partition
have the same size, hence the block size divides ``|S|`` ([1], p.23).
A ``G``-congruence is an equivalence relation ``~`` on the set ``S``
such that ``a ~ b`` implies ``g(a) ~ g(b)`` for all ``g`` in ``G``.
For a transitive group, the equivalence classes of a ``G``-congruence
and the blocks of a block system are the same thing ([1], p.23).
The algorithm below checks the group for transitivity, and then finds
the ``G``-congruence generated by the pairs ``(p_0, p_1), (p_0, p_2),
..., (p_0,p_{k-1})`` which is the same as finding the maximal block
system (i.e., the one with minimum block size) such that
``p_0, ..., p_{k-1}`` are in the same block ([1], p.83).
It is an implementation of Atkinson's algorithm, as suggested in [1],
and manipulates an equivalence relation on the set ``S`` using a
union-find data structure. The running time is just above
`O(|points||S|)`. ([1], pp. 83-87; [7]).
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> D = DihedralGroup(10)
>>> D.minimal_block([0, 5])
[0, 1, 2, 3, 4, 0, 1, 2, 3, 4]
>>> D.minimal_block([0, 1])
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
See Also
========
_union_find_rep, _union_find_merge, is_transitive, is_primitive
"""
if not self.is_transitive():
return False
n = self.degree
gens = self.generators
# initialize the list of equivalence class representatives
parents = list(range(n))
ranks = [1]*n
not_rep = []
k = len(points)
# the block size must divide the degree of the group
if k > self.max_div:
return [0]*n
for i in range(k - 1):
parents[points[i + 1]] = points[0]
not_rep.append(points[i + 1])
ranks[points[0]] = k
i = 0
len_not_rep = k - 1
while i < len_not_rep:
gamma = not_rep[i]
i += 1
for gen in gens:
# find has side effects: performs path compression on the list
# of representatives
delta = self._union_find_rep(gamma, parents)
# union has side effects: performs union by rank on the list
# of representatives
temp = self._union_find_merge(gen(gamma), gen(delta), ranks,
parents, not_rep)
if temp == -1:
return [0]*n
len_not_rep += temp
for i in range(n):
# force path compression to get the final state of the equivalence
# relation
self._union_find_rep(i, parents)
# rewrite result so that block representatives are minimal
new_reps = {}
return [new_reps.setdefault(r, i) for i, r in enumerate(parents)]
def conjugacy_class(self, x):
r"""Return the conjugacy class of an element in the group.
Explanation
===========
The conjugacy class of an element ``g`` in a group ``G`` is the set of
elements ``x`` in ``G`` that are conjugate with ``g``, i.e. for which
``g = xax^{-1}``
for some ``a`` in ``G``.
Note that conjugacy is an equivalence relation, and therefore that
conjugacy classes are partitions of ``G``. For a list of all the
conjugacy classes of the group, use the conjugacy_classes() method.
In a permutation group, each conjugacy class corresponds to a particular
`cycle structure': for example, in ``S_3``, the conjugacy classes are:
* the identity class, ``{()}``
* all transpositions, ``{(1 2), (1 3), (2 3)}``
* all 3-cycles, ``{(1 2 3), (1 3 2)}``
Examples
========
>>> from sympy.combinatorics import Permutation, SymmetricGroup
>>> S3 = SymmetricGroup(3)
>>> S3.conjugacy_class(Permutation(0, 1, 2))
{(0 1 2), (0 2 1)}
Notes
=====
This procedure computes the conjugacy class directly by finding the
orbit of the element under conjugation in G. This algorithm is only
feasible for permutation groups of relatively small order, but is like
the orbit() function itself in that respect.
"""
# Ref: "Computing the conjugacy classes of finite groups"; Butler, G.
# Groups '93 Galway/St Andrews; edited by Campbell, C. M.
new_class = {x}
last_iteration = new_class
while len(last_iteration) > 0:
this_iteration = set()
for y in last_iteration:
for s in self.generators:
conjugated = s * y * (~s)
if conjugated not in new_class:
this_iteration.add(conjugated)
new_class.update(last_iteration)
last_iteration = this_iteration
return new_class
def conjugacy_classes(self):
r"""Return the conjugacy classes of the group.
Explanation
===========
As described in the documentation for the .conjugacy_class() function,
conjugacy is an equivalence relation on a group G which partitions the
set of elements. This method returns a list of all these conjugacy
classes of G.
Examples
========
>>> from sympy.combinatorics import SymmetricGroup
>>> SymmetricGroup(3).conjugacy_classes()
[{(2)}, {(0 1 2), (0 2 1)}, {(0 2), (1 2), (2)(0 1)}]
"""
identity = _af_new(list(range(self.degree)))
known_elements = {identity}
classes = [known_elements.copy()]
for x in self.generate():
if x not in known_elements:
new_class = self.conjugacy_class(x)
classes.append(new_class)
known_elements.update(new_class)
return classes
def normal_closure(self, other, k=10):
r"""Return the normal closure of a subgroup/set of permutations.
Explanation
===========
If ``S`` is a subset of a group ``G``, the normal closure of ``A`` in ``G``
is defined as the intersection of all normal subgroups of ``G`` that
contain ``A`` ([1], p.14). Alternatively, it is the group generated by
the conjugates ``x^{-1}yx`` for ``x`` a generator of ``G`` and ``y`` a
generator of the subgroup ``\left\langle S\right\rangle`` generated by
``S`` (for some chosen generating set for ``\left\langle S\right\rangle``)
([1], p.73).
Parameters
==========
other
a subgroup/list of permutations/single permutation
k
an implementation-specific parameter that determines the number
of conjugates that are adjoined to ``other`` at once
Examples
========
>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... CyclicGroup, AlternatingGroup)
>>> S = SymmetricGroup(5)
>>> C = CyclicGroup(5)
>>> G = S.normal_closure(C)
>>> G.order()
60
>>> G.is_subgroup(AlternatingGroup(5))
True
See Also
========
commutator, derived_subgroup, random_pr
Notes
=====
The algorithm is described in [1], pp. 73-74; it makes use of the
generation of random elements for permutation groups by the product
replacement algorithm.
"""
if hasattr(other, 'generators'):
degree = self.degree
identity = _af_new(list(range(degree)))
if all(g == identity for g in other.generators):
return other
Z = PermutationGroup(other.generators[:])
base, strong_gens = Z.schreier_sims_incremental()
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
basic_orbits, basic_transversals = \
_orbits_transversals_from_bsgs(base, strong_gens_distr)
self._random_pr_init(r=10, n=20)
_loop = True
while _loop:
Z._random_pr_init(r=10, n=10)
for _ in range(k):
g = self.random_pr()
h = Z.random_pr()
conj = h^g
res = _strip(conj, base, basic_orbits, basic_transversals)
if res[0] != identity or res[1] != len(base) + 1:
gens = Z.generators
gens.append(conj)
Z = PermutationGroup(gens)
strong_gens.append(conj)
temp_base, temp_strong_gens = \
Z.schreier_sims_incremental(base, strong_gens)
base, strong_gens = temp_base, temp_strong_gens
strong_gens_distr = \
_distribute_gens_by_base(base, strong_gens)
basic_orbits, basic_transversals = \
_orbits_transversals_from_bsgs(base,
strong_gens_distr)
_loop = False
for g in self.generators:
for h in Z.generators:
conj = h^g
res = _strip(conj, base, basic_orbits,
basic_transversals)
if res[0] != identity or res[1] != len(base) + 1:
_loop = True
break
if _loop:
break
return Z
elif hasattr(other, '__getitem__'):
return self.normal_closure(PermutationGroup(other))
elif hasattr(other, 'array_form'):
return self.normal_closure(PermutationGroup([other]))
def orbit(self, alpha, action='tuples'):
r"""Compute the orbit of alpha `\{g(\alpha) | g \in G\}` as a set.
Explanation
===========
The time complexity of the algorithm used here is `O(|Orb|*r)` where
`|Orb|` is the size of the orbit and ``r`` is the number of generators of
the group. For a more detailed analysis, see [1], p.78, [2], pp. 19-21.
Here alpha can be a single point, or a list of points.
If alpha is a single point, the ordinary orbit is computed.
if alpha is a list of points, there are three available options:
'union' - computes the union of the orbits of the points in the list
'tuples' - computes the orbit of the list interpreted as an ordered
tuple under the group action ( i.e., g((1,2,3)) = (g(1), g(2), g(3)) )
'sets' - computes the orbit of the list interpreted as a sets
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([1, 2, 0, 4, 5, 6, 3])
>>> G = PermutationGroup([a])
>>> G.orbit(0)
{0, 1, 2}
>>> G.orbit([0, 4], 'union')
{0, 1, 2, 3, 4, 5, 6}
See Also
========
orbit_transversal
"""
return _orbit(self.degree, self.generators, alpha, action)
def orbit_rep(self, alpha, beta, schreier_vector=None):
"""Return a group element which sends ``alpha`` to ``beta``.
Explanation
===========
If ``beta`` is not in the orbit of ``alpha``, the function returns
``False``. This implementation makes use of the schreier vector.
For a proof of correctness, see [1], p.80
Examples
========
>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> G = AlternatingGroup(5)
>>> G.orbit_rep(0, 4)
(0 4 1 2 3)
See Also
========
schreier_vector
"""
if schreier_vector is None:
schreier_vector = self.schreier_vector(alpha)
if schreier_vector[beta] is None:
return False
k = schreier_vector[beta]
gens = [x._array_form for x in self.generators]
a = []
while k != -1:
a.append(gens[k])
beta = gens[k].index(beta) # beta = (~gens[k])(beta)
k = schreier_vector[beta]
if a:
return _af_new(_af_rmuln(*a))
else:
return _af_new(list(range(self._degree)))
def orbit_transversal(self, alpha, pairs=False):
r"""Computes a transversal for the orbit of ``alpha`` as a set.
Explanation
===========
For a permutation group `G`, a transversal for the orbit
`Orb = \{g(\alpha) | g \in G\}` is a set
`\{g_\beta | g_\beta(\alpha) = \beta\}` for `\beta \in Orb`.
Note that there may be more than one possible transversal.
If ``pairs`` is set to ``True``, it returns the list of pairs
`(\beta, g_\beta)`. For a proof of correctness, see [1], p.79
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> G = DihedralGroup(6)
>>> G.orbit_transversal(0)
[(5), (0 1 2 3 4 5), (0 5)(1 4)(2 3), (0 2 4)(1 3 5), (5)(0 4)(1 3), (0 3)(1 4)(2 5)]
See Also
========
orbit
"""
return _orbit_transversal(self._degree, self.generators, alpha, pairs)
def orbits(self, rep=False):
"""Return the orbits of ``self``, ordered according to lowest element
in each orbit.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation(1, 5)(2, 3)(4, 0, 6)
>>> b = Permutation(1, 5)(3, 4)(2, 6, 0)
>>> G = PermutationGroup([a, b])
>>> G.orbits()
[{0, 2, 3, 4, 6}, {1, 5}]
"""
return _orbits(self._degree, self._generators)
def order(self):
"""Return the order of the group: the number of permutations that
can be generated from elements of the group.
The number of permutations comprising the group is given by
``len(group)``; the length of each permutation in the group is
given by ``group.size``.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([1, 0, 2])
>>> G = PermutationGroup([a])
>>> G.degree
3
>>> len(G)
1
>>> G.order()
2
>>> list(G.generate())
[(2), (2)(0 1)]
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.order()
6
See Also
========
degree
"""
if self._order is not None:
return self._order
if self._is_sym:
n = self._degree
self._order = factorial(n)
return self._order
if self._is_alt:
n = self._degree
self._order = factorial(n)/2
return self._order
m = prod([len(x) for x in self.basic_transversals])
self._order = m
return m
def index(self, H):
"""
Returns the index of a permutation group.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation(1,2,3)
>>> b =Permutation(3)
>>> G = PermutationGroup([a])
>>> H = PermutationGroup([b])
>>> G.index(H)
3
"""
if H.is_subgroup(self):
return self.order()//H.order()
@property
def is_symmetric(self):
"""Return ``True`` if the group is symmetric.
Examples
========
>>> from sympy.combinatorics import SymmetricGroup
>>> g = SymmetricGroup(5)
>>> g.is_symmetric
True
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> g = PermutationGroup(
... Permutation(0, 1, 2, 3, 4),
... Permutation(2, 3))
>>> g.is_symmetric
True
Notes
=====
This uses a naive test involving the computation of the full
group order.
If you need more quicker taxonomy for large groups, you can use
:meth:`PermutationGroup.is_alt_sym`.
However, :meth:`PermutationGroup.is_alt_sym` may not be accurate
and is not able to distinguish between an alternating group and
a symmetric group.
See Also
========
is_alt_sym
"""
_is_sym = self._is_sym
if _is_sym is not None:
return _is_sym
n = self.degree
if n >= 8:
if self.is_transitive():
_is_alt_sym = self._eval_is_alt_sym_monte_carlo()
if _is_alt_sym:
if any(g.is_odd for g in self.generators):
self._is_sym, self._is_alt = True, False
return True
self._is_sym, self._is_alt = False, True
return False
return self._eval_is_alt_sym_naive(only_sym=True)
self._is_sym, self._is_alt = False, False
return False
return self._eval_is_alt_sym_naive(only_sym=True)
@property
def is_alternating(self):
"""Return ``True`` if the group is alternating.
Examples
========
>>> from sympy.combinatorics import AlternatingGroup
>>> g = AlternatingGroup(5)
>>> g.is_alternating
True
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> g = PermutationGroup(
... Permutation(0, 1, 2, 3, 4),
... Permutation(2, 3, 4))
>>> g.is_alternating
True
Notes
=====
This uses a naive test involving the computation of the full
group order.
If you need more quicker taxonomy for large groups, you can use
:meth:`PermutationGroup.is_alt_sym`.
However, :meth:`PermutationGroup.is_alt_sym` may not be accurate
and is not able to distinguish between an alternating group and
a symmetric group.
See Also
========
is_alt_sym
"""
_is_alt = self._is_alt
if _is_alt is not None:
return _is_alt
n = self.degree
if n >= 8:
if self.is_transitive():
_is_alt_sym = self._eval_is_alt_sym_monte_carlo()
if _is_alt_sym:
if all(g.is_even for g in self.generators):
self._is_sym, self._is_alt = False, True
return True
self._is_sym, self._is_alt = True, False
return False
return self._eval_is_alt_sym_naive(only_alt=True)
self._is_sym, self._is_alt = False, False
return False
return self._eval_is_alt_sym_naive(only_alt=True)
@classmethod
def _distinct_primes_lemma(cls, primes):
"""Subroutine to test if there is only one cyclic group for the
order."""
primes = sorted(primes)
l = len(primes)
for i in range(l):
for j in range(i+1, l):
if primes[j] % primes[i] == 1:
return None
return True
@property
def is_cyclic(self):
r"""
Return ``True`` if the group is Cyclic.
Examples
========
>>> from sympy.combinatorics.named_groups import AbelianGroup
>>> G = AbelianGroup(3, 4)
>>> G.is_cyclic
True
>>> G = AbelianGroup(4, 4)
>>> G.is_cyclic
False
Notes
=====
If the order of a group $n$ can be factored into the distinct
primes $p_1, p_2, \dots , p_s$ and if
.. math::
\forall i, j \in \{1, 2, \dots, s \}:
p_i \not \equiv 1 \pmod {p_j}
holds true, there is only one group of the order $n$ which
is a cyclic group [1]_. This is a generalization of the lemma
that the group of order $15, 35, \dots$ are cyclic.
And also, these additional lemmas can be used to test if a
group is cyclic if the order of the group is already found.
- If the group is abelian and the order of the group is
square-free, the group is cyclic.
- If the order of the group is less than $6$ and is not $4$, the
group is cyclic.
- If the order of the group is prime, the group is cyclic.
References
==========
.. [1] 1978: John S. Rose: A Course on Group Theory,
Introduction to Finite Group Theory: 1.4
"""
if self._is_cyclic is not None:
return self._is_cyclic
if len(self.generators) == 1:
self._is_cyclic = True
self._is_abelian = True
return True
if self._is_abelian is False:
self._is_cyclic = False
return False
order = self.order()
if order < 6:
self._is_abelian = True
if order != 4:
self._is_cyclic = True
return True
factors = factorint(order)
if all(v == 1 for v in factors.values()):
if self._is_abelian:
self._is_cyclic = True
return True
primes = list(factors.keys())
if PermutationGroup._distinct_primes_lemma(primes) is True:
self._is_cyclic = True
self._is_abelian = True
return True
if not self.is_abelian:
self._is_cyclic = False
return False
self._is_cyclic = all(
any(g**(order//p) != self.identity for g in self.generators)
for p, e in factors.items() if e > 1
)
return self._is_cyclic
@property
def is_dihedral(self):
r"""
Return ``True`` if the group is dihedral.
Examples
========
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.named_groups import SymmetricGroup, CyclicGroup
>>> G = PermutationGroup(Permutation(1, 6)(2, 5)(3, 4), Permutation(0, 1, 2, 3, 4, 5, 6))
>>> G.is_dihedral
True
>>> G = SymmetricGroup(3)
>>> G.is_dihedral
True
>>> G = CyclicGroup(6)
>>> G.is_dihedral
False
References
==========
.. [Di1] https://math.stackexchange.com/a/827273
.. [Di2] https://kconrad.math.uconn.edu/blurbs/grouptheory/dihedral.pdf
.. [Di3] https://kconrad.math.uconn.edu/blurbs/grouptheory/dihedral2.pdf
.. [Di4] https://en.wikipedia.org/wiki/Dihedral_group
"""
if self._is_dihedral is not None:
return self._is_dihedral
order = self.order()
if order % 2 == 1:
self._is_dihedral = False
return False
if order == 2:
self._is_dihedral = True
return True
if order == 4:
# The dihedral group of order 4 is the Klein 4-group.
self._is_dihedral = not self.is_cyclic
return self._is_dihedral
if self.is_abelian:
# The only abelian dihedral groups are the ones of orders 2 and 4.
self._is_dihedral = False
return False
# Now we know the group is of even order >= 6, and nonabelian.
n = order // 2
# Handle special cases where there are exactly two generators.
gens = self.generators
if len(gens) == 2:
x, y = gens
a, b = x.order(), y.order()
# Make a >= b
if a < b:
x, y, a, b = y, x, b, a
# Using Theorem 2.1 of [Di3]:
if a == 2 == b:
self._is_dihedral = True
return True
# Using Theorem 1.1 of [Di3]:
if a == n and b == 2 and y*x*y == ~x:
self._is_dihedral = True
return True
# Proceed with algorithm of [Di1]
# Find elements of orders 2 and n
order_2, order_n = [], []
for p in self.elements:
k = p.order()
if k == 2:
order_2.append(p)
elif k == n:
order_n.append(p)
if len(order_2) != n + 1 - (n % 2):
self._is_dihedral = False
return False
if not order_n:
self._is_dihedral = False
return False
x = order_n[0]
# Want an element y of order 2 that is not a power of x
# (i.e. that is not the 180-deg rotation, when n is even).
y = order_2[0]
if n % 2 == 0 and y == x**(n//2):
y = order_2[1]
self._is_dihedral = (y*x*y == ~x)
return self._is_dihedral
def pointwise_stabilizer(self, points, incremental=True):
r"""Return the pointwise stabilizer for a set of points.
Explanation
===========
For a permutation group `G` and a set of points
`\{p_1, p_2,\ldots, p_k\}`, the pointwise stabilizer of
`p_1, p_2, \ldots, p_k` is defined as
`G_{p_1,\ldots, p_k} =
\{g\in G | g(p_i) = p_i \forall i\in\{1, 2,\ldots,k\}\}` ([1],p20).
It is a subgroup of `G`.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> S = SymmetricGroup(7)
>>> Stab = S.pointwise_stabilizer([2, 3, 5])
>>> Stab.is_subgroup(S.stabilizer(2).stabilizer(3).stabilizer(5))
True
See Also
========
stabilizer, schreier_sims_incremental
Notes
=====
When incremental == True,
rather than the obvious implementation using successive calls to
``.stabilizer()``, this uses the incremental Schreier-Sims algorithm
to obtain a base with starting segment - the given points.
"""
if incremental:
base, strong_gens = self.schreier_sims_incremental(base=points)
stab_gens = []
degree = self.degree
for gen in strong_gens:
if [gen(point) for point in points] == points:
stab_gens.append(gen)
if not stab_gens:
stab_gens = _af_new(list(range(degree)))
return PermutationGroup(stab_gens)
else:
gens = self._generators
degree = self.degree
for x in points:
gens = _stabilizer(degree, gens, x)
return PermutationGroup(gens)
def make_perm(self, n, seed=None):
"""
Multiply ``n`` randomly selected permutations from
pgroup together, starting with the identity
permutation. If ``n`` is a list of integers, those
integers will be used to select the permutations and they
will be applied in L to R order: make_perm((A, B, C)) will
give CBA(I) where I is the identity permutation.
``seed`` is used to set the seed for the random selection
of permutations from pgroup. If this is a list of integers,
the corresponding permutations from pgroup will be selected
in the order give. This is mainly used for testing purposes.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a, b = [Permutation([1, 0, 3, 2]), Permutation([1, 3, 0, 2])]
>>> G = PermutationGroup([a, b])
>>> G.make_perm(1, [0])
(0 1)(2 3)
>>> G.make_perm(3, [0, 1, 0])
(0 2 3 1)
>>> G.make_perm([0, 1, 0])
(0 2 3 1)
See Also
========
random
"""
if is_sequence(n):
if seed is not None:
raise ValueError('If n is a sequence, seed should be None')
n, seed = len(n), n
else:
try:
n = int(n)
except TypeError:
raise ValueError('n must be an integer or a sequence.')
randomrange = _randrange(seed)
# start with the identity permutation
result = Permutation(list(range(self.degree)))
m = len(self)
for _ in range(n):
p = self[randomrange(m)]
result = rmul(result, p)
return result
def random(self, af=False):
"""Return a random group element
"""
rank = randrange(self.order())
return self.coset_unrank(rank, af)
def random_pr(self, gen_count=11, iterations=50, _random_prec=None):
"""Return a random group element using product replacement.
Explanation
===========
For the details of the product replacement algorithm, see
``_random_pr_init`` In ``random_pr`` the actual 'product replacement'
is performed. Notice that if the attribute ``_random_gens``
is empty, it needs to be initialized by ``_random_pr_init``.
See Also
========
_random_pr_init
"""
if self._random_gens == []:
self._random_pr_init(gen_count, iterations)
random_gens = self._random_gens
r = len(random_gens) - 1
# handle randomized input for testing purposes
if _random_prec is None:
s = randrange(r)
t = randrange(r - 1)
if t == s:
t = r - 1
x = choice([1, 2])
e = choice([-1, 1])
else:
s = _random_prec['s']
t = _random_prec['t']
if t == s:
t = r - 1
x = _random_prec['x']
e = _random_prec['e']
if x == 1:
random_gens[s] = _af_rmul(random_gens[s], _af_pow(random_gens[t], e))
random_gens[r] = _af_rmul(random_gens[r], random_gens[s])
else:
random_gens[s] = _af_rmul(_af_pow(random_gens[t], e), random_gens[s])
random_gens[r] = _af_rmul(random_gens[s], random_gens[r])
return _af_new(random_gens[r])
def random_stab(self, alpha, schreier_vector=None, _random_prec=None):
"""Random element from the stabilizer of ``alpha``.
The schreier vector for ``alpha`` is an optional argument used
for speeding up repeated calls. The algorithm is described in [1], p.81
See Also
========
random_pr, orbit_rep
"""
if schreier_vector is None:
schreier_vector = self.schreier_vector(alpha)
if _random_prec is None:
rand = self.random_pr()
else:
rand = _random_prec['rand']
beta = rand(alpha)
h = self.orbit_rep(alpha, beta, schreier_vector)
return rmul(~h, rand)
def schreier_sims(self):
"""Schreier-Sims algorithm.
Explanation
===========
It computes the generators of the chain of stabilizers
`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 as the product
`h_1*..*h_s`.
We use the incremental Schreier-Sims algorithm.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.schreier_sims()
>>> G.basic_transversals
[{0: (2)(0 1), 1: (2), 2: (1 2)},
{0: (2), 2: (0 2)}]
"""
if self._transversals:
return
self._schreier_sims()
return
def _schreier_sims(self, base=None):
schreier = self.schreier_sims_incremental(base=base, slp_dict=True)
base, strong_gens = schreier[:2]
self._base = base
self._strong_gens = strong_gens
self._strong_gens_slp = schreier[2]
if not base:
self._transversals = []
self._basic_orbits = []
return
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
basic_orbits, transversals, slps = _orbits_transversals_from_bsgs(base,\
strong_gens_distr, slp=True)
# rewrite the indices stored in slps in terms of strong_gens
for i, slp in enumerate(slps):
gens = strong_gens_distr[i]
for k in slp:
slp[k] = [strong_gens.index(gens[s]) for s in slp[k]]
self._transversals = transversals
self._basic_orbits = [sorted(x) for x in basic_orbits]
self._transversal_slp = slps
def schreier_sims_incremental(self, base=None, gens=None, slp_dict=False):
"""Extend a sequence of points and generating set to a base and strong
generating set.
Parameters
==========
base
The sequence of points to be extended to a base. Optional
parameter with default value ``[]``.
gens
The generating set to be extended to a strong generating set
relative to the base obtained. Optional parameter with default
value ``self.generators``.
slp_dict
If `True`, return a dictionary `{g: gens}` for each strong
generator `g` where `gens` is a list of strong generators
coming before `g` in `strong_gens`, such that the product
of the elements of `gens` is equal to `g`.
Returns
=======
(base, strong_gens)
``base`` is the base obtained, and ``strong_gens`` is the strong
generating set relative to it. The original parameters ``base``,
``gens`` remain unchanged.
Examples
========
>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> from sympy.combinatorics.testutil import _verify_bsgs
>>> A = AlternatingGroup(7)
>>> base = [2, 3]
>>> seq = [2, 3]
>>> base, strong_gens = A.schreier_sims_incremental(base=seq)
>>> _verify_bsgs(A, base, strong_gens)
True
>>> base[:2]
[2, 3]
Notes
=====
This version of the Schreier-Sims algorithm runs in polynomial time.
There are certain assumptions in the implementation - if the trivial
group is provided, ``base`` and ``gens`` are returned immediately,
as any sequence of points is a base for the trivial group. If the
identity is present in the generators ``gens``, it is removed as
it is a redundant generator.
The implementation is described in [1], pp. 90-93.
See Also
========
schreier_sims, schreier_sims_random
"""
if base is None:
base = []
if gens is None:
gens = self.generators[:]
degree = self.degree
id_af = list(range(degree))
# handle the trivial group
if len(gens) == 1 and gens[0].is_Identity:
if slp_dict:
return base, gens, {gens[0]: [gens[0]]}
return base, gens
# prevent side effects
_base, _gens = base[:], gens[:]
# remove the identity as a generator
_gens = [x for x in _gens if not x.is_Identity]
# make sure no generator fixes all base points
for gen in _gens:
if all(x == gen._array_form[x] for x in _base):
for new in id_af:
if gen._array_form[new] != new:
break
else:
assert None # can this ever happen?
_base.append(new)
# distribute generators according to basic stabilizers
strong_gens_distr = _distribute_gens_by_base(_base, _gens)
strong_gens_slp = []
# initialize the basic stabilizers, basic orbits and basic transversals
orbs = {}
transversals = {}
slps = {}
base_len = len(_base)
for i in range(base_len):
transversals[i], slps[i] = _orbit_transversal(degree, strong_gens_distr[i],
_base[i], pairs=True, af=True, slp=True)
transversals[i] = dict(transversals[i])
orbs[i] = list(transversals[i].keys())
# main loop: amend the stabilizer chain until we have generators
# for all stabilizers
i = base_len - 1
while i >= 0:
# this flag is used to continue with the main loop from inside
# a nested loop
continue_i = False
# test the generators for being a strong generating set
db = {}
for beta, u_beta in list(transversals[i].items()):
for j, gen in enumerate(strong_gens_distr[i]):
gb = gen._array_form[beta]
u1 = transversals[i][gb]
g1 = _af_rmul(gen._array_form, u_beta)
slp = [(i, g) for g in slps[i][beta]]
slp = [(i, j)] + slp
if g1 != u1:
# test if the schreier generator is in the i+1-th
# would-be basic stabilizer
y = True
try:
u1_inv = db[gb]
except KeyError:
u1_inv = db[gb] = _af_invert(u1)
schreier_gen = _af_rmul(u1_inv, g1)
u1_inv_slp = slps[i][gb][:]
u1_inv_slp.reverse()
u1_inv_slp = [(i, (g,)) for g in u1_inv_slp]
slp = u1_inv_slp + slp
h, j, slp = _strip_af(schreier_gen, _base, orbs, transversals, i, slp=slp, slps=slps)
if j <= base_len:
# new strong generator h at level j
y = False
elif h:
# h fixes all base points
y = False
moved = 0
while h[moved] == moved:
moved += 1
_base.append(moved)
base_len += 1
strong_gens_distr.append([])
if y is False:
# if a new strong generator is found, update the
# data structures and start over
h = _af_new(h)
strong_gens_slp.append((h, slp))
for l in range(i + 1, j):
strong_gens_distr[l].append(h)
transversals[l], slps[l] =\
_orbit_transversal(degree, strong_gens_distr[l],
_base[l], pairs=True, af=True, slp=True)
transversals[l] = dict(transversals[l])
orbs[l] = list(transversals[l].keys())
i = j - 1
# continue main loop using the flag
continue_i = True
if continue_i is True:
break
if continue_i is True:
break
if continue_i is True:
continue
i -= 1
strong_gens = _gens[:]
if slp_dict:
# create the list of the strong generators strong_gens and
# rewrite the indices of strong_gens_slp in terms of the
# elements of strong_gens
for k, slp in strong_gens_slp:
strong_gens.append(k)
for i in range(len(slp)):
s = slp[i]
if isinstance(s[1], tuple):
slp[i] = strong_gens_distr[s[0]][s[1][0]]**-1
else:
slp[i] = strong_gens_distr[s[0]][s[1]]
strong_gens_slp = dict(strong_gens_slp)
# add the original generators
for g in _gens:
strong_gens_slp[g] = [g]
return (_base, strong_gens, strong_gens_slp)
strong_gens.extend([k for k, _ in strong_gens_slp])
return _base, strong_gens
def schreier_sims_random(self, base=None, gens=None, consec_succ=10,
_random_prec=None):
r"""Randomized Schreier-Sims algorithm.
Explanation
===========
The randomized Schreier-Sims algorithm takes the sequence ``base``
and the generating set ``gens``, and extends ``base`` to a base, and
``gens`` to a strong generating set relative to that base with
probability of a wrong answer at most `2^{-consec\_succ}`,
provided the random generators are sufficiently random.
Parameters
==========
base
The sequence to be extended to a base.
gens
The generating set to be extended to a strong generating set.
consec_succ
The parameter defining the probability of a wrong answer.
_random_prec
An internal parameter used for testing purposes.
Returns
=======
(base, strong_gens)
``base`` is the base and ``strong_gens`` is the strong generating
set relative to it.
Examples
========
>>> from sympy.combinatorics.testutil import _verify_bsgs
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> S = SymmetricGroup(5)
>>> base, strong_gens = S.schreier_sims_random(consec_succ=5)
>>> _verify_bsgs(S, base, strong_gens) #doctest: +SKIP
True
Notes
=====
The algorithm is described in detail in [1], pp. 97-98. It extends
the orbits ``orbs`` and the permutation groups ``stabs`` to
basic orbits and basic stabilizers for the base and strong generating
set produced in the end.
The idea of the extension process
is to "sift" random group elements through the stabilizer chain
and amend the stabilizers/orbits along the way when a sift
is not successful.
The helper function ``_strip`` is used to attempt
to decompose a random group element according to the current
state of the stabilizer chain and report whether the element was
fully decomposed (successful sift) or not (unsuccessful sift). In
the latter case, the level at which the sift failed is reported and
used to amend ``stabs``, ``base``, ``gens`` and ``orbs`` accordingly.
The halting condition is for ``consec_succ`` consecutive successful
sifts to pass. This makes sure that the current ``base`` and ``gens``
form a BSGS with probability at least `1 - 1/\text{consec\_succ}`.
See Also
========
schreier_sims
"""
if base is None:
base = []
if gens is None:
gens = self.generators
base_len = len(base)
n = self.degree
# make sure no generator fixes all base points
for gen in gens:
if all(gen(x) == x for x in base):
new = 0
while gen._array_form[new] == new:
new += 1
base.append(new)
base_len += 1
# distribute generators according to basic stabilizers
strong_gens_distr = _distribute_gens_by_base(base, gens)
# initialize the basic stabilizers, basic transversals and basic orbits
transversals = {}
orbs = {}
for i in range(base_len):
transversals[i] = dict(_orbit_transversal(n, strong_gens_distr[i],
base[i], pairs=True))
orbs[i] = list(transversals[i].keys())
# initialize the number of consecutive elements sifted
c = 0
# start sifting random elements while the number of consecutive sifts
# is less than consec_succ
while c < consec_succ:
if _random_prec is None:
g = self.random_pr()
else:
g = _random_prec['g'].pop()
h, j = _strip(g, base, orbs, transversals)
y = True
# determine whether a new base point is needed
if j <= base_len:
y = False
elif not h.is_Identity:
y = False
moved = 0
while h(moved) == moved:
moved += 1
base.append(moved)
base_len += 1
strong_gens_distr.append([])
# if the element doesn't sift, amend the strong generators and
# associated stabilizers and orbits
if y is False:
for l in range(1, j):
strong_gens_distr[l].append(h)
transversals[l] = dict(_orbit_transversal(n,
strong_gens_distr[l], base[l], pairs=True))
orbs[l] = list(transversals[l].keys())
c = 0
else:
c += 1
# build the strong generating set
strong_gens = strong_gens_distr[0][:]
for gen in strong_gens_distr[1]:
if gen not in strong_gens:
strong_gens.append(gen)
return base, strong_gens
def schreier_vector(self, alpha):
"""Computes the schreier vector for ``alpha``.
Explanation
===========
The Schreier vector efficiently stores information
about the orbit of ``alpha``. It can later be used to quickly obtain
elements of the group that send ``alpha`` to a particular element
in the orbit. Notice that the Schreier vector depends on the order
in which the group generators are listed. For a definition, see [3].
Since list indices start from zero, we adopt the convention to use
"None" instead of 0 to signify that an element does not belong
to the orbit.
For the algorithm and its correctness, see [2], pp.78-80.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([2, 4, 6, 3, 1, 5, 0])
>>> b = Permutation([0, 1, 3, 5, 4, 6, 2])
>>> G = PermutationGroup([a, b])
>>> G.schreier_vector(0)
[-1, None, 0, 1, None, 1, 0]
See Also
========
orbit
"""
n = self.degree
v = [None]*n
v[alpha] = -1
orb = [alpha]
used = [False]*n
used[alpha] = True
gens = self.generators
r = len(gens)
for b in orb:
for i in range(r):
temp = gens[i]._array_form[b]
if used[temp] is False:
orb.append(temp)
used[temp] = True
v[temp] = i
return v
def stabilizer(self, alpha):
r"""Return the stabilizer subgroup of ``alpha``.
Explanation
===========
The stabilizer of `\alpha` is the group `G_\alpha =
\{g \in G | g(\alpha) = \alpha\}`.
For a proof of correctness, see [1], p.79.
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> G = DihedralGroup(6)
>>> G.stabilizer(5)
PermutationGroup([
(5)(0 4)(1 3)])
See Also
========
orbit
"""
return PermGroup(_stabilizer(self._degree, self._generators, alpha))
@property
def strong_gens(self):
r"""Return a strong generating set from the Schreier-Sims algorithm.
Explanation
===========
A generating set `S = \{g_1, g_2, \dots, g_t\}` for a permutation group
`G` is a strong generating set relative to the sequence of points
(referred to as a "base") `(b_1, b_2, \dots, b_k)` if, for
`1 \leq i \leq k` we have that the intersection of the pointwise
stabilizer `G^{(i+1)} := G_{b_1, b_2, \dots, b_i}` with `S` generates
the pointwise stabilizer `G^{(i+1)}`. The concepts of a base and
strong generating set and their applications are discussed in depth
in [1], pp. 87-89 and [2], pp. 55-57.
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> D = DihedralGroup(4)
>>> D.strong_gens
[(0 1 2 3), (0 3)(1 2), (1 3)]
>>> D.base
[0, 1]
See Also
========
base, basic_transversals, basic_orbits, basic_stabilizers
"""
if self._strong_gens == []:
self.schreier_sims()
return self._strong_gens
def subgroup(self, gens):
"""
Return the subgroup generated by `gens` which is a list of
elements of the group
"""
if not all(g in self for g in gens):
raise ValueError("The group does not contain the supplied generators")
G = PermutationGroup(gens)
return G
def subgroup_search(self, prop, base=None, strong_gens=None, tests=None,
init_subgroup=None):
"""Find the subgroup of all elements satisfying the property ``prop``.
Explanation
===========
This is done by a depth-first search with respect to base images that
uses several tests to prune the search tree.
Parameters
==========
prop
The property to be used. Has to be callable on group elements
and always return ``True`` or ``False``. It is assumed that
all group elements satisfying ``prop`` indeed form a subgroup.
base
A base for the supergroup.
strong_gens
A strong generating set for the supergroup.
tests
A list of callables of length equal to the length of ``base``.
These are used to rule out group elements by partial base images,
so that ``tests[l](g)`` returns False if the element ``g`` is known
not to satisfy prop base on where g sends the first ``l + 1`` base
points.
init_subgroup
if a subgroup of the sought group is
known in advance, it can be passed to the function as this
parameter.
Returns
=======
res
The subgroup of all elements satisfying ``prop``. The generating
set for this group is guaranteed to be a strong generating set
relative to the base ``base``.
Examples
========
>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... AlternatingGroup)
>>> from sympy.combinatorics.testutil import _verify_bsgs
>>> S = SymmetricGroup(7)
>>> prop_even = lambda x: x.is_even
>>> base, strong_gens = S.schreier_sims_incremental()
>>> G = S.subgroup_search(prop_even, base=base, strong_gens=strong_gens)
>>> G.is_subgroup(AlternatingGroup(7))
True
>>> _verify_bsgs(G, base, G.generators)
True
Notes
=====
This function is extremely lengthy and complicated and will require
some careful attention. The implementation is described in
[1], pp. 114-117, and the comments for the code here follow the lines
of the pseudocode in the book for clarity.
The complexity is exponential in general, since the search process by
itself visits all members of the supergroup. However, there are a lot
of tests which are used to prune the search tree, and users can define
their own tests via the ``tests`` parameter, so in practice, and for
some computations, it's not terrible.
A crucial part in the procedure is the frequent base change performed
(this is line 11 in the pseudocode) in order to obtain a new basic
stabilizer. The book mentiones that this can be done by using
``.baseswap(...)``, however the current implementation uses a more
straightforward way to find the next basic stabilizer - calling the
function ``.stabilizer(...)`` on the previous basic stabilizer.
"""
# initialize BSGS and basic group properties
def get_reps(orbits):
# get the minimal element in the base ordering
return [min(orbit, key = lambda x: base_ordering[x]) \
for orbit in orbits]
def update_nu(l):
temp_index = len(basic_orbits[l]) + 1 -\
len(res_basic_orbits_init_base[l])
# this corresponds to the element larger than all points
if temp_index >= len(sorted_orbits[l]):
nu[l] = base_ordering[degree]
else:
nu[l] = sorted_orbits[l][temp_index]
if base is None:
base, strong_gens = self.schreier_sims_incremental()
base_len = len(base)
degree = self.degree
identity = _af_new(list(range(degree)))
base_ordering = _base_ordering(base, degree)
# add an element larger than all points
base_ordering.append(degree)
# add an element smaller than all points
base_ordering.append(-1)
# compute BSGS-related structures
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
basic_orbits, transversals = _orbits_transversals_from_bsgs(base,
strong_gens_distr)
# handle subgroup initialization and tests
if init_subgroup is None:
init_subgroup = PermutationGroup([identity])
if tests is None:
trivial_test = lambda x: True
tests = []
for i in range(base_len):
tests.append(trivial_test)
# line 1: more initializations.
res = init_subgroup
f = base_len - 1
l = base_len - 1
# line 2: set the base for K to the base for G
res_base = base[:]
# line 3: compute BSGS and related structures for K
res_base, res_strong_gens = res.schreier_sims_incremental(
base=res_base)
res_strong_gens_distr = _distribute_gens_by_base(res_base,
res_strong_gens)
res_generators = res.generators
res_basic_orbits_init_base = \
[_orbit(degree, res_strong_gens_distr[i], res_base[i])\
for i in range(base_len)]
# initialize orbit representatives
orbit_reps = [None]*base_len
# line 4: orbit representatives for f-th basic stabilizer of K
orbits = _orbits(degree, res_strong_gens_distr[f])
orbit_reps[f] = get_reps(orbits)
# line 5: remove the base point from the representatives to avoid
# getting the identity element as a generator for K
orbit_reps[f].remove(base[f])
# line 6: more initializations
c = [0]*base_len
u = [identity]*base_len
sorted_orbits = [None]*base_len
for i in range(base_len):
sorted_orbits[i] = basic_orbits[i][:]
sorted_orbits[i].sort(key=lambda point: base_ordering[point])
# line 7: initializations
mu = [None]*base_len
nu = [None]*base_len
# this corresponds to the element smaller than all points
mu[l] = degree + 1
update_nu(l)
# initialize computed words
computed_words = [identity]*base_len
# line 8: main loop
while True:
# apply all the tests
while l < base_len - 1 and \
computed_words[l](base[l]) in orbit_reps[l] and \
base_ordering[mu[l]] < \
base_ordering[computed_words[l](base[l])] < \
base_ordering[nu[l]] and \
tests[l](computed_words):
# line 11: change the (partial) base of K
new_point = computed_words[l](base[l])
res_base[l] = new_point
new_stab_gens = _stabilizer(degree, res_strong_gens_distr[l],
new_point)
res_strong_gens_distr[l + 1] = new_stab_gens
# line 12: calculate minimal orbit representatives for the
# l+1-th basic stabilizer
orbits = _orbits(degree, new_stab_gens)
orbit_reps[l + 1] = get_reps(orbits)
# line 13: amend sorted orbits
l += 1
temp_orbit = [computed_words[l - 1](point) for point
in basic_orbits[l]]
temp_orbit.sort(key=lambda point: base_ordering[point])
sorted_orbits[l] = temp_orbit
# lines 14 and 15: update variables used minimality tests
new_mu = degree + 1
for i in range(l):
if base[l] in res_basic_orbits_init_base[i]:
candidate = computed_words[i](base[i])
if base_ordering[candidate] > base_ordering[new_mu]:
new_mu = candidate
mu[l] = new_mu
update_nu(l)
# line 16: determine the new transversal element
c[l] = 0
temp_point = sorted_orbits[l][c[l]]
gamma = computed_words[l - 1]._array_form.index(temp_point)
u[l] = transversals[l][gamma]
# update computed words
computed_words[l] = rmul(computed_words[l - 1], u[l])
# lines 17 & 18: apply the tests to the group element found
g = computed_words[l]
temp_point = g(base[l])
if l == base_len - 1 and \
base_ordering[mu[l]] < \
base_ordering[temp_point] < base_ordering[nu[l]] and \
temp_point in orbit_reps[l] and \
tests[l](computed_words) and \
prop(g):
# line 19: reset the base of K
res_generators.append(g)
res_base = base[:]
# line 20: recalculate basic orbits (and transversals)
res_strong_gens.append(g)
res_strong_gens_distr = _distribute_gens_by_base(res_base,
res_strong_gens)
res_basic_orbits_init_base = \
[_orbit(degree, res_strong_gens_distr[i], res_base[i]) \
for i in range(base_len)]
# line 21: recalculate orbit representatives
# line 22: reset the search depth
orbit_reps[f] = get_reps(orbits)
l = f
# line 23: go up the tree until in the first branch not fully
# searched
while l >= 0 and c[l] == len(basic_orbits[l]) - 1:
l = l - 1
# line 24: if the entire tree is traversed, return K
if l == -1:
return PermutationGroup(res_generators)
# lines 25-27: update orbit representatives
if l < f:
# line 26
f = l
c[l] = 0
# line 27
temp_orbits = _orbits(degree, res_strong_gens_distr[f])
orbit_reps[f] = get_reps(temp_orbits)
# line 28: update variables used for minimality testing
mu[l] = degree + 1
temp_index = len(basic_orbits[l]) + 1 - \
len(res_basic_orbits_init_base[l])
if temp_index >= len(sorted_orbits[l]):
nu[l] = base_ordering[degree]
else:
nu[l] = sorted_orbits[l][temp_index]
# line 29: set the next element from the current branch and update
# accordingly
c[l] += 1
if l == 0:
gamma = sorted_orbits[l][c[l]]
else:
gamma = computed_words[l - 1]._array_form.index(sorted_orbits[l][c[l]])
u[l] = transversals[l][gamma]
if l == 0:
computed_words[l] = u[l]
else:
computed_words[l] = rmul(computed_words[l - 1], u[l])
@property
def transitivity_degree(self):
r"""Compute the degree of transitivity of the group.
Explanation
===========
A permutation group `G` acting on `\Omega = \{0, 1, \dots, n-1\}` is
``k``-fold transitive, if, for any `k` points
`(a_1, a_2, \dots, a_k) \in \Omega` and any `k` points
`(b_1, b_2, \dots, b_k) \in \Omega` there exists `g \in G` such that
`g(a_1) = b_1, g(a_2) = b_2, \dots, g(a_k) = b_k`
The degree of transitivity of `G` is the maximum ``k`` such that
`G` is ``k``-fold transitive. ([8])
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([1, 2, 0])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.transitivity_degree
3
See Also
========
is_transitive, orbit
"""
if self._transitivity_degree is None:
n = self.degree
G = self
# if G is k-transitive, a tuple (a_0,..,a_k)
# can be brought to (b_0,...,b_(k-1), b_k)
# where b_0,...,b_(k-1) are fixed points;
# consider the group G_k which stabilizes b_0,...,b_(k-1)
# if G_k is transitive on the subset excluding b_0,...,b_(k-1)
# then G is (k+1)-transitive
for i in range(n):
orb = G.orbit(i)
if len(orb) != n - i:
self._transitivity_degree = i
return i
G = G.stabilizer(i)
self._transitivity_degree = n
return n
else:
return self._transitivity_degree
def _p_elements_group(self, p):
'''
For an abelian p-group, return the subgroup consisting of
all elements of order p (and the identity)
'''
gens = self.generators[:]
gens = sorted(gens, key=lambda x: x.order(), reverse=True)
gens_p = [g**(g.order()/p) for g in gens]
gens_r = []
for i in range(len(gens)):
x = gens[i]
x_order = x.order()
# x_p has order p
x_p = x**(x_order/p)
if i > 0:
P = PermutationGroup(gens_p[:i])
else:
P = PermutationGroup(self.identity)
if x**(x_order/p) not in P:
gens_r.append(x**(x_order/p))
else:
# replace x by an element of order (x.order()/p)
# so that gens still generates G
g = P.generator_product(x_p, original=True)
for s in g:
x = x*s**-1
x_order = x_order/p
# insert x to gens so that the sorting is preserved
del gens[i]
del gens_p[i]
j = i - 1
while j < len(gens) and gens[j].order() >= x_order:
j += 1
gens = gens[:j] + [x] + gens[j:]
gens_p = gens_p[:j] + [x] + gens_p[j:]
return PermutationGroup(gens_r)
def _sylow_alt_sym(self, p):
'''
Return a p-Sylow subgroup of a symmetric or an
alternating group.
Explanation
===========
The algorithm for this is hinted at in [1], Chapter 4,
Exercise 4.
For Sym(n) with n = p^i, the idea is as follows. Partition
the interval [0..n-1] into p equal parts, each of length p^(i-1):
[0..p^(i-1)-1], [p^(i-1)..2*p^(i-1)-1]...[(p-1)*p^(i-1)..p^i-1].
Find a p-Sylow subgroup of Sym(p^(i-1)) (treated as a subgroup
of ``self``) acting on each of the parts. Call the subgroups
P_1, P_2...P_p. The generators for the subgroups P_2...P_p
can be obtained from those of P_1 by applying a "shifting"
permutation to them, that is, a permutation mapping [0..p^(i-1)-1]
to the second part (the other parts are obtained by using the shift
multiple times). The union of this permutation and the generators
of P_1 is a p-Sylow subgroup of ``self``.
For n not equal to a power of p, partition
[0..n-1] in accordance with how n would be written in base p.
E.g. for p=2 and n=11, 11 = 2^3 + 2^2 + 1 so the partition
is [[0..7], [8..9], {10}]. To generate a p-Sylow subgroup,
take the union of the generators for each of the parts.
For the above example, {(0 1), (0 2)(1 3), (0 4), (1 5)(2 7)}
from the first part, {(8 9)} from the second part and
nothing from the third. This gives 4 generators in total, and
the subgroup they generate is p-Sylow.
Alternating groups are treated the same except when p=2. In this
case, (0 1)(s s+1) should be added for an appropriate s (the start
of a part) for each part in the partitions.
See Also
========
sylow_subgroup, is_alt_sym
'''
n = self.degree
gens = []
identity = Permutation(n-1)
# the case of 2-sylow subgroups of alternating groups
# needs special treatment
alt = p == 2 and all(g.is_even for g in self.generators)
# find the presentation of n in base p
coeffs = []
m = n
while m > 0:
coeffs.append(m % p)
m = m // p
power = len(coeffs)-1
# for a symmetric group, gens[:i] is the generating
# set for a p-Sylow subgroup on [0..p**(i-1)-1]. For
# alternating groups, the same is given by gens[:2*(i-1)]
for i in range(1, power+1):
if i == 1 and alt:
# (0 1) shouldn't be added for alternating groups
continue
gen = Permutation([(j + p**(i-1)) % p**i for j in range(p**i)])
gens.append(identity*gen)
if alt:
gen = Permutation(0, 1)*gen*Permutation(0, 1)*gen
gens.append(gen)
# the first point in the current part (see the algorithm
# description in the docstring)
start = 0
while power > 0:
a = coeffs[power]
# make the permutation shifting the start of the first
# part ([0..p^i-1] for some i) to the current one
for _ in range(a):
shift = Permutation()
if start > 0:
for i in range(p**power):
shift = shift(i, start + i)
if alt:
gen = Permutation(0, 1)*shift*Permutation(0, 1)*shift
gens.append(gen)
j = 2*(power - 1)
else:
j = power
for i, gen in enumerate(gens[:j]):
if alt and i % 2 == 1:
continue
# shift the generator to the start of the
# partition part
gen = shift*gen*shift
gens.append(gen)
start += p**power
power = power-1
return gens
def sylow_subgroup(self, p):
'''
Return a p-Sylow subgroup of the group.
The algorithm is described in [1], Chapter 4, Section 7
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> D = DihedralGroup(6)
>>> S = D.sylow_subgroup(2)
>>> S.order()
4
>>> G = SymmetricGroup(6)
>>> S = G.sylow_subgroup(5)
>>> S.order()
5
>>> G1 = AlternatingGroup(3)
>>> G2 = AlternatingGroup(5)
>>> G3 = AlternatingGroup(9)
>>> S1 = G1.sylow_subgroup(3)
>>> S2 = G2.sylow_subgroup(3)
>>> S3 = G3.sylow_subgroup(3)
>>> len1 = len(S1.lower_central_series())
>>> len2 = len(S2.lower_central_series())
>>> len3 = len(S3.lower_central_series())
>>> len1 == len2
True
>>> len1 < len3
True
'''
from sympy.combinatorics.homomorphisms import (
orbit_homomorphism, block_homomorphism)
if not isprime(p):
raise ValueError("p must be a prime")
def is_p_group(G):
# check if the order of G is a power of p
# and return the power
m = G.order()
n = 0
while m % p == 0:
m = m/p
n += 1
if m == 1:
return True, n
return False, n
def _sylow_reduce(mu, nu):
# reduction based on two homomorphisms
# mu and nu with trivially intersecting
# kernels
Q = mu.image().sylow_subgroup(p)
Q = mu.invert_subgroup(Q)
nu = nu.restrict_to(Q)
R = nu.image().sylow_subgroup(p)
return nu.invert_subgroup(R)
order = self.order()
if order % p != 0:
return PermutationGroup([self.identity])
p_group, n = is_p_group(self)
if p_group:
return self
if self.is_alt_sym():
return PermutationGroup(self._sylow_alt_sym(p))
# if there is a non-trivial orbit with size not divisible
# by p, the sylow subgroup is contained in its stabilizer
# (by orbit-stabilizer theorem)
orbits = self.orbits()
non_p_orbits = [o for o in orbits if len(o) % p != 0 and len(o) != 1]
if non_p_orbits:
G = self.stabilizer(list(non_p_orbits[0]).pop())
return G.sylow_subgroup(p)
if not self.is_transitive():
# apply _sylow_reduce to orbit actions
orbits = sorted(orbits, key=len)
omega1 = orbits.pop()
omega2 = orbits[0].union(*orbits)
mu = orbit_homomorphism(self, omega1)
nu = orbit_homomorphism(self, omega2)
return _sylow_reduce(mu, nu)
blocks = self.minimal_blocks()
if len(blocks) > 1:
# apply _sylow_reduce to block system actions
mu = block_homomorphism(self, blocks[0])
nu = block_homomorphism(self, blocks[1])
return _sylow_reduce(mu, nu)
elif len(blocks) == 1:
block = list(blocks)[0]
if any(e != 0 for e in block):
# self is imprimitive
mu = block_homomorphism(self, block)
if not is_p_group(mu.image())[0]:
S = mu.image().sylow_subgroup(p)
return mu.invert_subgroup(S).sylow_subgroup(p)
# find an element of order p
g = self.random()
g_order = g.order()
while g_order % p != 0 or g_order == 0:
g = self.random()
g_order = g.order()
g = g**(g_order // p)
if order % p**2 != 0:
return PermutationGroup(g)
C = self.centralizer(g)
while C.order() % p**n != 0:
S = C.sylow_subgroup(p)
s_order = S.order()
Z = S.center()
P = Z._p_elements_group(p)
h = P.random()
C_h = self.centralizer(h)
while C_h.order() % p*s_order != 0:
h = P.random()
C_h = self.centralizer(h)
C = C_h
return C.sylow_subgroup(p)
def _block_verify(self, L, alpha):
delta = sorted(self.orbit(alpha))
# p[i] will be the number of the block
# delta[i] belongs to
p = [-1]*len(delta)
blocks = [-1]*len(delta)
B = [[]] # future list of blocks
u = [0]*len(delta) # u[i] in L s.t. alpha^u[i] = B[0][i]
t = L.orbit_transversal(alpha, pairs=True)
for a, beta in t:
B[0].append(a)
i_a = delta.index(a)
p[i_a] = 0
blocks[i_a] = alpha
u[i_a] = beta
rho = 0
m = 0 # number of blocks - 1
while rho <= m:
beta = B[rho][0]
for g in self.generators:
d = beta^g
i_d = delta.index(d)
sigma = p[i_d]
if sigma < 0:
# define a new block
m += 1
sigma = m
u[i_d] = u[delta.index(beta)]*g
p[i_d] = sigma
rep = d
blocks[i_d] = rep
newb = [rep]
for gamma in B[rho][1:]:
i_gamma = delta.index(gamma)
d = gamma^g
i_d = delta.index(d)
if p[i_d] < 0:
u[i_d] = u[i_gamma]*g
p[i_d] = sigma
blocks[i_d] = rep
newb.append(d)
else:
# B[rho] is not a block
s = u[i_gamma]*g*u[i_d]**(-1)
return False, s
B.append(newb)
else:
for h in B[rho][1:]:
if h^g not in B[sigma]:
# B[rho] is not a block
s = u[delta.index(beta)]*g*u[i_d]**(-1)
return False, s
rho += 1
return True, blocks
def _verify(H, K, phi, z, alpha):
'''
Return a list of relators ``rels`` in generators ``gens`_h` that
are mapped to ``H.generators`` by ``phi`` so that given a finite
presentation <gens_k | rels_k> of ``K`` on a subset of ``gens_h``
<gens_h | rels_k + rels> is a finite presentation of ``H``.
Explanation
===========
``H`` should be generated by the union of ``K.generators`` and ``z``
(a single generator), and ``H.stabilizer(alpha) == K``; ``phi`` is a
canonical injection from a free group into a permutation group
containing ``H``.
The algorithm is described in [1], Chapter 6.
Examples
========
>>> from sympy.combinatorics import free_group, Permutation, PermutationGroup
>>> from sympy.combinatorics.homomorphisms import homomorphism
>>> from sympy.combinatorics.fp_groups import FpGroup
>>> H = PermutationGroup(Permutation(0, 2), Permutation (1, 5))
>>> K = PermutationGroup(Permutation(5)(0, 2))
>>> F = free_group("x_0 x_1")[0]
>>> gens = F.generators
>>> phi = homomorphism(F, H, F.generators, H.generators)
>>> rels_k = [gens[0]**2] # relators for presentation of K
>>> z= Permutation(1, 5)
>>> check, rels_h = H._verify(K, phi, z, 1)
>>> check
True
>>> rels = rels_k + rels_h
>>> G = FpGroup(F, rels) # presentation of H
>>> G.order() == H.order()
True
See also
========
strong_presentation, presentation, stabilizer
'''
orbit = H.orbit(alpha)
beta = alpha^(z**-1)
K_beta = K.stabilizer(beta)
# orbit representatives of K_beta
gammas = [alpha, beta]
orbits = list({tuple(K_beta.orbit(o)) for o in orbit})
orbit_reps = [orb[0] for orb in orbits]
for rep in orbit_reps:
if rep not in gammas:
gammas.append(rep)
# orbit transversal of K
betas = [alpha, beta]
transversal = {alpha: phi.invert(H.identity), beta: phi.invert(z**-1)}
for s, g in K.orbit_transversal(beta, pairs=True):
if s not in transversal:
transversal[s] = transversal[beta]*phi.invert(g)
union = K.orbit(alpha).union(K.orbit(beta))
while (len(union) < len(orbit)):
for gamma in gammas:
if gamma in union:
r = gamma^z
if r not in union:
betas.append(r)
transversal[r] = transversal[gamma]*phi.invert(z)
for s, g in K.orbit_transversal(r, pairs=True):
if s not in transversal:
transversal[s] = transversal[r]*phi.invert(g)
union = union.union(K.orbit(r))
break
# compute relators
rels = []
for b in betas:
k_gens = K.stabilizer(b).generators
for y in k_gens:
new_rel = transversal[b]
gens = K.generator_product(y, original=True)
for g in gens[::-1]:
new_rel = new_rel*phi.invert(g)
new_rel = new_rel*transversal[b]**-1
perm = phi(new_rel)
try:
gens = K.generator_product(perm, original=True)
except ValueError:
return False, perm
for g in gens:
new_rel = new_rel*phi.invert(g)**-1
if new_rel not in rels:
rels.append(new_rel)
for gamma in gammas:
new_rel = transversal[gamma]*phi.invert(z)*transversal[gamma^z]**-1
perm = phi(new_rel)
try:
gens = K.generator_product(perm, original=True)
except ValueError:
return False, perm
for g in gens:
new_rel = new_rel*phi.invert(g)**-1
if new_rel not in rels:
rels.append(new_rel)
return True, rels
def strong_presentation(self):
'''
Return a strong finite presentation of group. The generators
of the returned group are in the same order as the strong
generators of group.
The algorithm is based on Sims' Verify algorithm described
in [1], Chapter 6.
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> P = DihedralGroup(4)
>>> G = P.strong_presentation()
>>> P.order() == G.order()
True
See Also
========
presentation, _verify
'''
from sympy.combinatorics.fp_groups import (FpGroup,
simplify_presentation)
from sympy.combinatorics.free_groups import free_group
from sympy.combinatorics.homomorphisms import (block_homomorphism,
homomorphism, GroupHomomorphism)
strong_gens = self.strong_gens[:]
stabs = self.basic_stabilizers[:]
base = self.base[:]
# injection from a free group on len(strong_gens)
# generators into G
gen_syms = [('x_%d'%i) for i in range(len(strong_gens))]
F = free_group(', '.join(gen_syms))[0]
phi = homomorphism(F, self, F.generators, strong_gens)
H = PermutationGroup(self.identity)
while stabs:
alpha = base.pop()
K = H
H = stabs.pop()
new_gens = [g for g in H.generators if g not in K]
if K.order() == 1:
z = new_gens.pop()
rels = [F.generators[-1]**z.order()]
intermediate_gens = [z]
K = PermutationGroup(intermediate_gens)
# add generators one at a time building up from K to H
while new_gens:
z = new_gens.pop()
intermediate_gens = [z] + intermediate_gens
K_s = PermutationGroup(intermediate_gens)
orbit = K_s.orbit(alpha)
orbit_k = K.orbit(alpha)
# split into cases based on the orbit of K_s
if orbit_k == orbit:
if z in K:
rel = phi.invert(z)
perm = z
else:
t = K.orbit_rep(alpha, alpha^z)
rel = phi.invert(z)*phi.invert(t)**-1
perm = z*t**-1
for g in K.generator_product(perm, original=True):
rel = rel*phi.invert(g)**-1
new_rels = [rel]
elif len(orbit_k) == 1:
# `success` is always true because `strong_gens`
# and `base` are already a verified BSGS. Later
# this could be changed to start with a randomly
# generated (potential) BSGS, and then new elements
# would have to be appended to it when `success`
# is false.
success, new_rels = K_s._verify(K, phi, z, alpha)
else:
# K.orbit(alpha) should be a block
# under the action of K_s on K_s.orbit(alpha)
check, block = K_s._block_verify(K, alpha)
if check:
# apply _verify to the action of K_s
# on the block system; for convenience,
# add the blocks as additional points
# that K_s should act on
t = block_homomorphism(K_s, block)
m = t.codomain.degree # number of blocks
d = K_s.degree
# conjugating with p will shift
# permutations in t.image() to
# higher numbers, e.g.
# p*(0 1)*p = (m m+1)
p = Permutation()
for i in range(m):
p *= Permutation(i, i+d)
t_img = t.images
# combine generators of K_s with their
# action on the block system
images = {g: g*p*t_img[g]*p for g in t_img}
for g in self.strong_gens[:-len(K_s.generators)]:
images[g] = g
K_s_act = PermutationGroup(list(images.values()))
f = GroupHomomorphism(self, K_s_act, images)
K_act = PermutationGroup([f(g) for g in K.generators])
success, new_rels = K_s_act._verify(K_act, f.compose(phi), f(z), d)
for n in new_rels:
if n not in rels:
rels.append(n)
K = K_s
group = FpGroup(F, rels)
return simplify_presentation(group)
def presentation(self, eliminate_gens=True):
'''
Return an `FpGroup` presentation of the group.
The algorithm is described in [1], Chapter 6.1.
'''
from sympy.combinatorics.fp_groups import (FpGroup,
simplify_presentation)
from sympy.combinatorics.coset_table import CosetTable
from sympy.combinatorics.free_groups import free_group
from sympy.combinatorics.homomorphisms import homomorphism
if self._fp_presentation:
return self._fp_presentation
def _factor_group_by_rels(G, rels):
if isinstance(G, FpGroup):
rels.extend(G.relators)
return FpGroup(G.free_group, list(set(rels)))
return FpGroup(G, rels)
gens = self.generators
len_g = len(gens)
if len_g == 1:
order = gens[0].order()
# handle the trivial group
if order == 1:
return free_group([])[0]
F, x = free_group('x')
return FpGroup(F, [x**order])
if self.order() > 20:
half_gens = self.generators[0:(len_g+1)//2]
else:
half_gens = []
H = PermutationGroup(half_gens)
H_p = H.presentation()
len_h = len(H_p.generators)
C = self.coset_table(H)
n = len(C) # subgroup index
gen_syms = [('x_%d'%i) for i in range(len(gens))]
F = free_group(', '.join(gen_syms))[0]
# mapping generators of H_p to those of F
images = [F.generators[i] for i in range(len_h)]
R = homomorphism(H_p, F, H_p.generators, images, check=False)
# rewrite relators
rels = R(H_p.relators)
G_p = FpGroup(F, rels)
# injective homomorphism from G_p into self
T = homomorphism(G_p, self, G_p.generators, gens)
C_p = CosetTable(G_p, [])
C_p.table = [[None]*(2*len_g) for i in range(n)]
# initiate the coset transversal
transversal = [None]*n
transversal[0] = G_p.identity
# fill in the coset table as much as possible
for i in range(2*len_h):
C_p.table[0][i] = 0
gamma = 1
for alpha, x in product(range(n), range(2*len_g)):
beta = C[alpha][x]
if beta == gamma:
gen = G_p.generators[x//2]**((-1)**(x % 2))
transversal[beta] = transversal[alpha]*gen
C_p.table[alpha][x] = beta
C_p.table[beta][x + (-1)**(x % 2)] = alpha
gamma += 1
if gamma == n:
break
C_p.p = list(range(n))
beta = x = 0
while not C_p.is_complete():
# find the first undefined entry
while C_p.table[beta][x] == C[beta][x]:
x = (x + 1) % (2*len_g)
if x == 0:
beta = (beta + 1) % n
# define a new relator
gen = G_p.generators[x//2]**((-1)**(x % 2))
new_rel = transversal[beta]*gen*transversal[C[beta][x]]**-1
perm = T(new_rel)
nxt = G_p.identity
for s in H.generator_product(perm, original=True):
nxt = nxt*T.invert(s)**-1
new_rel = new_rel*nxt
# continue coset enumeration
G_p = _factor_group_by_rels(G_p, [new_rel])
C_p.scan_and_fill(0, new_rel)
C_p = G_p.coset_enumeration([], strategy="coset_table",
draft=C_p, max_cosets=n, incomplete=True)
self._fp_presentation = simplify_presentation(G_p)
return self._fp_presentation
def polycyclic_group(self):
"""
Return the PolycyclicGroup instance with below parameters:
Explanation
===========
* pc_sequence : Polycyclic sequence is formed by collecting all
the missing generators between the adjacent groups in the
derived series of given permutation group.
* pc_series : Polycyclic series is formed by adding all the missing
generators of ``der[i+1]`` in ``der[i]``, where ``der`` represents
the derived series.
* relative_order : A list, computed by the ratio of adjacent groups in
pc_series.
"""
from sympy.combinatorics.pc_groups import PolycyclicGroup
if not self.is_polycyclic:
raise ValueError("The group must be solvable")
der = self.derived_series()
pc_series = []
pc_sequence = []
relative_order = []
pc_series.append(der[-1])
der.reverse()
for i in range(len(der)-1):
H = der[i]
for g in der[i+1].generators:
if g not in H:
H = PermutationGroup([g] + H.generators)
pc_series.insert(0, H)
pc_sequence.insert(0, g)
G1 = pc_series[0].order()
G2 = pc_series[1].order()
relative_order.insert(0, G1 // G2)
return PolycyclicGroup(pc_sequence, pc_series, relative_order, collector=None)
def _orbit(degree, generators, alpha, action='tuples'):
r"""Compute the orbit of alpha `\{g(\alpha) | g \in G\}` as a set.
Explanation
===========
The time complexity of the algorithm used here is `O(|Orb|*r)` where
`|Orb|` is the size of the orbit and ``r`` is the number of generators of
the group. For a more detailed analysis, see [1], p.78, [2], pp. 19-21.
Here alpha can be a single point, or a list of points.
If alpha is a single point, the ordinary orbit is computed.
if alpha is a list of points, there are three available options:
'union' - computes the union of the orbits of the points in the list
'tuples' - computes the orbit of the list interpreted as an ordered
tuple under the group action ( i.e., g((1, 2, 3)) = (g(1), g(2), g(3)) )
'sets' - computes the orbit of the list interpreted as a sets
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> from sympy.combinatorics.perm_groups import _orbit
>>> a = Permutation([1, 2, 0, 4, 5, 6, 3])
>>> G = PermutationGroup([a])
>>> _orbit(G.degree, G.generators, 0)
{0, 1, 2}
>>> _orbit(G.degree, G.generators, [0, 4], 'union')
{0, 1, 2, 3, 4, 5, 6}
See Also
========
orbit, orbit_transversal
"""
if not hasattr(alpha, '__getitem__'):
alpha = [alpha]
gens = [x._array_form for x in generators]
if len(alpha) == 1 or action == 'union':
orb = alpha
used = [False]*degree
for el in alpha:
used[el] = True
for b in orb:
for gen in gens:
temp = gen[b]
if used[temp] == False:
orb.append(temp)
used[temp] = True
return set(orb)
elif action == 'tuples':
alpha = tuple(alpha)
orb = [alpha]
used = {alpha}
for b in orb:
for gen in gens:
temp = tuple([gen[x] for x in b])
if temp not in used:
orb.append(temp)
used.add(temp)
return set(orb)
elif action == 'sets':
alpha = frozenset(alpha)
orb = [alpha]
used = {alpha}
for b in orb:
for gen in gens:
temp = frozenset([gen[x] for x in b])
if temp not in used:
orb.append(temp)
used.add(temp)
return {tuple(x) for x in orb}
def _orbits(degree, generators):
"""Compute the orbits of G.
If ``rep=False`` it returns a list of sets else it returns a list of
representatives of the orbits
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import _orbits
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> _orbits(a.size, [a, b])
[{0, 1, 2}]
"""
orbs = []
sorted_I = list(range(degree))
I = set(sorted_I)
while I:
i = sorted_I[0]
orb = _orbit(degree, generators, i)
orbs.append(orb)
# remove all indices that are in this orbit
I -= orb
sorted_I = [i for i in sorted_I if i not in orb]
return orbs
def _orbit_transversal(degree, generators, alpha, pairs, af=False, slp=False):
r"""Computes a transversal for the orbit of ``alpha`` as a set.
Explanation
===========
generators generators of the group ``G``
For a permutation group ``G``, a transversal for the orbit
`Orb = \{g(\alpha) | g \in G\}` is a set
`\{g_\beta | g_\beta(\alpha) = \beta\}` for `\beta \in Orb`.
Note that there may be more than one possible transversal.
If ``pairs`` is set to ``True``, it returns the list of pairs
`(\beta, g_\beta)`. For a proof of correctness, see [1], p.79
if ``af`` is ``True``, the transversal elements are given in
array form.
If `slp` is `True`, a dictionary `{beta: slp_beta}` is returned
for `\beta \in Orb` where `slp_beta` is a list of indices of the
generators in `generators` s.t. if `slp_beta = [i_1 \dots i_n]`
`g_\beta = generators[i_n] \times \dots \times generators[i_1]`.
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> from sympy.combinatorics.perm_groups import _orbit_transversal
>>> G = DihedralGroup(6)
>>> _orbit_transversal(G.degree, G.generators, 0, False)
[(5), (0 1 2 3 4 5), (0 5)(1 4)(2 3), (0 2 4)(1 3 5), (5)(0 4)(1 3), (0 3)(1 4)(2 5)]
"""
tr = [(alpha, list(range(degree)))]
slp_dict = {alpha: []}
used = [False]*degree
used[alpha] = True
gens = [x._array_form for x in generators]
for x, px in tr:
px_slp = slp_dict[x]
for gen in gens:
temp = gen[x]
if used[temp] == False:
slp_dict[temp] = [gens.index(gen)] + px_slp
tr.append((temp, _af_rmul(gen, px)))
used[temp] = True
if pairs:
if not af:
tr = [(x, _af_new(y)) for x, y in tr]
if not slp:
return tr
return tr, slp_dict
if af:
tr = [y for _, y in tr]
if not slp:
return tr
return tr, slp_dict
tr = [_af_new(y) for _, y in tr]
if not slp:
return tr
return tr, slp_dict
def _stabilizer(degree, generators, alpha):
r"""Return the stabilizer subgroup of ``alpha``.
Explanation
===========
The stabilizer of `\alpha` is the group `G_\alpha =
\{g \in G | g(\alpha) = \alpha\}`.
For a proof of correctness, see [1], p.79.
degree : degree of G
generators : generators of G
Examples
========
>>> from sympy.combinatorics.perm_groups import _stabilizer
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> G = DihedralGroup(6)
>>> _stabilizer(G.degree, G.generators, 5)
[(5)(0 4)(1 3), (5)]
See Also
========
orbit
"""
orb = [alpha]
table = {alpha: list(range(degree))}
table_inv = {alpha: list(range(degree))}
used = [False]*degree
used[alpha] = True
gens = [x._array_form for x in generators]
stab_gens = []
for b in orb:
for gen in gens:
temp = gen[b]
if used[temp] is False:
gen_temp = _af_rmul(gen, table[b])
orb.append(temp)
table[temp] = gen_temp
table_inv[temp] = _af_invert(gen_temp)
used[temp] = True
else:
schreier_gen = _af_rmuln(table_inv[temp], gen, table[b])
if schreier_gen not in stab_gens:
stab_gens.append(schreier_gen)
return [_af_new(x) for x in stab_gens]
PermGroup = PermutationGroup
class SymmetricPermutationGroup(Basic):
"""
The class defining the lazy form of SymmetricGroup.
deg : int
"""
def __new__(cls, deg):
deg = _sympify(deg)
obj = Basic.__new__(cls, deg)
return obj
def __init__(self, *args, **kwargs):
self._deg = self.args[0]
self._order = None
def __contains__(self, i):
"""Return ``True`` if *i* is contained in SymmetricPermutationGroup.
Examples
========
>>> from sympy.combinatorics import Permutation, SymmetricPermutationGroup
>>> G = SymmetricPermutationGroup(4)
>>> Permutation(1, 2, 3) in G
True
"""
if not isinstance(i, Permutation):
raise TypeError("A SymmetricPermutationGroup contains only Permutations as "
"elements, not elements of type %s" % type(i))
return i.size == self.degree
def order(self):
"""
Return the order of the SymmetricPermutationGroup.
Examples
========
>>> from sympy.combinatorics import SymmetricPermutationGroup
>>> G = SymmetricPermutationGroup(4)
>>> G.order()
24
"""
if self._order is not None:
return self._order
n = self._deg
self._order = factorial(n)
return self._order
@property
def degree(self):
"""
Return the degree of the SymmetricPermutationGroup.
Examples
========
>>> from sympy.combinatorics import SymmetricPermutationGroup
>>> G = SymmetricPermutationGroup(4)
>>> G.degree
4
"""
return self._deg
@property
def identity(self):
'''
Return the identity element of the SymmetricPermutationGroup.
Examples
========
>>> from sympy.combinatorics import SymmetricPermutationGroup
>>> G = SymmetricPermutationGroup(4)
>>> G.identity()
(3)
'''
return _af_new(list(range(self._deg)))
class Coset(Basic):
"""A left coset of a permutation group with respect to an element.
Parameters
==========
g : Permutation
H : PermutationGroup
dir : "+" or "-", If not specified by default it will be "+"
here ``dir`` specified the type of coset "+" represent the
right coset and "-" represent the left coset.
G : PermutationGroup, optional
The group which contains *H* as its subgroup and *g* as its
element.
If not specified, it would automatically become a symmetric
group ``SymmetricPermutationGroup(g.size)`` and
``SymmetricPermutationGroup(H.degree)`` if ``g.size`` and ``H.degree``
are matching.``SymmetricPermutationGroup`` is a lazy form of SymmetricGroup
used for representation purpose.
"""
def __new__(cls, g, H, G=None, dir="+"):
g = _sympify(g)
if not isinstance(g, Permutation):
raise NotImplementedError
H = _sympify(H)
if not isinstance(H, PermutationGroup):
raise NotImplementedError
if G is not None:
G = _sympify(G)
if not isinstance(G, (PermutationGroup, SymmetricPermutationGroup)):
raise NotImplementedError
if not H.is_subgroup(G):
raise ValueError("{} must be a subgroup of {}.".format(H, G))
if g not in G:
raise ValueError("{} must be an element of {}.".format(g, G))
else:
g_size = g.size
h_degree = H.degree
if g_size != h_degree:
raise ValueError(
"The size of the permutation {} and the degree of "
"the permutation group {} should be matching "
.format(g, H))
G = SymmetricPermutationGroup(g.size)
if isinstance(dir, str):
dir = Symbol(dir)
elif not isinstance(dir, Symbol):
raise TypeError("dir must be of type basestring or "
"Symbol, not %s" % type(dir))
if str(dir) not in ('+', '-'):
raise ValueError("dir must be one of '+' or '-' not %s" % dir)
obj = Basic.__new__(cls, g, H, G, dir)
return obj
def __init__(self, *args, **kwargs):
self._dir = self.args[3]
@property
def is_left_coset(self):
"""
Check if the coset is left coset that is ``gH``.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup, Coset
>>> a = Permutation(1, 2)
>>> b = Permutation(0, 1)
>>> G = PermutationGroup([a, b])
>>> cst = Coset(a, G, dir="-")
>>> cst.is_left_coset
True
"""
return str(self._dir) == '-'
@property
def is_right_coset(self):
"""
Check if the coset is right coset that is ``Hg``.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup, Coset
>>> a = Permutation(1, 2)
>>> b = Permutation(0, 1)
>>> G = PermutationGroup([a, b])
>>> cst = Coset(a, G, dir="+")
>>> cst.is_right_coset
True
"""
return str(self._dir) == '+'
def as_list(self):
"""
Return all the elements of coset in the form of list.
"""
g = self.args[0]
H = self.args[1]
cst = []
if str(self._dir) == '+':
for h in H.elements:
cst.append(h*g)
else:
for h in H.elements:
cst.append(g*h)
return cst
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