Source code for sympy.combinatorics.perm_groups

from __future__ import print_function, division

from random import randrange, choice
from math import log

from sympy.core import Basic
from sympy.core.compatibility import range
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.functions.combinatorial.factorials import factorial
from sympy.ntheory import sieve
from sympy.utilities.iterables import has_variety, is_sequence, uniq
from sympy.utilities.randtest import _randrange
from itertools import islice

rmul = Permutation.rmul_with_af
_af_new = Permutation._af_new


[docs]class PermutationGroup(Basic): """The class defining a Permutation group. 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 >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.permutations import Cycle >>> from sympy.combinatorics.polyhedron import Polyhedron >>> from sympy.combinatorics.perm_groups import PermutationGroup The permutations corresponding to motion of the front, right and bottom face of a 2x2 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 2x2 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] http://en.wikipedia.org/wiki/Schreier_vector [4] http://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] http://en.wikipedia.org/wiki/Block_%28permutation_group_theory%29 [7] http://www.algorithmist.com/index.php/Union_Find [8] http://en.wikipedia.org/wiki/Multiply_transitive_group#Multiply_transitive_groups [9] http://en.wikipedia.org/wiki/Center_%28group_theory%29 [10] http://en.wikipedia.org/wiki/Centralizer_and_normalizer [11] http://groupprops.subwiki.org/wiki/Derived_subgroup [12] http://en.wikipedia.org/wiki/Nilpotent_group [13] http://www.math.colostate.edu/~hulpke/CGT/cgtnotes.pdf """ is_group = True def __new__(cls, *args, **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 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 kwargs.pop('dups', True): args = list(uniq([_af_new(list(a)) for a in args])) obj = Basic.__new__(cls, *args, **kwargs) obj._generators = args obj._order = None obj._center = [] obj._is_abelian = None obj._is_transitive = None obj._is_sym = None obj._is_alt = None obj._is_primitive = None obj._is_nilpotent = None obj._is_solvable = None obj._is_trivial = None obj._transitivity_degree = None obj._max_div = None obj._r = len(obj._generators) obj._degree = obj._generators[0].size # these attributes are assigned after running schreier_sims obj._base = [] obj._strong_gens = [] obj._basic_orbits = [] obj._transversals = [] # these attributes are assigned after running _random_pr_init obj._random_gens = [] return obj 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 __eq__(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 >>> from sympy.combinatorics.perm_groups import 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 == 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 __hash__(self): return super(PermutationGroup, self).__hash__() def __mul__(self, other): """Return the direct product of two permutation groups as a permutation group. 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.perm_groups import PermutationGroup >>> 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 """ 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. 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. 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] http://www.algorithmist.com/index.php/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. 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] http://www.algorithmist.com/index.php/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): """Return a base from the Schreier-Sims algorithm. For a permutation group `G`, a base is a sequence of points `B = (b_1, b_2, ..., 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
[docs] 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. If a base for a group `G` is given by `(b_1, b_2, ..., b_k)`, this function returns a base `(b_1, b_2, ..., b_{i+1}, b_i, ..., 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): """ Return the basic orbits relative to a base and strong generating set. If `(b_1, b_2, ..., b_k)` is a base for a group `G`, and `G^{(i)} = G_{b_1, b_2, ..., 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): """ Return a chain of stabilizers relative to a base and strong generating set. The ``i``-th basic stabilizer `G^{(i)}` relative to a base `(b_1, b_2, ..., b_k)` is `G_{b_1, b_2, ..., 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. 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
[docs] 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 T = g_stabs[len(h_stabs)]._elements t_len = len(T) l = len(h_stabs)-1 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 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)
[docs] 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 from itertools import chain, product 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 fp_groups) # 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
[docs] def center(self): r""" Return the center of a permutation group. 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.perm_groups import PermutationGroup >>> 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)
[docs] def centralizer(self, other): r""" Return the centralizer of a group/set/element. 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]))
[docs] def commutator(self, G, H): """ Return the commutator of two subgroups. 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
[docs] def coset_factor(self, g, factor_index=False): """Return ``G``'s (self's) coset factorization of ``g`` 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 >>> Permutation.print_cyclic = True >>> 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 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
[docs] def coset_rank(self, g): """rank using Schreier-Sims representation The coset rank of ``g`` is the ordering number in which it appears in the lexicographic listing according to the coset decomposition 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 >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> a = Permutation(0, 1, 3, 7, 6, 4)(2, 5) >>> b = Permutation(0, 1, 3, 2)(4, 5, 7, 6) >>> G = PermutationGroup([a, b]) >>> c = Permutation(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
[docs] 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._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. 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 >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import 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 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 {(3), (2 3), (3)(1 2), (1 2 3), (1 3 2), (1 3)} """ 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))
[docs] def derived_series(self): r"""Return the derived series for the group. 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 next = self.derived_subgroup() while not current.is_subgroup(next): res.append(next) current = next next = next.derived_subgroup() return res
[docs] def derived_subgroup(self): """Compute the derived subgroup. 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 >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> a = Permutation([1, 0, 2, 4, 3]) >>> b = Permutation([0, 1, 3, 2, 4]) >>> G = PermutationGroup([a, b]) >>> C = G.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] gens_inv = [_af_invert(p) for p in gens] 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 not ct in set_commutators: set_commutators.add(ct) cms = [_af_new(p) for p in set_commutators] G2 = self.normal_closure(cms) return G2
[docs] def generate(self, method="coset", af=False): """Return iterator to generate the elements of the group 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 Permutation >>> Permutation.print_cyclic = True >>> 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), (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)
[docs] def generate_dimino(self, af=False): """Yield group elements using Dimino's algorithm If af == True it yields the array form of the permutations References ========== [1] The Implementation of Various Algorithms for Permutation Groups in the Computer Algebra System: AXIOM, N.J. Doye, M.Sc. Thesis Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> a = Permutation([0, 2, 1, 3]) >>> b = Permutation([0, 2, 3, 1]) >>> g = PermutationGroup([a, b]) >>> list(g.generate_dimino(af=True)) [[0, 1, 2, 3], [0, 2, 1, 3], [0, 2, 3, 1], [0, 1, 3, 2], [0, 3, 2, 1], [0, 3, 1, 2]] """ idn = 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)
[docs] 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 >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> a = Permutation([0, 2, 1, 3]) >>> b = Permutation([0, 2, 3, 1]) >>> g = PermutationGroup([a, b]) >>> list(g.generate_schreier_sims(af=True)) [[0, 1, 2, 3], [0, 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 >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import 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
[docs] def contains(self, g, strict=True): """Test if permutation ``g`` belong to self, ``G``. 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 >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import 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, has, in """ 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_abelian(self): """Test if the group is Abelian. Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import 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
[docs] def is_alt_sym(self, eps=0.05, _random_prec=None): r"""Monte Carlo test for the symmetric/alternating group for degrees >= 8. 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. 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.perm_groups import PermutationGroup >>> 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 None: n = self.degree if n < 8: return False if not self.is_transitive(): return False 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) for i in range(N_eps): perm = self.random_pr() if _check_cycles_alt_sym(perm): return True return False else: for i in range(_random_prec['N_eps']): perm = _random_prec[i] if _check_cycles_alt_sym(perm): return True return False
@property def is_nilpotent(self): """Test if the group is nilpotent. 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
[docs] def is_normal(self, gr, strict=True): """Test if ``G=self`` is a normal subgroup of ``gr``. G is normal in gr if for each g2 in G, g1 in gr, ``g = g1*g2*g1**-1`` belongs to G It is sufficient to check this for each g1 in gr.generators and g2 in G.generators. Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> a = Permutation([1, 2, 0]) >>> b = Permutation([1, 0, 2]) >>> G = PermutationGroup([a, b]) >>> G1 = PermutationGroup([a, Permutation([2, 0, 1])]) >>> G1.is_normal(G) True """ d_self = self.degree d_gr = gr.degree 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
[docs] def is_primitive(self, randomized=True): """Test if a group is primitive. 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.perm_groups import PermutationGroup >>> 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 n = self.degree 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 and self.minimal_block([0, x]) != [0]*n: self._is_primitive = False return False self._is_primitive = True return True
@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: 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
[docs] 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.named_groups 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 not isinstance(G, PermutationGroup): return False if self == G: 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)
[docs] def is_transitive(self, strict=True): """Test if the group is transitive. 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.permutations import Permutation >>> from sympy.combinatorics.perm_groups import 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 >>> from sympy.combinatorics.perm_groups import 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
[docs] 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 next = self.commutator(self, current) while not current.is_subgroup(next): res.append(next) current = next next = self.commutator(self, current) return res
@property def max_div(self): """Maximum proper divisor of the degree of a permutation group. Notes ===== 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 >>> from sympy.combinatorics.perm_groups import 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
[docs] def minimal_block(self, points): r"""For a transitive group, finds the block system generated by ``points``. 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.perm_groups import PermutationGroup >>> from sympy.combinatorics.named_groups import DihedralGroup >>> D = DihedralGroup(10) >>> D.minimal_block([0, 5]) [0, 6, 2, 8, 4, 0, 6, 2, 8, 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: temp = 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(temp, parents) # union has side effects: performs union by rank on the list # of representatives temp = self._union_find_merge(gen(temp), 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) return parents
[docs] def normal_closure(self, other, k=10): r"""Return the normal closure of a subgroup/set of permutations. 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 i 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]))
[docs] def orbit(self, alpha, action='tuples'): r"""Compute the orbit of alpha `\{g(\alpha) | g \in G\}` as a set. 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 >>> from sympy.combinatorics.perm_groups import 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)
[docs] def orbit_rep(self, alpha, beta, schreier_vector=None): """Return a group element which sends ``alpha`` to ``beta``. 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 import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> 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)))
[docs] def orbit_transversal(self, alpha, pairs=False): r"""Computes a transversal for the orbit of ``alpha`` as a set. 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 import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> 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)
[docs] def orbits(self, rep=False): """Return the orbits of ``self``, ordered according to lowest element in each orbit. Examples ======== >>> from sympy.combinatorics.permutations import Permutation >>> from sympy.combinatorics.perm_groups import 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)
[docs] 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.permutations import Permutation >>> from sympy.combinatorics.perm_groups import 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 != 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 basic_transversals = self.basic_transversals m = 1 for x in basic_transversals: m *= len(x) self._order = m return m
[docs] def pointwise_stabilizer(self, points, incremental=True): r"""Return the pointwise stabilizer for a set of points. 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)
[docs] 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 >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import 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.') randrange = _randrange(seed) # start with the identity permutation result = Permutation(list(range(self.degree))) m = len(self) for i in range(n): p = self[randrange(m)] result = rmul(result, p) return result
[docs] def random(self, af=False): """Return a random group element """ rank = randrange(self.order()) return self.coset_unrank(rank, af)
[docs] def random_pr(self, gen_count=11, iterations=50, _random_prec=None): """Return a random group element using product replacement. 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])
[docs] 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)
[docs] def schreier_sims(self): """Schreier-Sims algorithm. 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.permutations import Permutation >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> a = Permutation([0, 2, 1]) >>> b = Permutation([1, 0, 2]) >>> G = PermutationGroup([a, b]) >>> G.schreier_sims() >>> G.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): base, strong_gens = self.schreier_sims_incremental(base=base) self._base = base self._strong_gens = strong_gens if not base: self._transversals = [] self._basic_orbits = [] return strong_gens_distr = _distribute_gens_by_base(base, strong_gens) basic_orbits, transversals = _orbits_transversals_from_bsgs(base,\ strong_gens_distr) self._transversals = transversals self._basic_orbits = [sorted(x) for x in basic_orbits]
[docs] def schreier_sims_incremental(self, base=None, gens=None): """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``. 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.perm_groups import PermutationGroup >>> 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: 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) # initialize the basic stabilizers, basic orbits and basic transversals orbs = {} transversals = {} base_len = len(_base) for i in range(base_len): transversals[i] = dict(_orbit_transversal(degree, strong_gens_distr[i], _base[i], pairs=True, af=True)) 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 gen in strong_gens_distr[i]: gb = gen._array_form[beta] u1 = transversals[i][gb] g1 = _af_rmul(gen._array_form, u_beta) 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) h, j = _strip_af(schreier_gen, _base, orbs, transversals, i) 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) for l in range(i + 1, j): strong_gens_distr[l].append(h) transversals[l] =\ dict(_orbit_transversal(degree, strong_gens_distr[l], _base[l], pairs=True, af=True)) 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 # build the strong generating set strong_gens = list(uniq(i for gens in strong_gens_distr for i in gens)) return _base, strong_gens
[docs] def schreier_sims_random(self, base=None, gens=None, consec_succ=10, _random_prec=None): r"""Randomized Schreier-Sims algorithm. 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.perm_groups import PermutationGroup >>> 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
[docs] def schreier_vector(self, alpha): """Computes the schreier vector for ``alpha``. 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 doesn't belong to the orbit. For the algorithm and its correctness, see [2], pp.78-80. Examples ======== >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> from sympy.combinatorics.permutations import Permutation >>> 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
[docs] def stabilizer(self, alpha): r"""Return the stabilizer subgroup of ``alpha``. 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 import Permutation >>> Permutation.print_cyclic = True >>> from sympy.combinatorics.perm_groups import PermutationGroup >>> from sympy.combinatorics.named_groups import DihedralGroup >>> G = DihedralGroup(6) >>> G.stabilizer(5) PermutationGroup([ (5)(0 4)(1 3), (5)]) See Also ======== orbit """ return PermGroup(_stabilizer(self._degree, self._generators, alpha))
@property def strong_gens(self): """Return a strong generating set from the Schreier-Sims algorithm. A generating set `S = \{g_1, g_2, ..., 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, ..., 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, ..., 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
[docs] 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 doesn't contain the supplied generators") G = PermutationGroup(gens) return G
@property def transitivity_degree(self): """Compute the degree of transitivity of the group. A permutation group `G` acting on `\Omega = \{0, 1, ..., n-1\}` is ``k``-fold transitive, if, for any k points `(a_1, a_2, ..., a_k)\in\Omega` and any k points `(b_1, b_2, ..., b_k)\in\Omega` there exists `g\in G` such that `g(a_1)=b_1, g(a_2)=b_2, ..., 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.perm_groups import PermutationGroup >>> from sympy.combinatorics.permutations import Permutation >>> 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 _orbit(degree, generators, alpha, action='tuples'): r"""Compute the orbit of alpha `\{g(\alpha) | g \in G\}` as a set. 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 >>> from sympy.combinatorics.perm_groups import PermutationGroup, _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.permutations import Permutation >>> from sympy.combinatorics.perm_groups import PermutationGroup, _orbits >>> a = Permutation([0, 2, 1]) >>> b = Permutation([1, 0, 2]) >>> _orbits(a.size, [a, b]) [{0, 1, 2}] """ seen = set() # elements that have already appeared in orbits 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): r"""Computes a transversal for the orbit of ``alpha`` as a set. 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. Examples ======== >>> from sympy.combinatorics import Permutation >>> Permutation.print_cyclic = True >>> 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)))] used = [False]*degree used[alpha] = True gens = [x._array_form for x in generators] for x, px in tr: for gen in gens: temp = gen[x] if used[temp] == False: 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] return tr if af: return [y for _, y in tr] return [_af_new(y) for _, y in tr] def _stabilizer(degree, generators, alpha): r"""Return the stabilizer subgroup of ``alpha``. 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 import Permutation >>> Permutation.print_cyclic = True >>> 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