perm_groups.py

from itertools import islice
from math import log
from random import choice, randrange

from ..core import Basic
from ..functions import factorial
from ..ntheory import sieve
from ..utilities import has_variety
from ..utilities.iterables import is_sequence, uniq
from ..utilities.randtest import _randrange
from .permutations import (Cycle, Permutation, _af_commutes_with, _af_invert,
                           _af_pow, _af_rmul, _af_rmuln)
from .util import (_base_ordering, _check_cycles_alt_sym,
                   _distribute_gens_by_base, _handle_precomputed_bsgs,
                   _orbits_transversals_from_bsgs, _strip, _strip_af,
                   _strong_gens_from_distr)


rmul = Permutation.rmul_with_af
_af_new = Permutation._af_new


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
    ========

    >>> Permutation.print_cyclic = True

    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
    ========

    diofant.combinatorics.polyhedron.Polyhedron,
    diofant.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://web.archive.org/web/20170105021515/http://www.algorithmist.com:80/index.php/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

    """

    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, ai in enumerate(args):
                if ai.size != degree:
                    args[i] = Permutation(ai, 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
        ========

        >>> p = Permutation(1, 2, 3)
        >>> Permutation(3) in PermutationGroup(p)
        True

        """
        if not isinstance(i, Permutation):
            raise TypeError('A PermutationGroup contains only Permutations as '
                            f'elements, not elements of type {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
        ========

        >>> 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

    __hash__ = Basic.__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
        ========

        >>> G = CyclicGroup(5)
        >>> H = G*G
        >>> H
        PermutationGroup([
            Permutation(9)(0, 1, 2, 3, 4),
            Permutation(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, g2i in enumerate(gens2):
            gens2[i] = [x + n1 for x in g2i]
        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] https://web.archive.org/web/20170105021515/http://www.algorithmist.com:80/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] https://web.archive.org/web/20170105021515/http://www.algorithmist.com:80/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
        ========

        >>> 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.

        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 diofant.combinatorics.testutil import _verify_bsgs
        >>> 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],
        [Permutation(0, 1, 2, 3), Permutation(3)(0, 1), Permutation(1, 3, 2),
         Permutation(2, 3), Permutation(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
        ========

        >>> 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
        ========

        >>> A = AlternatingGroup(4)
        >>> A.schreier_sims()
        >>> A.base
        [0, 1]
        >>> for g in A.basic_stabilizers:
        ...     print(g)
        ...
        PermutationGroup([
            Permutation(3)(0, 1, 2),
            Permutation(1, 2, 3)])
        PermutationGroup([
            Permutation(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
        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
        ========

        >>> A = AlternatingGroup(4)
        >>> A.basic_transversals
        [{0: Permutation(3),
          1: Permutation(3)(0, 1, 2),
          2: Permutation(3)(0, 2, 1),
          3: Permutation(0, 3, 1)},
         {1: Permutation(3),
          2: Permutation(1, 2, 3),
          3: Permutation(1, 3, 2)}]

        See Also
        ========

        strong_gens, base, basic_orbits, basic_stabilizers

        """
        if self._transversals == []:
            self.schreier_sims()
        return self._transversals

    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
        ========

        >>> 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.

        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
        ========

        >>> 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))

            def trivial_test(x):
                return 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)
        if hasattr(other, '__getitem__'):
            gens = list(other)
            return self.centralizer(PermutationGroup(gens))
        if hasattr(other, 'array_form'):
            return self.centralizer(PermutationGroup([other]))

    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
        ========

        >>> 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``

        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
        ========

        >>> 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, bi in enumerate(base):
            beta = h[bi]
            if beta == bi:
                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 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
        ========

        >>> 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])
        >>> c = Permutation(7)(2, 4)(3, 5)
        >>> G.coset_rank(c)
        16
        >>> G.coset_unrank(16)
        Permutation(7)(2, 4)(3, 5)

        See Also
        ========

        coset_factor

        """
        factors = self.coset_factor(g, True)
        if not factors:
            return
        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
        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
        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
        ========

        >>> Permutation.print_cyclic = True
        >>> a = Permutation([1, 0, 2])
        >>> G = PermutationGroup([a])
        >>> G.degree
        3
        >>> len(G)
        1
        >>> G.order()
        2
        >>> list(G.generate())
        [Permutation(2), Permutation(2)(0, 1)]

        See Also
        ========

        order

        """
        return self._degree

    @property
    def elements(self):
        """Returns all the elements of the permutation group in
        a list

        """
        return set(islice(self.generate(), None))

    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
        ========

        >>> 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

    def derived_subgroup(self):
        r"""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
        ========

        >>> Permutation.print_cyclic = True
        >>> 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

        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
        ========

        >>> Permutation.print_cyclic = True

        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([
            Permutation(0, 1)(2, 3),
            Permutation(3),
            Permutation(1, 2, 3),
            Permutation(1, 3, 2),
            Permutation(0, 3, 1),
            Permutation(0, 2, 3),
            Permutation(0, 3)(1, 2),
            Permutation(0, 1, 3),
            Permutation(3)(0, 2, 1),
            Permutation(0, 3, 2),
            Permutation(3)(0, 1, 2),
            Permutation(0, 2)(1, 3)])
        >>> _.is_group
        True

        """
        if method == 'coset':
            return self.generate_schreier_sims(af)
        if method == 'dimino':
            return self.generate_dimino(af)
        raise NotImplementedError(f'No generation defined for {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

        References
        ==========

        [1] The Implementation of Various Algorithms for Permutation Groups in
        the Computer Algebra System: AXIOM, N.J. Doye, M.Sc. Thesis

        Examples
        ========

        >>> Permutation.print_cyclic = True
        >>> 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)

    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
        ========

        >>> Permutation.print_cyclic = True
        >>> 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
        ========

        >>> Permutation.print_cyclic = True
        >>> a = Permutation([0, 2, 1])
        >>> b = Permutation([1, 0, 2])
        >>> G = PermutationGroup([a, b])
        >>> G.generators
        [Permutation(1, 2), Permutation(2)(0, 1)]

        """
        return self._generators

    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
        ========

        >>> Permutation.print_cyclic = True

        >>> 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, diofant.core.basic.Basic.has

        """
        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
        ========

        >>> Permutation.print_cyclic = True
        >>> 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 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
        ========

        >>> D = DihedralGroup(10)
        >>> D.is_alt_sym()
        False

        See Also
        ========

        diofant.combinatorics.util._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
        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
        ========

        >>> 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
            self._is_nilpotent = False
            return False
        return self._is_nilpotent

    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.generator and
        g2 g2 in G.generator

        Examples
        ========

        >>> Permutation.print_cyclic = True
        >>> 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

    def is_primitive(self, randomized=True):
        r"""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
        ========

        >>> 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 _ 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
        ========

        >>> 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
            self._is_solvable = False
            return False
        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
        ========

        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)

    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
        ========

        >>> 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
        ========

        >>> 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
        ========

        >>> 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 ``diofant.ntheory`` is used.
        This function is also used as an optimization tool for the functions
        ``minimal_block`` and ``_union_find_merge``.

        Examples
        ========

        >>> 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``.

        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
        ========

        >>> 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

    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
        ========

        >>> 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
        if hasattr(other, '__getitem__'):
            return self.normal_closure(PermutationGroup(other))
        if 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.

        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
        ========

        >>> 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``.

        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
        ========

        >>> Permutation.print_cyclic = True
        >>> G = AlternatingGroup(5)
        >>> G.orbit_rep(0, 4)
        Permutation(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))
        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.

        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
        ========

        >>> Permutation.print_cyclic = True
        >>> G = DihedralGroup(6)
        >>> G.orbit_transversal(0)
        [Permutation(5),
         Permutation(0, 1, 2, 3, 4, 5),
         Permutation(0, 5)(1, 4)(2, 3),
         Permutation(0, 2, 4)(1, 3, 5),
         Permutation(5)(0, 4)(1, 3),
         Permutation(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
        ========

        >>> 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
        ========

        >>> a = Permutation([1, 0, 2])
        >>> G = PermutationGroup([a])
        >>> G.degree
        3
        >>> len(G)
        1
        >>> G.order()
        2
        >>> list(G.generate())
        [Permutation(2), Permutation(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

        basic_transversals = self.basic_transversals
        m = 1
        for x in basic_transversals:
            m *= len(x)
        self._order = m
        return m

    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
        ========

        >>> 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:
            _, 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)
        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
        ========

        >>> Permutation.print_cyclic = True
        >>> a, b = [Permutation([1, 0, 3, 2]), Permutation([1, 3, 0, 2])]
        >>> G = PermutationGroup([a, b])
        >>> G.make_perm(1, [0])
        Permutation(0, 1)(2, 3)
        >>> G.make_perm(3, [0, 1, 0])
        Permutation(0, 2, 3, 1)
        >>> G.make_perm([0, 1, 0])
        Permutation(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 as exc:
                raise ValueError('n must be an integer or a sequence.') from exc
        randrange = _randrange(seed)

        # start with the identity permutation
        result = Permutation(list(range(self.degree)))
        m = len(self)
        for _ in range(n):
            p = self[randrange(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.

        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``.

        """
        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.

        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
        ========

        >>> a = Permutation([0, 2, 1])
        >>> b = Permutation([1, 0, 2])
        >>> G = PermutationGroup([a, b])
        >>> G.schreier_sims()
        >>> G.basic_transversals
        [{0: Permutation(2)(0, 1), 1: Permutation(2), 2: Permutation(1, 2)},
         {0: Permutation(2), 2: Permutation(0, 2)}]

        """
        if self._transversals:
            return
        base, strong_gens = self.schreier_sims_incremental()
        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]

    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 diofant.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
                _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])
        # 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])
                            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

    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 diofant.combinatorics.testutil import _verify_bsgs
        >>> S = SymmetricGroup(5)
        >>> base, strong_gens = S.schreier_sims_random(consec_succ=5)
        >>> _verify_bsgs(S, base, strong_gens)
        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])
        # 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])
                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``.

        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
        ========

        >>> 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)  # pylint: disable=modified-iterating-list
                    used[temp] = True
                    v[temp] = i
        return v

    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
        ========

        >>> Permutation.print_cyclic = True
        >>> G = DihedralGroup(6)
        >>> G.stabilizer(5)
        PermutationGroup([
            Permutation(5)(0, 4)(1, 3),
            Permutation(5)])

        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.

        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
        ========

        >>> D = DihedralGroup(4)
        >>> D.strong_gens
        [Permutation(0, 1, 2, 3), Permutation(0, 3)(1, 2), Permutation(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_search(self, prop, base=None, strong_gens=None, tests=None,
                        init_subgroup=None):
        """Find the subgroup of all elements satisfying the property ``prop``.

        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 diofant.combinatorics.testutil import _verify_bsgs
        >>> S = SymmetricGroup(7)
        >>> def prop_even(x):
        ...     return 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 lenghty 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:
            def trivial_test(x):
                return 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
            # accorndingly
            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.

        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
        ========

        >>> 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
        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
    ========

    >>> 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] is False:
                    orb.append(temp)
                    used[temp] = True
        return set(orb)
    if 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)  # pylint: disable=modified-iterating-list
                    used.add(temp)
        return set(orb)
    if 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)  # pylint: disable=modified-iterating-list
                    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
    ========

    >>> 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):
    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
    ========

    >>> Permutation.print_cyclic = True
    >>> G = DihedralGroup(6)
    >>> _orbit_transversal(G.degree, G.generators, 0, False)
        [Permutation(5),
         Permutation(0, 1, 2, 3, 4, 5),
         Permutation(0, 5)(1, 4)(2, 3),
         Permutation(0, 2, 4)(1, 3, 5),
         Permutation(5)(0, 4)(1, 3),
         Permutation(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] is False:
                tr.append((temp, _af_rmul(gen, px)))  # pylint: disable=modified-iterating-list
                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
    ========

    >>> Permutation.print_cyclic = True
    >>> G = DihedralGroup(6)
    >>> _stabilizer(G.degree, G.generators, 5)
    [Permutation(5)(0, 4)(1, 3), Permutation(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)  # pylint: disable=modified-iterating-list
                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