permgroup.py

# -*- coding: utf-8 -*-
r"""
Permutation groups

A permutation group is a finite group `G` whose elements are
permutations of a given finite set `X` (i.e., bijections
`X \longrightarrow X`) and whose group operation is the composition of
permutations. The number of elements of `X` is called the degree of `G`.

In Sage, a permutation is represented as either a string that
defines a permutation using disjoint cycle notation, or a list of
tuples, which represent disjoint cycles. That is::

    (a,...,b)(c,...,d)...(e,...,f)  <--> [(a,...,b), (c,...,d),..., (e,...,f)]
                      () = identity <--> []

You can make the "named" permutation groups (see
``permgp_named.py``) and use the following
constructions:

- permutation group generated by elements,
- ``direct_product_permgroups``, which takes a list of permutation
  groups and returns their direct product.

JOKE: Q: What's hot, chunky, and acts on a polygon? A: Dihedral
soup. Renteln, P. and Dundes, A. "Foolproof: A Sampling of
Mathematical Folk Humor." Notices Amer. Math. Soc. 52, 24-34,
2005.

Index of methods
----------------

Here are the method of a :func:`PermutationGroup`

{METHODS_OF_PermutationGroup_generic}

AUTHORS:

- David Joyner (2005-10-14): first version

- David Joyner (2005-11-17)

- William Stein (2005-11-26): rewrite to better wrap Gap

- David Joyner (2005-12-21)

- William Stein and David Joyner (2006-01-04): added conjugacy_class_representatives

- David Joyner (2006-03): reorganization into subdirectory perm_gps;
  added __contains__, has_element; fixed _cmp_; added subgroup
  class+methods, PGL,PSL,PSp, PSU classes,

- David Joyner (2006-06): added PGU, functionality to SymmetricGroup,
  AlternatingGroup, direct_product_permgroups

- David Joyner (2006-08): added degree,
  ramification_module_decomposition_modular_curve and
  ramification_module_decomposition_hurwitz_curve methods to PSL(2,q),
  MathieuGroup, is_isomorphic

- Bobby Moretti (2006)-10): Added KleinFourGroup, fixed bug in DihedralGroup

- David Joyner (2006-10): added is_subgroup (fixing a bug found by
  Kiran Kedlaya), is_solvable, normalizer, is_normal_subgroup, Suzuki

- David Kohel (2007-02): fixed __contains__ to not enumerate group
  elements, following the convention for __call__

- David Harvey, Mike Hansen, Nick Alexander, William Stein
  (2007-02,03,04,05): Various patches

- Nathan Dunfield (2007-05): added orbits

- David Joyner (2007-06): added subgroup method (suggested by David
  Kohel), composition_series, lower_central_series,
  upper_central_series, cayley_table, quotient_group, sylow_subgroup,
  is_cyclic, homology, homology_part, cohomology, cohomology_part,
  poincare_series, molien_series, is_simple, is_monomial,
  is_supersolvable, is_nilpotent, is_perfect, is_polycyclic,
  is_elementary_abelian, is_pgroup, gens_small,
  isomorphism_type_info_simple_group. moved all the"named"
  groups to a new file.

- Nick Alexander (2007-07): move is_isomorphic to isomorphism_to, add
  from_gap_list

- William Stein (2007-07): put is_isomorphic back (and make it better)

- David Joyner (2007-08): fixed bugs in composition_series,
  upper/lower_central_series, derived_series,

- David Joyner (2008-06): modified is_normal (reported by
  W. J. Palenstijn), and added normalizes

- David Joyner (2008-08): Added example to docstring of cohomology.

- Simon King (2009-04): __cmp__ methods for PermutationGroup_generic and PermutationGroup_subgroup

- Nicolas Borie (2009): Added orbit, transversals, stabiliser and strong_generating_system methods

- Christopher Swenson (2012): Added a special case to compute the order efficiently.
  (This patch Copyright 2012 Google Inc. All Rights Reserved. )

- Javier Lopez Pena (2013): Added conjugacy classes.

REFERENCES:

- Cameron, P., Permutation Groups. New York: Cambridge University
  Press, 1999.

- Wielandt, H., Finite Permutation Groups.  New York: Academic Press,
  1964.

- Dixon, J. and Mortimer, B., Permutation Groups, Springer-Verlag,
  Berlin/New York, 1996.

.. note::

   Though Suzuki groups are okay, Ree groups should *not* be wrapped
   as permutation groups - the construction is too slow - unless (for
   small values or the parameter) they are made using explicit
   generators.

"""
#*****************************************************************************
#       Copyright (C) 2006 William Stein <wstein@gmail.com>
#                          David Joyner <wdjoyner@gmail.com>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#                  http://www.gnu.org/licenses/
#*****************************************************************************
from __future__ import absolute_import
from six.moves import range
from six import integer_types

from functools import wraps

from sage.misc.randstate import current_randstate
from sage.groups.group import FiniteGroup

from sage.rings.all import QQ, Integer
from sage.interfaces.expect import is_ExpectElement
from sage.interfaces.gap import gap, GapElement
from sage.groups.perm_gps.permgroup_element import PermutationGroupElement, standardize_generator
from sage.groups.abelian_gps.abelian_group import AbelianGroup
from sage.misc.cachefunc import cached_method
from sage.groups.class_function import ClassFunction
from sage.sets.finite_enumerated_set import FiniteEnumeratedSet
from sage.categories.all import FiniteEnumeratedSets
from sage.groups.conjugacy_classes import ConjugacyClassGAP
from sage.structure.richcmp import (richcmp_method, richcmp_not_equal,
                                    richcmp, rich_to_bool, op_EQ)


def load_hap():
    r"""
    Load the GAP hap package into the default GAP interpreter interface.

    EXAMPLES::

        sage: sage.groups.perm_gps.permgroup.load_hap() # optional - gap_packages
    """
    from sage.features.gap import GapPackage
    GapPackage("hap", spkg="gap_packages").require()
    gap.load_package("hap")


def hap_decorator(f):
    """
    A decorator for permutation group methods that require HAP.  It
    checks to see that HAP is installed as well as checks that the
    argument ``p`` is either 0 or prime.

    EXAMPLES::

        sage: from sage.groups.perm_gps.permgroup import hap_decorator
        sage: def foo(self, n, p=0): print("Done")
        sage: foo = hap_decorator(foo)
        sage: foo(None, 3)    #optional - gap_packages
        Done
        sage: foo(None, 3, 0) # optional - gap_packages
        Done
        sage: foo(None, 3, 5) # optional - gap_packages
        Done
        sage: foo(None, 3, 4) #optional - gap_packages
        Traceback (most recent call last):
        ...
        ValueError: p must be 0 or prime
    """
    @wraps(f)
    def wrapped(self, n, p=0):
        load_hap()
        from sage.arith.all import is_prime
        if not (p == 0 or is_prime(p)):
            raise ValueError("p must be 0 or prime")

        return f(self, n, p=p)
    return wrapped

def direct_product_permgroups(P):
    """
    Takes the direct product of the permutation groups listed in ``P``.

    EXAMPLES::

        sage: G1 = AlternatingGroup([1,2,4,5])
        sage: G2 = AlternatingGroup([3,4,6,7])
        sage: D = direct_product_permgroups([G1,G2,G1])
        sage: D.order()
        1728
        sage: D = direct_product_permgroups([G1])
        sage: D==G1
        True
        sage: direct_product_permgroups([])
        Symmetric group of order 0! as a permutation group
    """
    n = len(P)
    if n == 0:
        from sage.groups.perm_gps.permgroup_named import SymmetricGroup
        return SymmetricGroup(0)
    elif n == 1:
        return P[0]
    else:
        G = gap.DirectProduct(*P)
        return PermutationGroup(gap_group=G)

def from_gap_list(G, src):
    r"""
    Convert a string giving a list of GAP permutations into a list of
    elements of ``G``.

    EXAMPLES::

        sage: from sage.groups.perm_gps.permgroup import from_gap_list
        sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
        sage: L = from_gap_list(G, "[(1,2,3)(4,5), (3,4)]"); L
        [(1,2,3)(4,5), (3,4)]
        sage: L[0].parent() is G
        True
        sage: L[1].parent() is G
        True
    """
    # src is a list of strings, each of which is a permutation of
    # integers in cycle notation. It may contain \n and spaces.
    src = [str(g)[1:].split(")(")
           for g in str(src).replace(" ","").replace("\n","")[1:-2].split("),")]

    # src is a list of list of strings. Each string is a list of
    # integers separated by ','
    src = [G([tuple(G._domain_from_gap[int(x)] for x in cycle.split(","))
              for cycle in g])
           for g in src]

    # src is now a list of group elements
    return src

def PermutationGroup(gens=None, gap_group=None, domain=None, canonicalize=True, category=None):
    """
    Return the permutation group associated to `x` (typically a
    list of generators).

    INPUT:


    -  ``gens`` - list of generators (default: ``None``)

    -  ``gap_group`` - a gap permutation group (default: ``None``)

    -  ``canonicalize`` - bool (default: ``True``); if ``True``,
       sort generators and remove duplicates


    OUTPUT:

    - A permutation group.

    EXAMPLES::

        sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
        sage: G
        Permutation Group with generators [(3,4), (1,2,3)(4,5)]

    We can also make permutation groups from PARI groups::

        sage: H = pari('x^4 - 2*x^3 - 2*x + 1').polgalois()
        sage: G = PariGroup(H, 4); G
        PARI group [8, -1, 3, "D(4)"] of degree 4
        sage: H = PermutationGroup(G); H          # optional - database_gap
        Transitive group number 3 of degree 4
        sage: H.gens()                            # optional - database_gap
        [(1,2,3,4), (1,3)]

    We can also create permutation groups whose generators are Gap
    permutation objects::

        sage: p = gap('(1,2)(3,7)(4,6)(5,8)'); p
        (1,2)(3,7)(4,6)(5,8)
        sage: PermutationGroup([p])
        Permutation Group with generators [(1,2)(3,7)(4,6)(5,8)]

    Permutation groups can work on any domain. In the following
    examples, the permutations are specified in list notation,
    according to the order of the elements of the domain::

        sage: list(PermutationGroup([['b','c','a']], domain=['a','b','c']))
        [(), ('a','b','c'), ('a','c','b')]
        sage: list(PermutationGroup([['b','c','a']], domain=['b','c','a']))
        [()]
        sage: list(PermutationGroup([['b','c','a']], domain=['a','c','b']))
        [(), ('a','b')]

    There is an underlying gap object that implements each
    permutation group::

        sage: G = PermutationGroup([[(1,2,3,4)]])
        sage: G._gap_()
        Group( [ (1,2,3,4) ] )
        sage: gap(G)
        Group( [ (1,2,3,4) ] )
        sage: gap(G) is G._gap_()
        True
        sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
        sage: current_randstate().set_seed_gap()
        sage: G._gap_().DerivedSeries()
        [ Group( [ (3,4), (1,2,3)(4,5) ] ), Group( [ (1,5)(3,4), (1,5)(2,4), (1,5,3) ] ) ]

    TESTS::

        sage: r = Permutation("(1,7,9,3)(2,4,8,6)")
        sage: f = Permutation("(1,3)(4,6)(7,9)")
        sage: PermutationGroup([r,f]) #See Trac #12597
        Permutation Group with generators [(1,3)(4,6)(7,9), (1,7,9,3)(2,4,8,6)]

        sage: PermutationGroup(SymmetricGroup(5))
        Traceback (most recent call last):
        ...
        TypeError: gens must be a tuple, list, or GapElement
    """
    if not is_ExpectElement(gens) and hasattr(gens, '_permgroup_'):
        return gens._permgroup_()
    if gens is not None and not isinstance(gens, (tuple, list, GapElement)):
        raise TypeError("gens must be a tuple, list, or GapElement")
    return PermutationGroup_generic(gens=gens, gap_group=gap_group, domain=domain,
                                    canonicalize=canonicalize, category=category)


@richcmp_method
class PermutationGroup_generic(FiniteGroup):
    """
    A generic permutation group.

    EXAMPLES::

        sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
        sage: G
        Permutation Group with generators [(3,4), (1,2,3)(4,5)]
        sage: G.center()
        Subgroup of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]) generated by [()]
        sage: G.group_id()          # optional - database_gap
        [120, 34]
        sage: n = G.order(); n
        120
        sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
        sage: TestSuite(G).run()
    """
    def __init__(self, gens=None, gap_group=None, canonicalize=True, domain=None, category=None):
        r"""
        Initialize ``self``.

        INPUT:

        -  ``gens`` - list of generators (default: ``None``)

        -  ``gap_group`` - a gap permutation group (default: ``None``)

        -  ``canonicalize`` - bool (default: ``True``); if ``True``,
           sort generators and remove duplicates

        OUTPUT:

        - A permutation group.

        EXAMPLES:

        We explicitly construct the alternating group on four
        elements::

            sage: A4 = PermutationGroup([[(1,2,3)],[(2,3,4)]]); A4
            Permutation Group with generators [(2,3,4), (1,2,3)]
            sage: A4.__init__([[(1,2,3)],[(2,3,4)]]); A4
            Permutation Group with generators [(2,3,4), (1,2,3)]
            sage: A4.center()
            Subgroup of (Permutation Group with generators [(2,3,4), (1,2,3)]) generated by [()]
            sage: A4.category()
            Category of finite enumerated permutation groups
            sage: TestSuite(A4).run()

        TESTS::

            sage: TestSuite(PermutationGroup([[]])).run()
            sage: TestSuite(PermutationGroup([])).run()
            sage: TestSuite(PermutationGroup([(0,1)])).run()
        """
        from sage.categories.permutation_groups import PermutationGroups
        category = PermutationGroups().FinitelyGenerated().Finite().or_subcategory(category)
        super(PermutationGroup_generic, self).__init__(category=category)
        if (gens is None and gap_group is None):
            raise ValueError("you must specify gens or gap_group")

        #Handle the case where only the GAP group is specified.
        if gens is None:
            if isinstance(gap_group, str):
                gap_group = gap(gap_group)
            gens = [gen for gen in gap_group.GeneratorsOfGroup()]

        if domain is None:
            gens = [standardize_generator(x) for x in gens]
            domain = set()
            for x in gens:
                for cycle in x:
                    domain = domain.union(cycle)
            domain = sorted(domain)

            #Here we need to check if all of the points are integers
            #to make the domain contain all integers up to the max.
            #This is needed for backward compatibility
            if all(isinstance(p, (Integer,) + integer_types) for p in domain):
                domain = list(range(min([1] + domain), max([1] + domain)+1))

        if domain not in FiniteEnumeratedSets():
            domain = FiniteEnumeratedSet(domain)

        self._domain = domain
        self._deg = len(self._domain)
        self._domain_to_gap = dict((key, i+1) for i, key in enumerate(self._domain))
        self._domain_from_gap = dict((i+1, key) for i, key in enumerate(self._domain))

        if not gens:  # length 0
            gens = [()]
        gens = [self.element_class(x, self, check=False) for x in gens]
        if canonicalize:
            gens = sorted(set(gens))
        self._gens = gens

    def construction(self):
        """
        Return the construction of ``self``.

        EXAMPLES::

            sage: P1 = PermutationGroup([[(1,2)]])
            sage: P1.construction()
            (PermutationGroupFunctor[(1,2)], Permutation Group with generators [()])

            sage: PermutationGroup([]).construction() is None
            True

        This allows us to perform computations like the following::

            sage: P1 = PermutationGroup([[(1,2)]]); p1 = P1.gen()
            sage: P2 = PermutationGroup([[(1,3)]]); p2 = P2.gen()
            sage: p = p1*p2; p
            (1,2,3)
            sage: p.parent()
            Permutation Group with generators [(1,2), (1,3)]
            sage: p.parent().domain()
            {1, 2, 3}

        Note that this will merge permutation groups with different
        domains::

            sage: g1 = PermutationGroupElement([(1,2),(3,4,5)])
            sage: g2 = PermutationGroup([('a','b')], domain=['a', 'b']).gens()[0]
            sage: g2
            ('a','b')
            sage: p = g1*g2; p
            (1,2)(3,4,5)('a','b')
        """
        gens = self.gens()
        if len(gens) == 1 and gens[0].is_one():
            return None
        else:
            from sage.categories.pushout import PermutationGroupFunctor
            return (PermutationGroupFunctor(gens, self.domain()),
                    PermutationGroup([]))

    @cached_method
    def _has_natural_domain(self):
        """
        Returns True if the underlying domain is of the form (1,...,n)

        EXAMPLES::

            sage: SymmetricGroup(3)._has_natural_domain()
            True
            sage: SymmetricGroup((1,2,3))._has_natural_domain()
            True
            sage: SymmetricGroup((1,3))._has_natural_domain()
            False
            sage: SymmetricGroup((3,2,1))._has_natural_domain()
            False
        """
        domain = self.domain()
        natural_domain = FiniteEnumeratedSet(list(range(1, len(domain)+1)))
        return domain == natural_domain

    def _gap_init_(self):
        r"""
        Returns a string showing how to declare / initialize ``self`` in Gap.
        Stored in the ``self._gap_string`` attribute.

        EXAMPLES:

        The ``_gap_init_`` method shows how you
        would define the Sage ``PermutationGroup_generic``
        object in Gap::

            sage: A4 = PermutationGroup([[(1,2,3)],[(2,3,4)]]); A4
            Permutation Group with generators [(2,3,4), (1,2,3)]
            sage: A4._gap_init_()
            'Group([PermList([1, 3, 4, 2]), PermList([2, 3, 1, 4])])'
        """
        return 'Group([%s])'%(', '.join([g._gap_init_() for g in self.gens()]))

    def _magma_init_(self, magma):
        r"""
        Returns a string showing how to declare / initialize self in Magma.

        EXAMPLES:

        We explicitly construct the alternating group on four
        elements. In Magma, one would type the string below to construct
        the group::

            sage: A4 = PermutationGroup([[(1,2,3)],[(2,3,4)]]); A4
            Permutation Group with generators [(2,3,4), (1,2,3)]
            sage: A4._magma_init_(magma)
            'PermutationGroup<4 | (2,3,4), (1,2,3)>'

            sage: S = SymmetricGroup(['a', 'b', 'c'])
            sage: S._magma_init_(magma)
            'PermutationGroup<3 | (1,2,3), (1,2)>'
        """
        g = ', '.join([g._gap_cycle_string() for g in self.gens()])
        return 'PermutationGroup<%s | %s>'%(self.degree(), g)

    def __richcmp__(self, right, op):
        """
        Compare ``self`` and ``right``.

        The comparison extends the subgroup relation. Hence, it is first checked
        whether one of the groups is subgroup of the other. If this is not the
        case then the ordering is whatever it is in Gap.

        .. NOTE::

            The comparison does not provide a total ordering, as can be seen
            in the examples below.

        EXAMPLES::

            sage: G1 = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
            sage: G2 = PermutationGroup([[(1,2,3),(4,5)]])
            sage: G1 > G2 # since G2 is a subgroup of G1
            True
            sage: G1 < G2
            False

        The following example shows that the comparison does not yield a total
        ordering::

            sage: H1 = PermutationGroup([[(1,2)],[(5,6)]])
            sage: H2 = PermutationGroup([[(3,4)]])
            sage: H3 = PermutationGroup([[(1,2)]])
            sage: H1 < H2 # according to Gap's ordering
            True
            sage: H2 < H3 # according to Gap's ordering
            True
            sage: H3 < H1 # since H3 is a subgroup of H1
            True
        """
        if not isinstance(right, PermutationGroup_generic):
            return NotImplemented

        if self is right:
            return rich_to_bool(op, 0)

        gSelf = self._gap_()
        gRight = right._gap_()
        gapcmp = gSelf._cmp_(gRight)
        if not gapcmp:
            return rich_to_bool(op, 0)

        if gSelf.IsSubgroup(gRight):
            return rich_to_bool(op, 1)
        if gRight.IsSubgroup(gSelf):
            return rich_to_bool(op, -1)
        return rich_to_bool(op, gapcmp)

    Element = PermutationGroupElement

    def _element_constructor_(self, x, check=True):
        """
        Coerce ``x`` into this permutation group.

        The input can be either a string that defines a permutation in
        cycle notation, a permutation group element, a list of integers
        that gives the permutation as a mapping, a list of tuples, or the
        integer 1.

        EXAMPLES:

        We illustrate each way to make a permutation in `S_4`::

            sage: G = SymmetricGroup(4)
            sage: G((1,2,3,4))
            (1,2,3,4)
            sage: G([(1,2),(3,4)])
            (1,2)(3,4)
            sage: G('(1,2)(3,4)')
            (1,2)(3,4)
            sage: G('(1,2)(3)(4)')
            (1,2)
            sage: G(((1,2,3),(4,)))
            (1,2,3)
            sage: G(((1,2,3,4),))
            (1,2,3,4)
            sage: G([1,2,4,3])
            (3,4)
            sage: G([2,3,4,1])
            (1,2,3,4)
            sage: G(G((1,2,3,4)))
            (1,2,3,4)
            sage: G(1)
            ()

        Some more examples::

            sage: G = PermutationGroup([(1,2,3,4)])
            sage: G([(1,3), (2,4)])
            (1,3)(2,4)
            sage: G(G.0^3)
            (1,4,3,2)
            sage: G(1)
            ()
            sage: G((1,4,3,2))
            (1,4,3,2)
            sage: G([(1,2)])
            Traceback (most recent call last):
            ...
            TypeError: permutation [(1, 2)] not in Permutation Group with generators [(1,2,3,4)]
        """
        if isinstance(x, integer_types + (Integer,)) and x == 1:
            return self.identity()

        if isinstance(x, PermutationGroupElement):
            x_parent = x.parent()
            if (isinstance(x_parent, PermutationGroup_subgroup)
                and x_parent._ambient_group is self):
                return self.element_class(x.cycle_tuples(), self, check=False)

            from sage.groups.perm_gps.permgroup_named import SymmetricGroup
            compatible_domains = all(point in self._domain_to_gap
                                     for point in x_parent.domain())
            if compatible_domains and (isinstance(self, SymmetricGroup)
                                       or x._gap_() in self._gap_()):
                return self.element_class(x.cycle_tuples(), self, check=False)

        return self.element_class(x, self, check=check)

    def _coerce_map_from_(self, G):
        r"""
        Return if there is a coercion map from ``G`` into ``self``.

        EXAMPLES:

        We have coercion maps from subgroups::

            sage: G = SymmetricGroup(5)
            sage: H = G.subgroup([[(1,2,3)], [(4,5)]])
            sage: G._coerce_map_from_(H)
            True
            sage: K = H.subgroup([[(1,2,3)]])
            sage: H.has_coerce_map_from(K)
            True
            sage: G.has_coerce_map_from(K)
            True
            sage: G = PermutationGroup([[(1,2)], [(3,4,5)], [(2,3)]])
            sage: G.has_coerce_map_from(K)
            True

        We illustrate some arithmetic that involves coercion
        of elements in different permutation groups::

            sage: g1 = PermutationGroupElement([(1,2),(3,4,5)])
            sage: g1.parent()
            Symmetric group of order 5! as a permutation group
            sage: g2 = PermutationGroupElement([(1,2)])
            sage: g2.parent()
            Symmetric group of order 2! as a permutation group
            sage: g1*g2
            (3,4,5)
            sage: g2*g2
            ()
            sage: g2*g1
            (3,4,5)

        We try to convert in a non-permutation::

            sage: G = PermutationGroup([[(1,2,3,4)], [(1,2)]])
            sage: G(2)
            Traceback (most recent call last):
            ...
            TypeError: 'sage.rings.integer.Integer' object is not iterable
        """
        if isinstance(G, PermutationGroup_subgroup):
            if G._ambient_group is self:
                return True
            if self.has_coerce_map_from(G._ambient_group):
                return self._coerce_map_via([G._ambient_group], G)
        if isinstance(G, PermutationGroup_generic):
            if G.is_subgroup(self):
                return True
        return super(PermutationGroup_generic, self)._coerce_map_from_(G)

    def list(self):
        """
        Return list of all elements of this group.

        EXAMPLES::

            sage: G = PermutationGroup([[(1,2,3,4)], [(1,2)]])
            sage: G.list()
            [(), (1,2), (1,2,3,4), (1,3)(2,4), (1,3,4), (2,3,4), (1,4,3,2),
             (1,3,2,4), (1,3,4,2), (1,2,4,3), (1,4,2,3), (2,4,3), (1,4,3),
             (1,4)(2,3), (1,4,2), (1,3,2), (1,3), (3,4), (2,4), (1,4), (2,3),
             (1,2)(3,4), (1,2,3), (1,2,4)]

            sage: G = PermutationGroup([[('a','b')]], domain=('a', 'b')); G
            Permutation Group with generators [('a','b')]
            sage: G.list()
            [(), ('a','b')]

        TESTS:

        Test :trac:`9155`::

            sage: G = SymmetricGroup(2)
            sage: elements = G.list()
            sage: elements.remove(G("()"))
            sage: elements
            [(1,2)]
            sage: G.list()
            [(), (1,2)]
        """
        return [x for x in self]

    def __contains__(self, item):
        """
        Returns boolean value of ``item in self``.

        EXAMPLES::

            sage: G = SymmetricGroup(16)
            sage: g = G.gen(0)
            sage: h = G.gen(1)
            sage: g^7*h*g*h in G
            True
            sage: G = SymmetricGroup(4)
            sage: g = G((1,2,3,4))
            sage: h = G((1,2))
            sage: H = PermutationGroup([[(1,2,3,4)], [(1,2),(3,4)]])
            sage: g in H
            True
            sage: h in H
            False

            sage: G = PermutationGroup([[('a','b')]], domain=('a', 'b'))
            sage: [('a', 'b')] in G
            True
        """
        if isinstance(item, integer_types + (Integer,)):
            return item == 1
        try:
            item = self._element_constructor_(item, check=True)
        except Exception:
            return False
        return True

    def has_element(self, item):
        """
        Returns boolean value of ``item in self`` - however *ignores*
        parentage.

        EXAMPLES::

            sage: G = CyclicPermutationGroup(4)
            sage: gens = G.gens()
            sage: H = DihedralGroup(4)
            sage: g = G([(1,2,3,4)]); g
            (1,2,3,4)
            sage: G.has_element(g)
            True
            sage: h = H([(1,2),(3,4)]); h
            (1,2)(3,4)
            sage: G.has_element(h)
            False
        """
        return item in self

    def __iter__(self):
        """
        Return an iterator over the elements of this group.

        EXAMPLES::

            sage: G = PermutationGroup([[(1,2,3)], [(1,2)]])
            sage: [a for a in G]
            [(), (1,2), (1,2,3), (2,3), (1,3,2), (1,3)]

        Test that it is possible to iterate through moderately large groups
        (:trac:`18239`)::

            sage: p = [(i,i+1) for i in range(1,601,2)]
            sage: q = [tuple(range(1+i,601,3)) for i in range(3)]
            sage: A = PermutationGroup([p,q])
            sage: A.cardinality()
            60000
            sage: for x in A:    # long time - 2 secs
            ....:     pass       # long time
        """
        from sage.sets.recursively_enumerated_set import RecursivelyEnumeratedSet
        return iter(RecursivelyEnumeratedSet(seeds=[self.one()],
                successors=lambda g: (g._mul_(h) for h in self.gens())))

    def gens(self):
        """
        Return tuple of generators of this group. These need not be
        minimal, as they are the generators used in defining this group.

        EXAMPLES::

            sage: G = PermutationGroup([[(1,2,3)], [(1,2)]])
            sage: G.gens()
            [(1,2), (1,2,3)]

        Note that the generators need not be minimal, though duplicates are
        removed::

            sage: G = PermutationGroup([[(1,2)], [(1,3)], [(2,3)], [(1,2)]])
            sage: G.gens()
            [(2,3), (1,2), (1,3)]

        We can use index notation to access the generators returned by
        ``self.gens``::

            sage: G = PermutationGroup([[(1,2,3,4), (5,6)], [(1,2)]])
            sage: g = G.gens()
            sage: g[0]
            (1,2)
            sage: g[1]
            (1,2,3,4)(5,6)

        TESTS:

        We make sure that the trivial group gets handled correctly::

            sage: SymmetricGroup(1).gens()
            [()]
        """
        return self._gens


    def gens_small(self):
        """
        For this group, returns a generating set which has few elements.
        As neither irredundancy nor minimal length is proven, it is fast.

        EXAMPLES::

            sage: R = "(25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24)" ## R = right
            sage: U = "( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19)" ## U = top
            sage: L = "( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35)" ## L = left
            sage: F = "(17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11)" ## F = front
            sage: B = "(33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27)" ## B = back or rear
            sage: D = "(41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40)" ## D = down or bottom
            sage: G = PermutationGroup([R,L,U,F,B,D])
            sage: len(G.gens_small())
            2

        The output may be unpredictable, due to the use of randomized
        algorithms in GAP. Note that both the following answers are equally valid.

        ::

            sage: G = PermutationGroup([[('a','b')], [('b', 'c')], [('a', 'c')]])
            sage: G.gens_small() # random
            [('b','c'), ('a','c','b')] ## (on 64-bit Linux)
            [('a','b'), ('a','c','b')] ## (on Solaris)
            sage: len(G.gens_small()) == 2
            True
        """
        gens = self._gap_().SmallGeneratingSet()
        return [self.element_class(x, self, check=False) for x in gens]

    def gen(self, i=None):
        r"""
        Returns the i-th generator of ``self``; that is, the i-th element of
        the list ``self.gens()``.

        The argument `i` may be omitted if there is only one generator (but
        this will raise an error otherwise).

        EXAMPLES:

        We explicitly construct the alternating group on four
        elements::

            sage: A4 = PermutationGroup([[(1,2,3)],[(2,3,4)]]); A4
            Permutation Group with generators [(2,3,4), (1,2,3)]
            sage: A4.gens()
            [(2,3,4), (1,2,3)]
            sage: A4.gen(0)
            (2,3,4)
            sage: A4.gen(1)
            (1,2,3)
            sage: A4.gens()[0]; A4.gens()[1]
            (2,3,4)
            (1,2,3)

            sage: P1 = PermutationGroup([[(1,2)]]); P1.gen()
            (1,2)
        """
        gens = self.gens()
        if i is None:
             if len(gens) == 1:
                  return gens[0]
             else:
                  raise ValueError("You must specify which generator you want")
        else:
             return gens[i]

    def ngens(self):
        """
        Return the number of generators of ``self``.

        EXAMPLES::

            sage: A4 = PermutationGroup([[(1,2,3)], [(2,3,4)]]); A4
            Permutation Group with generators [(2,3,4), (1,2,3)]
            sage: A4.ngens()
            2
        """
        return len(self.gens())

    def identity(self):
        """
        Return the identity element of this group.

        EXAMPLES::

            sage: G = PermutationGroup([[(1,2,3),(4,5)]])
            sage: e = G.identity()
            sage: e
            ()
            sage: g = G.gen(0)
            sage: g*e
            (1,2,3)(4,5)
            sage: e*g
            (1,2,3)(4,5)

            sage: S = SymmetricGroup(['a','b','c'])
            sage: S.identity()
            ()
        """
        return self.element_class([], self, check=True)

    def exponent(self):
        r"""
        Computes the exponent of the group.

        The exponent `e` of a group `G` is the LCM of the orders of its
        elements, that is, `e` is the smallest integer such that `g^e=1` for all
        `g \in G`.

        EXAMPLES::

            sage: G = AlternatingGroup(4)
            sage: G.exponent()
            6

        """
        return Integer(self._gap_().Exponent())

    def largest_moved_point(self):
        """
        Return the largest point moved by a permutation in this group.

        EXAMPLES::

            sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
            sage: G.largest_moved_point()
            4
            sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]])
            sage: G.largest_moved_point()
            10

        ::

            sage: G = PermutationGroup([[('a','b','c'),('d','e')]])
            sage: G.largest_moved_point()
            'e'

        .. warning::

           The name of this function is not good; this function
           should be deprecated in term of degree::

                sage: P = PermutationGroup([[1,2,3,4]])
                sage: P.largest_moved_point()
                4
                sage: P.cardinality()
                1
        """
        try:
            return self._domain_from_gap[Integer(self._gap_().LargestMovedPoint())]
        except KeyError:
            return self.degree()

    def degree(self):
        """
        Returns the degree of this permutation group.

        EXAMPLES::

            sage: S = SymmetricGroup(['a','b','c'])
            sage: S.degree()
            3
            sage: G = PermutationGroup([(1,3),(4,5)])
            sage: G.degree()
            5

        Note that you can explicitly specify the domain to get a
        permutation group of smaller degree::

            sage: G = PermutationGroup([(1,3),(4,5)], domain=[1,3,4,5])
            sage: G.degree()
            4
        """
        return Integer(len(self._domain))

    def domain(self):
        r"""
        Returns the underlying set that this permutation group acts
        on.

        EXAMPLES::

            sage: P = PermutationGroup([(1,2),(3,5)])
            sage: P.domain()
            {1, 2, 3, 4, 5}
            sage: S = SymmetricGroup(['a', 'b', 'c'])
            sage: S.domain()
            {'a', 'b', 'c'}
        """
        return self._domain

    def _domain_gap(self, domain=None):
        """
        Returns a GAP string representation of the underlying set
        that this group acts on.  See also :meth:`domain`.

        EXAMPLES::

            sage: P = PermutationGroup([(1,2),(3,5)])
            sage: P._domain_gap()
            '[1, 2, 3, 4, 5]'
        """
        if domain is None:
            return repr(list(range(1, self.degree()+1)))
        else:
            try:
                return repr([self._domain_to_gap[point] for point in domain])
            except KeyError:
                raise ValueError("domain must be a subdomain of self.domain()")

    @cached_method
    def smallest_moved_point(self):
        """
        Return the smallest point moved by a permutation in this group.

        EXAMPLES::

            sage: G = PermutationGroup([[(3,4)], [(2,3,4)]])
            sage: G.smallest_moved_point()
            2
            sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]])
            sage: G.smallest_moved_point()
            1

        Note that this function uses the ordering from the domain::

            sage: S = SymmetricGroup(['a','b','c'])
            sage: S.smallest_moved_point()
            'a'
        """
        return self._domain_from_gap[Integer(self._gap_().SmallestMovedPoint())]

    def representative_action(self,x,y):
        r"""
        Return an element of self that maps `x` to `y` if it exists.

        This method wraps the gap function ``RepresentativeAction``, which can
        also return elements that map a given set of points on another set of
        points.

        INPUT:

        - ``x,y`` -- two elements of the domain.

        EXAMPLES::

            sage: G = groups.permutation.Cyclic(14)
            sage: g = G.representative_action(1,10)
            sage: all(g(x) == 1+((x+9-1)%14) for x in G.domain())
            True

        TESTS::

            sage: g = graphs.PetersenGraph()
            sage: g.relabel(list("abcdefghik"))
            sage: g.vertices()
            ['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'k']
            sage: ag = g.automorphism_group()
            sage: a = ag.representative_action('a','b')
            sage: g == g.relabel(a,inplace=False)
            True
            sage: a('a') == 'b'
            True
        """
        ans = self._gap_().RepresentativeAction(self._domain_to_gap[x],
                                                self._domain_to_gap[y])
        return self.element_class(ans, self, check=False)

    @cached_method
    def orbits(self):
        """
        Returns the orbits of the elements of the domain under the
        default group action.

        EXAMPLES::

            sage: G = PermutationGroup([ [(3,4)], [(1,3)] ])
            sage: G.orbits()
            [[1, 3, 4], [2]]
            sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]])
            sage: G.orbits()
            [[1, 2, 3, 4, 10], [5], [6], [7], [8], [9]]

            sage: G = PermutationGroup([ [('c','d')], [('a','c')],[('b',)]])
            sage: G.orbits()
            [['a', 'c', 'd'], ['b']]

        The answer is cached::

            sage: G.orbits() is G.orbits()
            True

        AUTHORS:

        - Nathan Dunfield
        """
        return [[self._domain_from_gap[x] for x in orbit] for orbit in
                self._gap_().Orbits(self._domain_gap()).sage()]

    @cached_method
    def orbit(self, point, action="OnPoints"):
        """
        Return the orbit of a point under a group action.

        INPUT:

        - ``point`` -- can be a point or any of the list above, depending on the
          action to be considered.

        - ``action`` -- string. if ``point`` is an element from the domain, a
          tuple of elements of the domain, a tuple of tuples [...], this
          variable describes how the group is acting.

          The actions currently available through this method are
          ``"OnPoints"``, ``"OnTuples"``, ``"OnSets"``, ``"OnPairs"``,
          ``"OnSetsSets"``, ``"OnSetsDisjointSets"``,
          ``"OnSetsTuples"``, ``"OnTuplesSets"``,
          ``"OnTuplesTuples"``. They are taken from GAP's list of
          group actions, see ``gap.help('Group Actions')``.

          It is set to ``"OnPoints"`` by default. See below for examples.

        OUTPUT:

        The orbit of ``point`` as a tuple. Each entry is an image
        under the action of the permutation group, if necessary
        converted to the corresponding container. That is, if
        ``action='OnSets'`` then each entry will be a set even if
        ``point`` was given by a list/tuple/iterable.

        EXAMPLES::

            sage: G = PermutationGroup([ [(3,4)], [(1,3)] ])
            sage: G.orbit(3)
            (3, 4, 1)
            sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]])
            sage: G.orbit(3)
            (3, 4, 10, 1, 2)
            sage: G = PermutationGroup([ [('c','d')], [('a','c')] ])
            sage: G.orbit('a')
            ('a', 'c', 'd')

        Action of `S_3` on sets::

            sage: S3 = groups.permutation.Symmetric(3)
            sage: S3.orbit((1,2), action = "OnSets")
            ({1, 2}, {2, 3}, {1, 3})

        On tuples::

            sage: S3.orbit((1,2), action = "OnTuples")
            ((1, 2), (2, 3), (2, 1), (3, 1), (1, 3), (3, 2))

        Action of `S_4` on sets of disjoint sets::

            sage: S4 = groups.permutation.Symmetric(4)
            sage: S4.orbit(((1,2),(3,4)), action = "OnSetsDisjointSets")
            ({{1, 2}, {3, 4}}, {{2, 3}, {1, 4}}, {{1, 3}, {2, 4}})

        Action of `S_4` (on a nonstandard domain) on tuples of sets::

            sage: S4 = PermutationGroup([ [('c','d')], [('a','c')], [('a','b')] ])
            sage: S4.orbit((('a','c'),('b','d')),"OnTuplesSets")
            (({'a', 'c'}, {'b', 'd'}),
             ({'a', 'd'}, {'c', 'b'}),
             ({'c', 'b'}, {'a', 'd'}),
             ({'b', 'd'}, {'a', 'c'}),
             ({'c', 'd'}, {'a', 'b'}),
             ({'a', 'b'}, {'c', 'd'}))

        Action of `S_4` (on a very nonstandard domain) on tuples of sets::

            sage: S4 = PermutationGroup([ [((11,(12,13)),'d')],
            ....:         [((12,(12,11)),(11,(12,13)))], [((12,(12,11)),'b')] ])
            sage: S4.orbit((( (11,(12,13)), (12,(12,11))),('b','d')),"OnTuplesSets")
            (({(11, (12, 13)), (12, (12, 11))}, {'b', 'd'}),
             ({'d', (12, (12, 11))}, {(11, (12, 13)), 'b'}),
             ({(11, (12, 13)), 'b'}, {'d', (12, (12, 11))}),
             ({(11, (12, 13)), 'd'}, {'b', (12, (12, 11))}),
             ({'b', 'd'}, {(11, (12, 13)), (12, (12, 11))}),
             ({'b', (12, (12, 11))}, {(11, (12, 13)), 'd'}))
        """
        from sage.sets.set import Set
        actions = {
            "OnPoints"           : [],
            "OnSets"             : [Set],
            "OnPairs"            : [tuple],
            "OnTuples"           : [tuple],
            "OnSetsSets"         : [Set, Set],
            "OnSetsDisjointSets" : [Set, Set],
            "OnSetsTuples"       : [Set, tuple],
            "OnTuplesSets"       : [tuple, Set],
            "OnTuplesTuples"     : [tuple, tuple],
            }

        def input_for_gap(x, depth, container):
            if depth == len(container):
                try:
                    return self._domain_to_gap[x]
                except KeyError:
                    raise ValueError('{0} is not part of the domain'.format(x))
            x = [input_for_gap(xx, depth+1, container) for xx in x]
            if container[depth] is Set:
                x.sort()
            return x

        def gap_to_output(x, depth, container):
            if depth == len(container):
                return self._domain_from_gap[x]
            else:
                x = [gap_to_output(xx, depth+1, container) for xx in x]
                return container[depth](x)
        try:
            container = actions[action]
        except KeyError:
            raise NotImplementedError("This action is not implemented (yet?).")
        point = input_for_gap(point, 0, container)
        result = self._gap_().Orbit(point, action).sage()
        result = [gap_to_output(x, 0, container) for x in result]
        return tuple(result)

    def transversals(self, point):
        """
        If G is a permutation group acting on the set `X = \{1, 2, ...., n\}`
        and H is the stabilizer subgroup of <integer>, a right
        (respectively left) transversal is a set containing exactly
        one element from each right (respectively left) coset of H. This
        method returns a right transversal of ``self`` by the stabilizer
        of ``self`` on <integer> position.

        EXAMPLES::

            sage: G = PermutationGroup([ [(3,4)], [(1,3)] ])
            sage: G.transversals(1)
            [(), (1,3,4), (1,4,3)]
            sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]])
            sage: G.transversals(1)
            [(), (1,2)(3,4), (1,3,2,10,4), (1,4,2,10,3), (1,10,4,3,2)]

            sage: G = PermutationGroup([ [('c','d')], [('a','c')] ])
            sage: G.transversals('a')
            [(), ('a','c','d'), ('a','d','c')]
        """
        G = self._gap_()
        return [self(G.RepresentativeAction(self._domain_to_gap[point], self._domain_to_gap[i]))
                for i in self.orbit(point)]

    def stabilizer(self, point, action="OnPoints"):
        """
        Return the subgroup of ``self`` which stabilize the given position.
        ``self`` and its stabilizers must have same degree.

        INPUT:

        - ``point`` -- a point of the :meth:`domain`, or a set of points
          depending on the value of ``action``.

        - ``action`` (string; default ``"OnPoints"``) -- should the group be
          considered to act on points (``action="OnPoints"``) or on sets of
          points (``action="OnSets"``) ? In the latter case, the first argument
          must be a subset of :meth:`domain`.

        EXAMPLES::

            sage: G = PermutationGroup([ [(3,4)], [(1,3)] ])
            sage: G.stabilizer(1)
            Subgroup of (Permutation Group with generators [(3,4), (1,3)]) generated by [(3,4)]
            sage: G.stabilizer(3)
            Subgroup of (Permutation Group with generators [(3,4), (1,3)]) generated by [(1,4)]

        The stabilizer of a set of points::

            sage: s10 = groups.permutation.Symmetric(10)
            sage: s10.stabilizer([1..3],"OnSets").cardinality()
            30240
            sage: factorial(3)*factorial(7)
            30240

        ::

            sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]])
            sage: G.stabilizer(10)
            Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4,10)]) generated by [(2,3,4), (1,2)(3,4)]
            sage: G.stabilizer(1)
            Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4,10)]) generated by [(2,3)(4,10), (2,10,4)]
            sage: G = PermutationGroup([[(2,3,4)],[(6,7)]])
            sage: G.stabilizer(1)
            Subgroup of (Permutation Group with generators [(6,7), (2,3,4)]) generated by [(6,7), (2,3,4)]
            sage: G.stabilizer(2)
            Subgroup of (Permutation Group with generators [(6,7), (2,3,4)]) generated by [(6,7)]
            sage: G.stabilizer(3)
            Subgroup of (Permutation Group with generators [(6,7), (2,3,4)]) generated by [(6,7)]
            sage: G.stabilizer(4)
            Subgroup of (Permutation Group with generators [(6,7), (2,3,4)]) generated by [(6,7)]
            sage: G.stabilizer(5)
            Subgroup of (Permutation Group with generators [(6,7), (2,3,4)]) generated by [(6,7), (2,3,4)]
            sage: G.stabilizer(6)
            Subgroup of (Permutation Group with generators [(6,7), (2,3,4)]) generated by [(2,3,4)]
            sage: G.stabilizer(7)
            Subgroup of (Permutation Group with generators [(6,7), (2,3,4)]) generated by [(2,3,4)]
            sage: G.stabilizer(8)
            Traceback (most recent call last):
            ...
            ValueError: 8 does not belong to the domain

        ::

            sage: G = PermutationGroup([ [('c','d')], [('a','c')] ], domain='abcd')
            sage: G.stabilizer('a')
            Subgroup of (Permutation Group with generators [('c','d'), ('a','c')]) generated by [('c','d')]
            sage: G.stabilizer('b')
            Subgroup of (Permutation Group with generators [('c','d'), ('a','c')]) generated by [('c','d'), ('a','c')]
            sage: G.stabilizer('c')
            Subgroup of (Permutation Group with generators [('c','d'), ('a','c')]) generated by [('a','d')]
            sage: G.stabilizer('d')
            Subgroup of (Permutation Group with generators [('c','d'), ('a','c')]) generated by [('a','c')]

        TESTS::

            sage: G.stabilizer(['a'],"OnMonkeys")
            Traceback (most recent call last):
            ...
            ValueError: 'action' must be equal to 'OnPoints' or to 'OnSets'.
        """
        if (action != "OnPoints" and
            action != "OnSets"):
            raise ValueError("'action' must be equal to 'OnPoints' or to 'OnSets'.")
        try:
            if action == "OnPoints":
                point = self._domain_to_gap[point]
            else:
                point = sorted([self._domain_to_gap[x] for x in point])
        except KeyError as x:
            raise ValueError("{} does not belong to the domain".format(x))

        return self.subgroup(gap_group=gap.Stabilizer(self, point, action))

    def base(self, seed=None):
        r"""
        Returns a (minimum) base of this permutation group. A base $B$
        of a permutation group is a subset of the domain of the group
        such that the only group element stabilizing all of $B$ is the
        identity.

        The argument `seed` is optional and must be a subset of the domain
        of `base`. When used, an attempt to create a base containing all or part
        of `seed` will be made.

        EXAMPLES::

            sage: G = PermutationGroup([(1,2,3),(6,7,8)])
            sage: G.base()
            [1, 6]
            sage: G.base([2])
            [2, 6]

            sage: H = PermutationGroup([('a','b','c'),('a','y')])
            sage: H.base()
            ['a', 'b', 'c']

            sage: S = SymmetricGroup(13)
            sage: S.base()
            [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]

            sage: S = MathieuGroup(12)
            sage: S.base()
            [1, 2, 3, 4, 5]
            sage: S.base([1,3,5,7,9,11]) # create a base for M12 with only odd integers
            [1, 3, 5, 7, 9]
        """
        if seed is None:
            seed = self.domain()

        try:
            seed = [self._domain_to_gap[point] for point in seed]
        except KeyError:
            raise ValueError("seed must be a subset of self.domain()")

        return [self._domain_from_gap[x] for x in self._gap_().StabChain(seed).BaseStabChain().sage()]

    def strong_generating_system(self, base_of_group=None):
        """
        Return a Strong Generating System of ``self`` according the given
        base for the right action of ``self`` on itself.

        ``base_of_group`` is a list of the  positions on which ``self`` acts,
        in any order. The algorithm returns a list of transversals and each
        transversal is a list of permutations. By default, ``base_of_group``
        is ``[1, 2, 3, ..., d]`` where `d` is the degree of the group.

        For ``base_of_group`` =
        `[ \mathrm{pos}_1, \mathrm{pos}_2, \dots , \mathrm{pos}_d]`
        let `G_i` be the subgroup of `G` = ``self`` which stabilizes
        `\mathrm{pos}_1, \mathrm{pos}_2, \dots , \mathrm{pos}_i`, so

        .. MATH::

           G = G_0 \supset G_1 \supset G_2 \supset \dots \supset G_n = \{e\}

        Then the algorithm returns `[ G_i.\mathrm{transversals}(\mathrm{pos}_{i+1})]_{1 \leq i \leq n}`

        INPUT:

        - ``base_of_group`` (optional) -- default: ``[1, 2, 3, ..., d]``
          -- a list containing the integers
          `1, 2, \dots , d` in any order (`d` is the degree of ``self``)

        OUTPUT:

        - A list of lists of permutations from the group, which form a strong
          generating system.

        EXAMPLES::

            sage: G = PermutationGroup([[(7,8)],[(3,4)],[(4,5)]])
            sage: G.strong_generating_system()
            [[()], [()], [(), (3,4,5), (3,5)], [(), (4,5)], [()], [()], [(), (7,8)], [()]]
            sage: G = PermutationGroup([[(1,2,3,4)],[(1,2)]])
            sage: G.strong_generating_system()
            [[(), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)], [(), (2,3,4), (2,4,3)], [(), (3,4)], [()]]
            sage: G = PermutationGroup([[(1,2,3)],[(4,5,7)],[(1,4,6)]])
            sage: G.strong_generating_system()
            [[(), (1,2,3), (1,4,6), (1,3,2), (1,5,7,4,6), (1,6,4), (1,7,5,4,6)], [(), (2,6,3), (2,5,7,6,3), (2,3,6), (2,7,5,6,3), (2,4,7,6,3)], [(), (3,6,7), (3,5,6), (3,7,6), (3,4,7,5,6)], [(), (4,5)(6,7), (4,7)(5,6), (4,6)(5,7)], [(), (5,7,6), (5,6,7)], [()], [()]]
            sage: G = PermutationGroup([[(1,2,3)],[(2,3,4)],[(3,4,5)]])
            sage: G.strong_generating_system([5,4,3,2,1])
            [[(), (1,5,3,4,2), (1,5,4,3,2), (1,5)(2,3), (1,5,2)], [(), (1,3)(2,4), (1,2)(3,4), (1,4)(2,3)], [(), (1,3,2), (1,2,3)], [()], [()]]
            sage: G = PermutationGroup([[(3,4)]])
            sage: G.strong_generating_system()
            [[()], [()], [(), (3,4)], [()]]
            sage: G.strong_generating_system(base_of_group=[3,1,2,4])
            [[(), (3,4)], [()], [()], [()]]
            sage: G = TransitiveGroup(12,17)                # optional - database_gap
            sage: G.strong_generating_system()              # optional - database_gap
            [[(), (1,4,11,2)(3,6,5,8)(7,10,9,12), (1,8,3,2)(4,11,10,9)(5,12,7,6), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (1,12,7,2)(3,10,9,8)(4,11,6,5), (1,11)(2,8)(3,5)(4,10)(6,12)(7,9), (1,10,11,8)(2,3,12,5)(4,9,6,7), (1,3)(2,8)(4,10)(5,7)(6,12)(9,11), (1,2,3,8)(4,9,10,11)(5,6,7,12), (1,6,7,8)(2,3,4,9)(5,10,11,12), (1,5,9)(3,11,7), (1,9,5)(3,7,11)], [(), (2,6,10)(4,12,8), (2,10,6)(4,8,12)], [()], [()], [()], [()], [()], [()], [()], [()], [()], [()]]

        TESTS::

            sage: G = SymmetricGroup(10)
            sage: H = PermutationGroup([G.random_element() for i in range(randrange(1,3,1))])
            sage: prod(len(x) for x in H.strong_generating_system()) == H.cardinality()
            True
        """
        sgs = []
        stab = self
        if base_of_group is None:
            base_of_group = self.domain()
        for j in base_of_group:
            sgs.append(stab.transversals(j))
            stab = stab.stabilizer(j)
        return sgs


    def _repr_(self):
        r"""
        Returns a string describing ``self``.

        EXAMPLES:

        We explicitly construct the alternating group on four
        elements. Note that the ``AlternatingGroup`` class has
        its own representation string::

            sage: A4 = PermutationGroup([[(1,2,3)],[(2,3,4)]]); A4
            Permutation Group with generators [(2,3,4), (1,2,3)]
            sage: A4._repr_()
            'Permutation Group with generators [(2,3,4), (1,2,3)]'
            sage: AlternatingGroup(4)._repr_()
            'Alternating group of order 4!/2 as a permutation group'
        """
        return "Permutation Group with generators %s"%self.gens()

    def _latex_(self):
        r"""
        Method for describing ``self`` in LaTeX. Encapsulates
        ``self.gens()`` in angle brackets to denote that ``self``
        is generated by these elements. Called by the
        ``latex()`` function.

        EXAMPLES:

        We explicitly construct the alternating group on four
        elements::

            sage: A4 = PermutationGroup([[(1,2,3)],[(2,3,4)]]); A4
            Permutation Group with generators [(2,3,4), (1,2,3)]
            sage: latex(A4)
            \langle (2,3,4), (1,2,3) \rangle
            sage: A4._latex_()
            '\\langle (2,3,4), (1,2,3) \\rangle'

            sage: S = SymmetricGroup(['a','b','c'])
            sage: latex(S)
            \langle (\text{\texttt{a}},\text{\texttt{b}},\text{\texttt{c}}), (\text{\texttt{a}},\text{\texttt{b}}) \rangle
        """
        return '\\langle ' + \
               ', '.join([x._latex_() for x in self.gens()]) + ' \\rangle'

    def _order(self):
        """
        This handles a few special cases of computing the subgroup order much
        faster than GAP.

        This currently operates very quickly for stabilizer subgroups of
        permutation groups, for one.

        Will return None if the we could not easily compute it.

        Author: Christopher Swenson

        EXAMPLES::

            sage: G = SymmetricGroup(10).subgroup([(i, 10) for i in range(1, 10) if i != 4])
            sage: G._order()
            362880

        TESTS::

            sage: [SymmetricGroup(n).stabilizer(1)._gap_().Size() for n in [4..10]]
            [6, 24, 120, 720, 5040, 40320, 362880]
            sage: special_gens = [
            ....:     [(3,4), (2,4)],
            ....:     [(4,5), (3,5), (2,5)],
            ....:     [(5,6), (4,6), (3,6), (2,6)],
            ....:     [(6,7), (5,7), (4,7), (3,7), (2,7)],
            ....:     [(7,8), (6,8), (5,8), (4,8), (3,8), (2,8)],
            ....:     [(8,9), (7,9), (6,9), (5,9), (4,9), (3,9), (2,9)],
            ....:     [(9,10), (8,10), (7,10), (6,10), (5,10), (4,10), (3,10), (2,10)]]
            sage: [SymmetricGroup(n).subgroup(gen)._order() for gen in special_gens]
            [6, 24, 120, 720, 5040, 40320, 362880]

        Checks that output is a Sage integer (:trac:`23778`)::

            sage: P = PermutationGroup(['(1,2)','(1,3)'])
            sage: P.cardinality()
            6
            sage: isinstance(P.cardinality(), Integer)
            True
        """
        gens = self.gens()
        # This special case only works with more than 1 generator.
        if not gens or len(gens) < 2:
            return None
        # Special case: certain subgroups of the symmetric group for which Gap reports
        # generators of the form ((1, 2), (1, 3), ...)
        # This means that this group is isomorphic to a smaller symmetric group
        # S_n, where n is the number of generators supported.
        #
        # The code that follows checks that the following assumptions hold:
        #     * All generators are transpositions (i.e., permutations
        #     that switch two elements and leave everything else fixed),
        #     * All generators share a common element.
        #
        # We then know that this group is isomorphic to S_n,
        # and therefore its order is n!.

        # Check that all generators are order 2 and have length-1 cycle tuples.
        cycle_tuples = []
        for g in gens:
            if g.order() != 2:
                return None
            c = g.cycle_tuples()
            if len(c) != 1:
                return None
            cycle_tuples.append(c)

        # Find the common element.
        g0 = cycle_tuples[0][0]
        g1 = cycle_tuples[1][0]
        a, b = g0
        if a not in g1 and b not in g1:
            return None
        if a in g1:
            elem = a
        else:
            elem = b
        # Count the number of unique elements in the generators.
        unique = set()
        for c in cycle_tuples:
            a, b = c[0]
            if a != elem and b != elem:
                return None
            unique.add(a)
            unique.add(b)

        # Compute the order.
        return Integer(len(unique)).factorial()

    def order(self):
        """
        Return the number of elements of this group.
        See also: G.degree()

        EXAMPLES::

            sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]])
            sage: G.order()
            12
            sage: G = PermutationGroup([()])
            sage: G.order()
            1
            sage: G = PermutationGroup([])
            sage: G.order()
            1

        ``cardinality`` is just an alias::

            sage: PermutationGroup([(1,2,3)]).cardinality()
            3
        """
        gens = self.gens()

        if not gens:
            return Integer(1)
        elif len(gens) == 1:
            return gens[0].order()

        subgroup_order = self._order()
        if subgroup_order is not None:
          return subgroup_order

        return Integer(self._gap_().Size())

    cardinality = order

    def random_element(self):
        """
        Return a random element of this group.

        EXAMPLES::

            sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]])
            sage: a = G.random_element()
            sage: a in G
            True
            sage: a.parent() is G
            True
            sage: a^6
            ()
        """
        current_randstate().set_seed_gap()
        return self(self._gap_().Random(), check=False)

    def group_id(self):
        """
        Return the ID code of this group, which is a list of two integers.
        Requires "optional" database_gap package.

        EXAMPLES::

            sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]])
            sage: G.group_id()    # optional - database_gap
            [12, 4]
        """
        from sage.features.gap import SmallGroupsLibrary
        SmallGroupsLibrary().require()

        return [Integer(n) for n in self._gap_().IdGroup()]

    def id(self):
        """
        (Same as ``self.group_id()``.) Return the ID code of this group, which
        is a list of two integers. Requires "optional" database_gap package.

        EXAMPLES::

            sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]])
            sage: G.group_id()    # optional - database_gap
            [12, 4]
        """
        return self.group_id()

    def group_primitive_id(self):
        """
        Return the index of this group in the GAP database of primitive groups.

        Requires "optional" database_gap package.

        OUTPUT:

        A positive integer, following GAP's conventions. A
        ``ValueError`` is raised if the group is not primitive.

        EXAMPLES::

            sage: G = PermutationGroup([[(1,2,3,4,5)], [(1,5),(2,4)]])
            sage: G.group_primitive_id()    # optional - database_gap
            2
            sage: G.degree()
            5

        From the information of the degree and the identification number,
        you can recover the isomorphism class of your group in the GAP
        database::

            sage: H = PrimitiveGroup(5,2)  # optional - database_gap
            sage: G == H                     # optional - database_gap
            False
            sage: G.is_isomorphic(H)         # optional - database_gap
            True
        """
        if not self.is_primitive():
            raise ValueError('Group is not primitive')

        from sage.features.gap import PrimitiveGroupsLibrary
        PrimitiveGroupsLibrary().require()

        return Integer(self._gap_().PrimitiveIdentification())

    def center(self):
        """
        Return the subgroup of elements that commute with every element
        of this group.

        EXAMPLES::

            sage: G = PermutationGroup([[(1,2,3,4)]])
            sage: G.center()
            Subgroup of (Permutation Group with generators [(1,2,3,4)]) generated by [(1,2,3,4)]
            sage: G = PermutationGroup([[(1,2,3,4)], [(1,2)]])
            sage: G.center()
            Subgroup of (Permutation Group with generators [(1,2), (1,2,3,4)]) generated by [()]
        """
        return self.subgroup(gap_group=self._gap_().Center())

    def socle(self):
        r"""
        Returns the socle of ``self``. The socle of a group $G$ is
        the subgroup generated by all minimal normal subgroups.

        EXAMPLES::

            sage: G=SymmetricGroup(4)
            sage: G.socle()
            Subgroup of (Symmetric group of order 4! as a permutation group) generated by [(1,2)(3,4), (1,4)(2,3)]
            sage: G.socle().socle()
            Subgroup of (Subgroup of (Symmetric group of order 4! as a permutation group) generated by [(1,2)(3,4), (1,4)(2,3)]) generated by [(1,2)(3,4), (1,4)(2,3)]
        """
        return self.subgroup(gap_group=self._gap_().Socle())

    def frattini_subgroup(self):
        r"""
        Returns the Frattini subgroup of ``self``.

        The Frattini subgroup of a group $G$ is the intersection of all maximal
        subgroups of `G`.

        EXAMPLES::

            sage: G=PermutationGroup([[(1,2,3,4)],[(2,4)]])
            sage: G.frattini_subgroup()
            Subgroup of (Permutation Group with generators [(2,4), (1,2,3,4)]) generated by [(1,3)(2,4)]
            sage: G=SymmetricGroup(4)
            sage: G.frattini_subgroup()
            Subgroup of (Symmetric group of order 4! as a permutation group) generated by [()]

        """
        return self.subgroup(gap_group=self._gap_().FrattiniSubgroup())

    def fitting_subgroup(self):
        r"""
        Returns the Fitting subgroup of ``self``.

        The Fitting subgroup of a group $G$ is the largest nilpotent normal
        subgroup of `G`.

        EXAMPLES::

            sage: G=PermutationGroup([[(1,2,3,4)],[(2,4)]])
            sage: G.fitting_subgroup()
            Subgroup of (Permutation Group with generators [(2,4), (1,2,3,4)]) generated by [(2,4), (1,2,3,4), (1,3)]
            sage: G=PermutationGroup([[(1,2,3,4)],[(1,2)]])
            sage: G.fitting_subgroup()
            Subgroup of (Permutation Group with generators [(1,2), (1,2,3,4)]) generated by [(1,2)(3,4), (1,3)(2,4)]

        """
        return self.subgroup(gap_group=self._gap_().FittingSubgroup())

    def solvable_radical(self):
        r"""
        Returns the solvable radical of ``self``. The solvable
        radical (or just radical) of a group $G$ is the largest
        solvable normal subgroup of $G$.

        EXAMPLES::

            sage: G=SymmetricGroup(4)
            sage: G.solvable_radical()
            Subgroup of (Symmetric group of order 4! as a permutation group) generated by [(1,2), (1,2,3,4)]
            sage: G=SymmetricGroup(5)
            sage: G.solvable_radical()
            Subgroup of (Symmetric group of order 5! as a permutation group) generated by [()]

        """
        return self.subgroup(gap_group=self._gap_().RadicalGroup())

    def intersection(self, other):
        r"""
        Returns the permutation group that is the intersection of
        ``self`` and ``other``.

        INPUT:

        - ``other`` - a permutation group.

        OUTPUT:

        A permutation group that is the set-theoretic intersection of ``self``
        with ``other``.  The groups are viewed as subgroups of a symmetric
        group big enough to contain both group's symbol sets.  So there is
        no strict notion of the two groups being subgroups of a common parent.

        EXAMPLES::

            sage: H = DihedralGroup(4)

            sage: K = CyclicPermutationGroup(4)
            sage: H.intersection(K)
            Permutation Group with generators [(1,2,3,4)]

            sage: L = DihedralGroup(5)
            sage: H.intersection(L)
            Permutation Group with generators [(1,4)(2,3)]

            sage: M = PermutationGroup(["()"])
            sage: H.intersection(M)
            Permutation Group with generators [()]

        Some basic properties. ::

            sage: H = DihedralGroup(4)
            sage: L = DihedralGroup(5)
            sage: H.intersection(L) == L.intersection(H)
            True
            sage: H.intersection(H) == H
            True

        The group ``other`` is verified as such.  ::

            sage: H = DihedralGroup(4)
            sage: H.intersection('junk')
            Traceback (most recent call last):
            ...
            TypeError: junk is not a permutation group
        """
        from sage.categories.finite_permutation_groups import FinitePermutationGroups
        if other not in FinitePermutationGroups():
            raise TypeError("{0} is not a permutation group".format(other))
        return PermutationGroup(gap_group=gap.Intersection(self, other))

    def conjugacy_class(self, g):
        r"""
        Return the conjugacy class of ``g`` inside the group ``self``.

        INPUT:

        - ``g`` -- an element of the permutation group ``self``

        OUTPUT:

        The conjugacy class of ``g`` in the group ``self``. If ``self`` is
        the group denoted by `G`, this method computes the set
        `\{x^{-1}gx\ \vert\ x \in G \}`

        EXAMPLES::

            sage: G = DihedralGroup(3)
            sage: g = G.gen(0)
            sage: G.conjugacy_class(g)
            Conjugacy class of (1,2,3) in Dihedral group of order 6 as a permutation group
        """
        return ConjugacyClassGAP(self, g)

    def conjugacy_classes(self):
        r"""
        Return a list with all the conjugacy classes of ``self``.

        EXAMPLES::

            sage: G = DihedralGroup(3)
            sage: G.conjugacy_classes()
            [Conjugacy class of () in Dihedral group of order 6 as a permutation group,
             Conjugacy class of (2,3) in Dihedral group of order 6 as a permutation group,
             Conjugacy class of (1,2,3) in Dihedral group of order 6 as a permutation group]
        """
        cl = self._gap_().ConjugacyClasses()
        n = Integer(cl.Length())
        L = gap("List([1..Length(%s)], i->Representative(%s[i]))"%(cl.name(),  cl.name()))
        return [ConjugacyClassGAP(self,self.element_class(L[i], self, check=False)) for i in range(1,n+1)]

    def conjugate(self, g):
        r"""
        Returns the group formed by conjugating ``self`` with ``g``.

        INPUT:

        - ``g`` - a permutation group element, or an object that converts
          to a permutation group element, such as a list of integers or
          a string of cycles.

        OUTPUT:

        If ``self`` is the group denoted by `H`, then this method computes
        the group

        .. MATH::

            g^{-1}Hg = \{g^{-1}hg\vert h\in H\}

        which is the group `H` conjugated by `g`.

        There are no restrictions on ``self`` and ``g`` belonging to
        a common permutation group, and correspondingly, there is no
        relationship (such as a common parent) between ``self`` and the
        output group.

        EXAMPLES::

            sage: G = DihedralGroup(6)
            sage: a = PermutationGroupElement("(1,2,3,4)")
            sage: G.conjugate(a)
            Permutation Group with generators [(1,4)(2,6)(3,5), (1,5,6,2,3,4)]

        The element performing the conjugation can be specified in
        several ways.  ::

            sage: G = DihedralGroup(6)
            sage: strng = "(1,2,3,4)"
            sage: G.conjugate(strng)
            Permutation Group with generators [(1,4)(2,6)(3,5), (1,5,6,2,3,4)]
            sage: G = DihedralGroup(6)
            sage: lst = [2,3,4,1]
            sage: G.conjugate(lst)
            Permutation Group with generators [(1,4)(2,6)(3,5), (1,5,6,2,3,4)]
            sage: G = DihedralGroup(6)
            sage: cycles = [(1,2,3,4)]
            sage: G.conjugate(cycles)
            Permutation Group with generators [(1,4)(2,6)(3,5), (1,5,6,2,3,4)]

        Conjugation is a group automorphism, so conjugate groups
        will be isomorphic. ::

            sage: G = DiCyclicGroup(6)
            sage: G.degree()
            11
            sage: cycle = [i+1 for i in range(1,11)] + [1]
            sage: C = G.conjugate(cycle)
            sage: G.is_isomorphic(C)
            True

        The conjugating element may be from a symmetric group with
        larger degree than the group being conjugated.  ::

            sage: G = AlternatingGroup(5)
            sage: G.degree()
            5
            sage: g = "(1,3)(5,6,7)"
            sage: H = G.conjugate(g); H
            Permutation Group with generators [(1,4,6,3,2), (1,4,6)]
            sage: H.degree()
            6

        The conjugating element is checked. ::

            sage: G = SymmetricGroup(3)
            sage: G.conjugate("junk")
            Traceback (most recent call last):
            ...
            TypeError: junk does not convert to a permutation group element
        """
        try:
            g = PermutationGroupElement(g)
        except Exception:
            raise TypeError("{0} does not convert to a permutation group element".format(g))
        return PermutationGroup(gap_group=gap.ConjugateGroup(self, g))

    def direct_product(self, other, maps=True):
        """
        Wraps GAP's ``DirectProduct``, ``Embedding``, and ``Projection``.

        Sage calls GAP's ``DirectProduct``, which chooses an efficient
        representation for the direct product. The direct product of
        permutation groups will be a permutation group again. For a direct
        product ``D``, the GAP operation ``Embedding(D,i)`` returns the
        homomorphism embedding the i-th factor into ``D``. The GAP operation
        ``Projection(D,i)`` gives the projection of ``D`` onto the i-th factor.
        This method returns a 5-tuple: a permutation group and 4 morphisms.

        INPUT:

        -  ``self, other`` - permutation groups

        OUTPUT:

        -  ``D`` - a direct product of the inputs, returned as
           a permutation group as well

        -  ``iota1`` - an embedding of ``self`` into ``D``

        -  ``iota2`` - an embedding of ``other`` into ``D``

        -  ``pr1`` - the projection of ``D`` onto ``self`` (giving a
           splitting 1 - other - D - self - 1)

        -  ``pr2`` - the projection of ``D`` onto ``other`` (giving a
           splitting 1 - self - D - other - 1)

        EXAMPLES::

            sage: G = CyclicPermutationGroup(4)
            sage: D = G.direct_product(G,False)
            sage: D
            Permutation Group with generators [(5,6,7,8), (1,2,3,4)]
            sage: D,iota1,iota2,pr1,pr2 = G.direct_product(G)
            sage: D; iota1; iota2; pr1; pr2
            Permutation Group with generators [(5,6,7,8), (1,2,3,4)]
            Permutation group morphism:
              From: Cyclic group of order 4 as a permutation group
              To:   Permutation Group with generators [(5,6,7,8), (1,2,3,4)]
              Defn: Embedding( Group( [ (1,2,3,4), (5,6,7,8) ] ), 1 )
            Permutation group morphism:
              From: Cyclic group of order 4 as a permutation group
              To:   Permutation Group with generators [(5,6,7,8), (1,2,3,4)]
              Defn: Embedding( Group( [ (1,2,3,4), (5,6,7,8) ] ), 2 )
            Permutation group morphism:
              From: Permutation Group with generators [(5,6,7,8), (1,2,3,4)]
              To:   Cyclic group of order 4 as a permutation group
              Defn: Projection( Group( [ (1,2,3,4), (5,6,7,8) ] ), 1 )
            Permutation group morphism:
              From: Permutation Group with generators [(5,6,7,8), (1,2,3,4)]
              To:   Cyclic group of order 4 as a permutation group
              Defn: Projection( Group( [ (1,2,3,4), (5,6,7,8) ] ), 2 )
            sage: g=D([(1,3),(2,4)]); g
            (1,3)(2,4)
            sage: d=D([(1,4,3,2),(5,7),(6,8)]); d
            (1,4,3,2)(5,7)(6,8)
            sage: iota1(g); iota2(g); pr1(d); pr2(d)
            (1,3)(2,4)
            (5,7)(6,8)
            (1,4,3,2)
            (1,3)(2,4)
        """
        G = self._gap_().DirectProduct(other)
        D = PermutationGroup(gap_group=G)
        if not maps:
            return D
        else:
            from sage.groups.perm_gps.permgroup_morphism import PermutationGroupMorphism_from_gap
            iota1 = PermutationGroupMorphism_from_gap(self,  D, G.Embedding(1))
            iota2 = PermutationGroupMorphism_from_gap(other, D, G.Embedding(2))
            pr1 = PermutationGroupMorphism_from_gap(D, self, G.Projection(1))
            pr2 = PermutationGroupMorphism_from_gap(D, other, G.Projection(2))
            return D, iota1, iota2, pr1, pr2

    def semidirect_product(self, N, mapping, check=True):
        r"""
        The semidirect product of ``self`` with ``N``.

        INPUT:

        - ``N`` - A group which is acted on by ``self`` and
          naturally embeds as a normal subgroup of the returned semidirect
          product.

        - ``mapping`` - A pair of lists that together define a
          homomorphism, `\phi :` self `\rightarrow` Aut(N), by giving,
          in the second list, the images of the generators of ``self``
          in the order given in the first list.

        - ``check`` - A boolean that, if set to False, will skip the
          initial tests which are made on ``mapping``. This may be beneficial
          for large ``N``, since in such cases the injectivity test can be
          expensive. Set to True by default.

        OUTPUT:

        The semidirect product of ``self`` and ``N`` defined by the
        action of ``self`` on ``N`` given in ``mapping`` (note that a
        homomorphism from A to the automorphism group of B is
        equivalent to an action of A on the B's underlying set). The
        semidirect product of two groups, `H` and `N`, is a construct
        similar to the direct product in so far as the elements are
        the Cartesian product of the elements of `H` and the elements
        of `N`. The operation, however, is built upon an action of `H`
        on `N`, and is defined as such:

        .. MATH::

                (h_1,n_1)(h_2,n_2) = (h_{1}h_{2}, n_{1}^{h_2}n_2)

        This function is a wrapper for GAP's ``SemidirectProduct``
        command. The permutation group returned is built upon a
        permutation representation of the semidirect product of ``self``
        and ``N`` on a set of size `\mid N \mid`. The generators of
        ``N`` are given as their right regular representations, while the
        generators of ``self`` are defined by the underlying action of
        ``self`` on ``N``. It should be noted that the defining action is
        not always faithful, and in this case the inputted representations
        of the generators of ``self`` are placed on additional letters
        and adjoined to the output's generators of ``self``.


        EXAMPLES:

        Perhaps the most common example of a semidirect product comes
        from the family of dihedral groups. Each dihedral group is the
        semidirect product of $C_2$ with $C_n$, where, by convention,
        $3 \leq n$. In this case, the nontrivial element of $C_2$ acts
        on $C_n$ so as to send each element to its inverse. ::

            sage: C2 = CyclicPermutationGroup(2)
            sage: C8 = CyclicPermutationGroup(8)
            sage: alpha = PermutationGroupMorphism_im_gens(C8,C8,[(1,8,7,6,5,4,3,2)])
            sage: S = C2.semidirect_product(C8,[[(1,2)],[alpha]])
            sage: S == DihedralGroup(8)
            False
            sage: S.is_isomorphic(DihedralGroup(8))
            True
            sage: S.gens()
            [(3,4,5,6,7,8,9,10), (1,2)(4,10)(5,9)(6,8)]

        A more complicated example can be drawn from [TW1980]_.
        It is there given that a semidirect product of $D_4$ and $C_3$
        is isomorphic to one of $C_2$ and the dicyclic group of order
        12. This nonabelian group of order 24 has very similar
        structure to the dicyclic and dihedral groups of order 24, the
        three being the only groups of order 24 with a two-element
        center and 9 conjugacy classes. ::

            sage: D4 = DihedralGroup(4)
            sage: C3 = CyclicPermutationGroup(3)
            sage: alpha1 = PermutationGroupMorphism_im_gens(C3,C3,[(1,3,2)])
            sage: alpha2 = PermutationGroupMorphism_im_gens(C3,C3,[(1,2,3)])
            sage: S1 = D4.semidirect_product(C3,[[(1,2,3,4),(1,3)],[alpha1,alpha2]])
            sage: C2 = CyclicPermutationGroup(2)
            sage: Q = DiCyclicGroup(3)
            sage: a = Q.gens()[0]; b=Q.gens()[1].inverse()
            sage: alpha = PermutationGroupMorphism_im_gens(Q,Q,[a,b])
            sage: S2 = C2.semidirect_product(Q,[[(1,2)],[alpha]])
            sage: S1.is_isomorphic(S2)
            True
            sage: S1.is_isomorphic(DihedralGroup(12))
            False
            sage: S1.is_isomorphic(DiCyclicGroup(6))
            False
            sage: S1.center()
            Subgroup of (Permutation Group with generators
            [(5,6,7), (1,2,3,4)(6,7), (1,3)]) generated by [(1,3)(2,4)]
            sage: len(S1.conjugacy_classes_representatives())
            9

        If your normal subgroup is large, and you are confident that
        your inputs will successfully create a semidirect product, then
        it is beneficial, for the sake of time efficiency, to set the
        ``check`` parameter to ``False``.  ::

            sage: C2 = CyclicPermutationGroup(2)
            sage: C2000 = CyclicPermutationGroup(500)
            sage: alpha = PermutationGroupMorphism(C2000,C2000,[C2000.gen().inverse()])
            sage: S = C2.semidirect_product(C2000,[[(1,2)],[alpha]],check=False)

        TESTS::

            sage: C3 = CyclicPermutationGroup(3)
            sage: D4 = DihedralGroup(4)
            sage: alpha = PermutationGroupMorphism(C3,C3,[C3("(1,3,2)")])
            sage: alpha1 = PermutationGroupMorphism(C3,C3,[C3("(1,2,3)")])

            sage: s = D4.semidirect_product('junk', [[(1,2,3,4),(1,2)], [alpha, alpha1]])
            Traceback (most recent call last):
            ...
            TypeError: junk is not a permutation group

            sage: s = D4.semidirect_product(C3, [[(1,2,3,4),(1,2)], [alpha, alpha1]])
            Traceback (most recent call last):
            ...
            ValueError: the generator list must generate the calling group, [(1, 2, 3, 4), (1, 2)]
            does not generate Dihedral group of order 8 as a permutation group

            sage: s = D4.semidirect_product(C3, [[(1,2,3,4),(1,3)], [alpha]])
            Traceback (most recent call last):
            ...
            ValueError: the list of generators and the list of morphisms must be of equal length

            sage: alpha2 = PermutationGroupMorphism(C3, D4, [D4("()")])
            sage: s = D4.semidirect_product(C3, [[(1,2,3,4),(1,3)], [alpha, alpha2]])
            Traceback (most recent call last):
            ...
            ValueError: an element of the automorphism list is not an endomorphism (and is therefore not an automorphism)

            sage: alpha3 = PermutationGroupMorphism(C3,C3,[C3("()")])
            sage: s = D4.semidirect_product(C3, [[(1,2,3,4),(1,3)], [alpha, alpha3]])
            Traceback (most recent call last):
            ...
            ValueError: an element of the automorphism list is not an injection (and is therefore not an automorphism)

        AUTHOR:

        - Kevin Halasz (2012-8-12)

        """
        if check:
            from sage.categories.finite_permutation_groups import FinitePermutationGroups
            if N not in FinitePermutationGroups():
                raise TypeError("{0} is not a permutation group".format(N))
            if not PermutationGroup(gens = mapping[0]) == self:
                msg = 'the generator list must generate the calling group, {0} does not generate {1}'
                raise ValueError(msg.format(mapping[0], self._repr_()))
            if len(mapping[0]) != len(mapping[1]):
                msg = 'the list of generators and the list of morphisms must be of equal length'
                raise ValueError(msg)
            if not all([a.is_endomorphism() for a in mapping[1]]):
                msg = 'an element of the automorphism list is not an endomorphism (and is therefore not an automorphism)'
                raise ValueError(msg)
            if not all([a.kernel().order() == 1 for a in mapping[1]]):
                msg = 'an element of the automorphism list is not an injection (and is therefore not an automorphism)'
                raise ValueError(msg)

        # create a parallel list of the automorphisms of N in GAP
        gap.eval('N := Group(' + str(N.gens()) + ')')
        gens_string = ",".join([str(x) for x in N.gens()])
        homomorphism_cmd = 'alpha := GroupHomomorphismByImages(N, N, [{0}],[{1}])'
        gap.eval('morphisms := []')
        for alpha in mapping[1]:
            images_string = ",".join([str(alpha(n)) for n in N.gens()])
            gap.eval(homomorphism_cmd.format(gens_string, images_string))
            gap.eval('Add(morphisms, alpha)')
        # create the necessary homomorphism from self into the
        # automorphism group of N in GAP
        gap.eval('H := Group({0})'.format(mapping[0]))
        gap.eval('phi := GroupHomomorphismByImages(H, AutomorphismGroup(N),{},morphisms)'.format(mapping[0]))
        gap.eval('sdp := SemidirectProduct(H, phi, N)')
        return PermutationGroup(gap_group = 'sdp')

    def holomorph(self):
        r"""
        The holomorph of a group as a permutation group.

        The holomorph of a group `G` is the semidirect product
        `G \rtimes_{id} Aut(G)`, where `id` is the identity function
        on `Aut(G)`, the automorphism group of `G`.

        See :wikipedia:`Holomorph (mathematics)`

        OUTPUT:

        Returns the holomorph of a given group as permutation group
        via a wrapping of GAP's semidirect product function.

        EXAMPLES:

        Thomas and Wood's 'Group Tables' (Shiva Publishing, 1980) tells
        us that the holomorph of `C_5` is the unique group of order 20
        with a trivial center. ::

            sage: C5 = CyclicPermutationGroup(5)
            sage: A = C5.holomorph()
            sage: A.order()
            20
            sage: A.is_abelian()
            False
            sage: A.center()
            Subgroup of (Permutation Group with generators
            [(5,6,7,8,9), (1,2,4,3)(6,7,9,8)]) generated by [()]
            sage: A
            Permutation Group with generators [(5,6,7,8,9), (1,2,4,3)(6,7,9,8)]

        Noting that the automorphism group of `D_4` is itself `D_4`, it
        can easily be shown that the holomorph is indeed an internal
        semidirect product of these two groups. ::

            sage: D4 = DihedralGroup(4)
            sage: H = D4.holomorph()
            sage: H.gens()
            [(3,8)(4,7), (2,3,5,8), (2,5)(3,8), (1,4,6,7)(2,3,5,8), (1,8)(2,7)(3,6)(4,5)]
            sage: G = H.subgroup([H.gens()[0],H.gens()[1],H.gens()[2]])
            sage: N = H.subgroup([H.gens()[3],H.gens()[4]])
            sage: N.is_normal(H)
            True
            sage: G.is_isomorphic(D4)
            True
            sage: N.is_isomorphic(D4)
            True
            sage: G.intersection(N)
            Permutation Group with generators [()]
            sage: L = [H(x)*H(y) for x in G for y in N]; L.sort()
            sage: L1 = H.list(); L1.sort()
            sage: L == L1
            True

        Author:

        - Kevin Halasz (2012-08-14)
        """

        gap.eval('G := Group(' + str(self.gens()) + ')')
        gap.eval('aut := AutomorphismGroup(G)')
        gap.eval('alpha := InverseGeneralMapping(NiceMonomorphism(aut))')
        gap.eval('product := SemidirectProduct(NiceObject(aut),alpha,G)')
        return PermutationGroup(gap_group = 'product')

    def subgroup(self, gens=None, gap_group=None, domain=None, category=None, canonicalize=True, check=True):
        """
        Wraps the ``PermutationGroup_subgroup`` constructor. The argument
        ``gens`` is a list of elements of ``self``.

        EXAMPLES::

            sage: G = PermutationGroup([(1,2,3),(3,4,5)])
            sage: g = G((1,2,3))
            sage: G.subgroup([g])
            Subgroup of (Permutation Group with generators [(3,4,5), (1,2,3)]) generated by [(1,2,3)]
        """
        return PermutationGroup_subgroup(self, gens=gens, gap_group=gap_group, domain=None,
                                         category=category, canonicalize=canonicalize, check=check)

    def as_finitely_presented_group(self, reduced=False):
        """
        Return a finitely presented group isomorphic to ``self``.

        This method acts as wrapper for the GAP function ``IsomorphismFpGroupByGenerators``,
        which yields an isomorphism from a given group to a finitely presented group.

        INPUT:

        - ``reduced`` -- Default ``False``, if ``True`` :meth:`FinitelyPresentedGroup.simplified
          <sage.groups.finitely_presented.FinitelyPresentedGroup.simplified>`
          is called, attempting to simplify the presentation of the finitely presented group
          to be returned.

        OUTPUT:

        Finite presentation of self, obtained by taking the image
        of the isomorphism returned by the GAP function, ``IsomorphismFpGroupByGenerators``.

        ALGORITHM:

        Uses GAP.

        EXAMPLES::

            sage: CyclicPermutationGroup(50).as_finitely_presented_group()
            Finitely presented group < a | a^50 >
            sage: DihedralGroup(4).as_finitely_presented_group()
            Finitely presented group < a, b | b^2, a^4, (b*a)^2 >
            sage: GeneralDihedralGroup([2,2]).as_finitely_presented_group()
            Finitely presented group < a, b, c | a^2, b^2, c^2, (c*b)^2, (c*a)^2, (b*a)^2 >

        GAP algorithm is not guaranteed to produce minimal or canonical presentation::

            sage: G = PermutationGroup(['(1,2,3,4,5)', '(1,5)(2,4)'])
            sage: G.is_isomorphic(DihedralGroup(5))
            True
            sage: K = G.as_finitely_presented_group(); K
            Finitely presented group < a, b | b^2, (b*a)^2, b*a^-3*b*a^2 >
            sage: K.as_permutation_group().is_isomorphic(DihedralGroup(5))
            True

        We can attempt to reduce the output presentation::

            sage: PermutationGroup(['(1,2,3,4,5)','(1,3,5,2,4)']).as_finitely_presented_group()
            Finitely presented group < a, b | b^-2*a^-1, b*a^-2 >
            sage: PermutationGroup(['(1,2,3,4,5)','(1,3,5,2,4)']).as_finitely_presented_group(reduced=True)
            Finitely presented group < a | a^5 >

        TESTS::

            sage: PermutationGroup([]).as_finitely_presented_group()
            Finitely presented group < a | a >
            sage: S = SymmetricGroup(6)
            sage: perm_ls = [S.random_element() for i in range(3)]
            sage: G = PermutationGroup(perm_ls)
            sage: G.as_finitely_presented_group().as_permutation_group().is_isomorphic(G)
            True

        `D_9` is the only non-Abelian group of order 18
        with an automorphism group of order 54 [TW1980]_::

            sage: D = DihedralGroup(9).as_finitely_presented_group().gap()
            sage: D.Order(), D.IsAbelian(), D.AutomorphismGroup().Order()
            (18, false, 54)

        `S_3` is the only non-Abelian group of order 6 [TW1980]_::

            sage: S = SymmetricGroup(3).as_finitely_presented_group().gap()
            sage: S.Order(), S.IsAbelian()
            (6, false)

        We can manually construct a permutation representation using GAP
        coset enumeration methods::

            sage: D = GeneralDihedralGroup([3,3,4]).as_finitely_presented_group().gap()
            sage: ctab = D.CosetTable(D.Subgroup([]))
            sage: gen_ls = gap.List(ctab, gap.PermList)
            sage: PermutationGroup(gen_ls).is_isomorphic(GeneralDihedralGroup([3,3,4]))
            True
            sage: A = AlternatingGroup(5).as_finitely_presented_group().gap()
            sage: ctab = A.CosetTable(A.Subgroup([]))
            sage: gen_ls = gap.List(ctab, gap.PermList)
            sage: PermutationGroup(gen_ls).is_isomorphic(AlternatingGroup(5))
            True

        AUTHORS:

        - Davis Shurbert (2013-06-21): initial version
        """
        from sage.groups.free_group import FreeGroup, _lexi_gen
        from sage.groups.finitely_presented import FinitelyPresentedGroup
        from sage.libs.gap.libgap import libgap

        image_fp = libgap.Image( libgap.IsomorphismFpGroupByGenerators(self, self.gens()))
        image_gens = image_fp.FreeGeneratorsOfFpGroup()
        name_itr = _lexi_gen() # Python generator object for variable names
        F = FreeGroup([next(name_itr) for x in image_gens])

        # Convert GAP relators to Sage relators using the Tietze word of each defining relation.
        ret_rls = tuple([F(rel_word.TietzeWordAbstractWord(image_gens).sage())
                        for rel_word in image_fp.RelatorsOfFpGroup()])
        ret_fp = FinitelyPresentedGroup(F,ret_rls)
        if reduced:
            ret_fp = ret_fp.simplified()
        return ret_fp

    def quotient(self, N):
        """
        Returns the quotient of this permutation group by the normal
        subgroup `N`, as a permutation group.

        Wraps the GAP operator "/".

        EXAMPLES::

            sage: G = PermutationGroup([(1,2,3), (2,3)])
            sage: N = PermutationGroup([(1,2,3)])
            sage: G.quotient(N)
            Permutation Group with generators [(1,2)]
            sage: G.quotient(G)
            Permutation Group with generators [()]
        """
        Q = self._gap_() / N._gap_()
        # Return Q as a permutation group
        # This is currently done using the right regular representation
        # FIXME: GAP certainly knows of a better way!
        phi = Q.RegularActionHomomorphism()
        return PermutationGroup(gap_group=phi.Image())

    def commutator(self, other=None):
        r"""
        Returns the commutator subgroup of a group, or of a pair of groups.

        INPUT:

        - ``other`` - default: ``None`` - a permutation group.

        OUTPUT:

        Let $G$ denote ``self``.  If ``other`` is ``None`` then this method
        returns the subgroup of $G$ generated by the set of commutators,

        .. MATH::

            \{[g_1,g_2]\vert g_1, g_2\in G\} = \{g_1^{-1}g_2^{-1}g_1g_2\vert g_1, g_2\in G\}

        Let $H$ denote ``other``, in the case that it is not ``None``.  Then
        this method returns the group generated by the set of commutators,

        .. MATH::

            \{[g,h]\vert g\in G\, h\in H\} = \{g^{-1}h^{-1}gh\vert  g\in G\, h\in H\}

        The two groups need only be permutation groups, there is no notion
        of requiring them to explicitly be subgroups of some other group.

        .. note::

            For the identical statement, the generators of the returned
            group can vary from one execution to the next.

        EXAMPLES::

            sage: G = DiCyclicGroup(4)
            sage: G.commutator()
            Permutation Group with generators [(1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)]

            sage: G = SymmetricGroup(5)
            sage: H = CyclicPermutationGroup(5)
            sage: C = G.commutator(H)
            sage: C.is_isomorphic(AlternatingGroup(5))
            True

        An abelian group will have a trivial commutator.  ::

            sage: G = CyclicPermutationGroup(10)
            sage: G.commutator()
            Permutation Group with generators [()]

        The quotient of a group by its commutator is always abelian.  ::

            sage: G = DihedralGroup(20)
            sage: C = G.commutator()
            sage: Q = G.quotient(C)
            sage: Q.is_abelian()
            True

        When forming commutators from two groups, the order of the
        groups does not matter.  ::

            sage: D = DihedralGroup(3)
            sage: S = SymmetricGroup(2)
            sage: C1 = D.commutator(S); C1
            Permutation Group with generators [(1,2,3)]
            sage: C2 = S.commutator(D); C2
            Permutation Group with generators [(1,3,2)]
            sage: C1 == C2
            True

        This method calls two different functions in GAP, so
        this tests that their results are consistent.  The
        commutator groups may have different generators, but the
        groups are equal. ::

            sage: G = DiCyclicGroup(3)
            sage: C = G.commutator(); C
            Permutation Group with generators [(5,7,6)]
            sage: CC = G.commutator(G); CC
            Permutation Group with generators [(5,6,7)]
            sage: C == CC
            True

        The second group is checked.  ::

            sage: G = SymmetricGroup(2)
            sage: G.commutator('junk')
            Traceback (most recent call last):
            ...
            TypeError: junk is not a permutation group
        """
        if other is None:
            return PermutationGroup(gap_group=gap.DerivedSubgroup(self))
        else:
            from sage.categories.finite_permutation_groups import FinitePermutationGroups
            if other not in FinitePermutationGroups():
                raise TypeError("{0} is not a permutation group".format(other))
            return PermutationGroup(gap_group=gap.CommutatorSubgroup(self, other))

    @hap_decorator
    def cohomology(self, n, p = 0):
        r"""
        Computes the group cohomology `H^n(G, F)`, where `F = \ZZ`
        if `p=0` and `F = \ZZ / p \ZZ` if `p > 0` is a prime.

        Wraps HAP's ``GroupHomology`` function, written by Graham Ellis.

        REQUIRES: GAP package HAP (in gap_packages-\*.spkg).

        EXAMPLES::

            sage: G = SymmetricGroup(4)
            sage: G.cohomology(1,2)                            # optional - gap_packages
            Multiplicative Abelian group isomorphic to C2
            sage: G = SymmetricGroup(3)
            sage: G.cohomology(5)                              # optional - gap_packages
            Trivial Abelian group
            sage: G.cohomology(5,2)                            # optional - gap_packages
            Multiplicative Abelian group isomorphic to C2
            sage: G.homology(5,3)                              # optional - gap_packages
            Trivial Abelian group
            sage: G.homology(5,4)                              # optional - gap_packages
            Traceback (most recent call last):
            ...
            ValueError: p must be 0 or prime

        This computes `H^4(S_3, \ZZ)` and
        `H^4(S_3, \ZZ / 2 \ZZ)`, respectively.

        AUTHORS:

        - David Joyner and Graham Ellis

        REFERENCES:

        - G. Ellis, 'Computing group resolutions', J. Symbolic
          Computation. Vol.38, (2004)1077-1118 (Available at
          http://hamilton.nuigalway.ie/).

        - D. Joyner, 'A primer on computational group homology and
          cohomology', http://front.math.ucdavis.edu/0706.0549.
        """
        if p == 0:
            L = self._gap_().GroupCohomology(n).sage()
        else:
            L = self._gap_().GroupCohomology(n, p).sage()
        return AbelianGroup(len(L), L)

    @hap_decorator
    def cohomology_part(self, n, p = 0):
        """
        Computes the p-part of the group cohomology `H^n(G, F)`,
        where `F = \ZZ` if `p=0` and `F = \ZZ / p \ZZ` if
        `p > 0` is a prime.

        Wraps HAP's Homology function, written
        by Graham Ellis, applied to the `p`-Sylow subgroup of
        `G`.

        REQUIRES: GAP package HAP (in gap_packages-\*.spkg).

        EXAMPLES::

            sage: G = SymmetricGroup(5)
            sage: G.cohomology_part(7,2)                   # optional - gap_packages
            Multiplicative Abelian group isomorphic to C2 x C2 x C2
            sage: G = SymmetricGroup(3)
            sage: G.cohomology_part(2,3)                   # optional - gap_packages
            Multiplicative Abelian group isomorphic to C3

        AUTHORS:

        - David Joyner and Graham Ellis
        """
        if p == 0:
            return AbelianGroup(1, [1])
        else:
            G = self._gap_()
            S = G.SylowSubgroup(p)
            R = S.ResolutionFiniteGroup(n+1)
            HR = R.HomToIntegers()
            L = HR.Cohomology(n).sage()
            return AbelianGroup(len(L), L)

    @hap_decorator
    def homology(self, n, p = 0):
        r"""
        Computes the group homology `H_n(G, F)`, where
        `F = \ZZ` if `p=0` and `F = \ZZ / p \ZZ` if
        `p > 0` is a prime. Wraps HAP's ``GroupHomology`` function,
        written by Graham Ellis.

        REQUIRES: GAP package HAP (in gap_packages-\*.spkg).

        AUTHORS:

        - David Joyner and Graham Ellis

        The example below computes `H_7(S_5, \ZZ)`,
        `H_7(S_5, \ZZ / 2 \ZZ)`,
        `H_7(S_5, \ZZ / 3 \ZZ)`, and
        `H_7(S_5, \ZZ / 5 \ZZ)`, respectively. To compute the
        `2`-part of `H_7(S_5, \ZZ)`, use the ``homology_part``
        function.

        EXAMPLES::

            sage: G = SymmetricGroup(5)
            sage: G.homology(7)                              # optional - gap_packages
            Multiplicative Abelian group isomorphic to C2 x C2 x C4 x C3 x C5
            sage: G.homology(7,2)                              # optional - gap_packages
            Multiplicative Abelian group isomorphic to C2 x C2 x C2 x C2 x C2
            sage: G.homology(7,3)                              # optional - gap_packages
            Multiplicative Abelian group isomorphic to C3
            sage: G.homology(7,5)                              # optional - gap_packages
            Multiplicative Abelian group isomorphic to C5

        REFERENCES:

        - G. Ellis, "Computing group resolutions", J. Symbolic
          Computation. Vol.38, (2004)1077-1118 (Available at
          http://hamilton.nuigalway.ie/.

        - D. Joyner, "A primer on computational group homology and cohomology",
          http://front.math.ucdavis.edu/0706.0549
        """
        if p == 0:
            L = self._gap_().GroupHomology(n).sage()
        else:
            L = self._gap_().GroupHomology(n, p).sage()
        return AbelianGroup(len(L), L)

    @hap_decorator
    def homology_part(self, n, p = 0):
        r"""
        Computes the `p`-part of the group homology
        `H_n(G, F)`, where `F = \ZZ` if `p=0` and
        `F = \ZZ / p \ZZ` if `p > 0` is a prime. Wraps HAP's
        ``Homology`` function, written by Graham Ellis, applied to the
        `p`-Sylow subgroup of `G`.

        REQUIRES: GAP package HAP (in gap_packages-\*.spkg).

        EXAMPLES::

            sage: G = SymmetricGroup(5)
            sage: G.homology_part(7,2)                              # optional - gap_packages
            Multiplicative Abelian group isomorphic to C2 x C2 x C2 x C2 x C4

        AUTHORS:

        - David Joyner and Graham Ellis
        """
        if p == 0:
            return AbelianGroup(1, [1])
        else:
            S = self._gap_().SylowSubgroup(p)
            R = S.ResolutionFiniteGroup(n+1)
            TR = R.TensorWithIntegers()
            L = TR.Homology(n).sage()
            return AbelianGroup(len(L), L)

    def character_table(self):
        r"""
        Returns the matrix of values of the irreducible characters of a
        permutation group `G` at the conjugacy classes of
        `G`.

        The columns represent the conjugacy classes of
        `G` and the rows represent the different irreducible
        characters in the ordering given by GAP.

        EXAMPLES::

            sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3)]])
            sage: G.order()
            12
            sage: G.character_table()
            [         1          1          1          1]
            [         1 -zeta3 - 1      zeta3          1]
            [         1      zeta3 -zeta3 - 1          1]
            [         3          0          0         -1]
            sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3)]])
            sage: CT = gap(G).CharacterTable()

        Type ``print(gap.eval("Display(%s)"%CT.name()))`` to display this
        nicely.

        ::

            sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
            sage: G.order()
            8
            sage: G.character_table()
            [ 1  1  1  1  1]
            [ 1 -1 -1  1  1]
            [ 1 -1  1 -1  1]
            [ 1  1 -1 -1  1]
            [ 2  0  0  0 -2]
            sage: CT = gap(G).CharacterTable()

        Again, type ``print(gap.eval("Display(%s)"%CT.name()))`` to display this
        nicely.

        ::

            sage: SymmetricGroup(2).character_table()
            [ 1 -1]
            [ 1  1]
            sage: SymmetricGroup(3).character_table()
            [ 1 -1  1]
            [ 2  0 -1]
            [ 1  1  1]
            sage: SymmetricGroup(5).character_table()
            [ 1 -1  1  1 -1 -1  1]
            [ 4 -2  0  1  1  0 -1]
            [ 5 -1  1 -1 -1  1  0]
            [ 6  0 -2  0  0  0  1]
            [ 5  1  1 -1  1 -1  0]
            [ 4  2  0  1 -1  0 -1]
            [ 1  1  1  1  1  1  1]
            sage: list(AlternatingGroup(6).character_table())
            [(1, 1, 1, 1, 1, 1, 1), (5, 1, 2, -1, -1, 0, 0), (5, 1, -1, 2, -1, 0, 0), (8, 0, -1, -1, 0, zeta5^3 + zeta5^2 + 1, -zeta5^3 - zeta5^2), (8, 0, -1, -1, 0, -zeta5^3 - zeta5^2, zeta5^3 + zeta5^2 + 1), (9, 1, 0, 0, 1, -1, -1), (10, -2, 1, 1, 0, 0, 0)]

        Suppose that you have a class function `f(g)` on
        `G` and you know the values `v_1, \ldots, v_n` on
        the conjugacy class elements in
        ``conjugacy_classes_representatives(G)`` =
        `[g_1, \ldots, g_n]`. Since the irreducible characters
        `\rho_1, \ldots, \rho_n` of `G` form an
        `E`-basis of the space of all class functions (`E`
        a "sufficiently large" cyclotomic field), such a class function is
        a linear combination of these basis elements,
        `f = c_1 \rho_1 + \cdots + c_n \rho_n`. To find
        the coefficients `c_i`, you simply solve the linear system
        ``character_table_values(G)`` `[v_1, ..., v_n] = [c_1, ..., c_n]`,
        where `[v_1, \ldots, v_n]` = ``character_table_values(G)`` `^{(-1)}[c_1, ..., c_n]`.

        AUTHORS:

        - David Joyner and William Stein (2006-01-04)

        """
        G    = self._gap_()
        cl   = G.ConjugacyClasses()
        n    = Integer(cl.Length())
        irrG = G.Irr()
        ct   = [[irrG[i+1, j+1] for j in range(n)] for i in range(n)]

        from sage.rings.all import CyclotomicField
        e = irrG.Flat().Conductor()
        K = CyclotomicField(e)
        ct = [[K(x) for x in v] for v in ct]

        # Finally return the result as a matrix.
        from sage.matrix.all import MatrixSpace
        MS = MatrixSpace(K, n)
        return MS(ct)

    def irreducible_characters(self):
        r"""
        Returns a list of the irreducible characters of ``self``.

        EXAMPLES::

            sage: irr = SymmetricGroup(3).irreducible_characters()
            sage: [x.values() for x in irr]
            [[1, -1, 1], [2, 0, -1], [1, 1, 1]]
        """
        return [ClassFunction(self, irr) for irr in self._gap_().Irr()]

    def trivial_character(self):
        r"""
        Returns the trivial character of ``self``.

        EXAMPLES::

            sage: SymmetricGroup(3).trivial_character()
            Character of Symmetric group of order 3! as a permutation group
        """
        values = [1]*self._gap_().NrConjugacyClasses().sage()
        return self.character(values)

    def character(self, values):
        r"""
        Returns a group character from ``values``, where ``values`` is
        a list of the values of the character evaluated on the conjugacy
        classes.

        EXAMPLES::

            sage: G = AlternatingGroup(4)
            sage: n = len(G.conjugacy_classes_representatives())
            sage: G.character([1]*n)
            Character of Alternating group of order 4!/2 as a permutation group
        """
        return ClassFunction(self, values)

    def conjugacy_classes_representatives(self):
        """
        Returns a complete list of representatives of conjugacy classes in
        a permutation group `G`.

        The ordering is that given by GAP.

        EXAMPLES::

            sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
            sage: cl = G.conjugacy_classes_representatives(); cl
            [(), (2,4), (1,2)(3,4), (1,2,3,4), (1,3)(2,4)]
            sage: cl[3] in G
            True

        ::

            sage: G = SymmetricGroup(5)
            sage: G.conjugacy_classes_representatives()
            [(), (1,2), (1,2)(3,4), (1,2,3), (1,2,3)(4,5), (1,2,3,4), (1,2,3,4,5)]

        ::

            sage: S = SymmetricGroup(['a','b','c'])
            sage: S.conjugacy_classes_representatives()
            [(), ('a','b'), ('a','b','c')]

        AUTHORS:

        - David Joyner and William Stein (2006-01-04)
        """
        cl = self._gap_().ConjugacyClasses()
        return [self(rep.Representative(), check=False) for rep in cl]

    def conjugacy_classes_subgroups(self):
        """
        Returns a complete list of representatives of conjugacy classes of
        subgroups in a permutation group `G`.

        The ordering is that given by GAP.

        EXAMPLES::

            sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
            sage: cl = G.conjugacy_classes_subgroups()
            sage: cl
            [Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [()], Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(1,2)(3,4)], Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(1,3)(2,4)], Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(2,4)], Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(1,2)(3,4), (1,4)(2,3)], Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(2,4), (1,3)(2,4)], Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(1,2,3,4), (1,3)(2,4)], Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(2,4), (1,2)(3,4), (1,4)(2,3)]]

        ::

            sage: G = SymmetricGroup(3)
            sage: G.conjugacy_classes_subgroups()
            [Subgroup of (Symmetric group of order 3! as a permutation group) generated by [()], Subgroup of (Symmetric group of order 3! as a permutation group) generated by [(2,3)], Subgroup of (Symmetric group of order 3! as a permutation group) generated by [(1,2,3)], Subgroup of (Symmetric group of order 3! as a permutation group) generated by [(2,3), (1,2,3)]]

        AUTHORS:

        - David Joyner (2006-10)
        """
        cl = self._gap_().ConjugacyClassesSubgroups()
        return [self.subgroup(gap_group=sub.Representative()) for sub in cl]

    def subgroups(self):
        r"""
        Returns a list of all the subgroups of ``self``.

        OUTPUT:

        Each possible subgroup of ``self`` is contained once
        in the returned list.  The list is in order, according
        to the size of the subgroups, from the trivial subgroup
        with one element on through up to the whole group.
        Conjugacy classes of subgroups are contiguous in the list.

        .. warning::

            For even relatively small groups this method can
            take a very long time to execute, or create vast
            amounts of output.  Likely both.  Its purpose is
            instructional, as it can be useful for studying
            small groups.  The 156 subgroups of the full
            symmetric group on 5 symbols of order 120, `S_5`,
            can be computed in about a minute on commodity hardware
            in 2011. The 64 subgroups of the cyclic group of order
            `30030 = 2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13` takes
            about twice as long.

            For faster results, which still exhibit the structure of
            the possible subgroups, use
            :meth:`conjugacy_classes_subgroups`.

        EXAMPLES::

            sage: G = SymmetricGroup(3)
            sage: G.subgroups()
            [Subgroup of (Symmetric group of order 3! as a permutation group) generated by [()],
             Subgroup of (Symmetric group of order 3! as a permutation group) generated by [(2,3)],
             Subgroup of (Symmetric group of order 3! as a permutation group) generated by [(1,2)],
             Subgroup of (Symmetric group of order 3! as a permutation group) generated by [(1,3)],
             Subgroup of (Symmetric group of order 3! as a permutation group) generated by [(1,2,3)],
             Subgroup of (Symmetric group of order 3! as a permutation group) generated by [(2,3), (1,2,3)]]

            sage: G = CyclicPermutationGroup(14)
            sage: G.subgroups()
            [Subgroup of (Cyclic group of order 14 as a permutation group) generated by [()],
             Subgroup of (Cyclic group of order 14 as a permutation group) generated by [(1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)],
             Subgroup of (Cyclic group of order 14 as a permutation group) generated by [(1,3,5,7,9,11,13)(2,4,6,8,10,12,14)],
             Subgroup of (Cyclic group of order 14 as a permutation group) generated by [(1,2,3,4,5,6,7,8,9,10,11,12,13,14)]]

        AUTHOR:

        - Rob Beezer (2011-01-24)
        """
        all_sg = []
        ccs = self._gap_().ConjugacyClassesSubgroups()
        for cc in ccs:
            for h in cc.Elements():
                all_sg.append(self.subgroup(gap_group=h))
        return all_sg

    @cached_method
    def _regular_subgroup_gap(self):
        r"""
        Return a conjugacy class of regular subgroups, if there is one, as a
        GAP element.

        This allows finding such a group without constructing it in Sage.
        The result is cached, so constructing the obtained subgroup later is
        possible without recomputing it.

        EXAMPLES:

        The symmetric group on 4 elements has a regular subgroup::

            sage: S4 = groups.permutation.Symmetric(4)
            sage: S4._regular_subgroup_gap() # random
            ConjugacyClassSubgroups(SymmetricGroup( [ 1 .. 4 ] ),Group(
            [ (1,4)(2,3), (1,3)(2,4) ] ))

        """
        gap = self._gap_().parent()
        C = gap.new("""
            First(ConjugacyClassesSubgroups(%s),
                x -> IsRegular(Representative(x), [1..%d]))
        """ % (self._gap_().name(), self.degree()))
        # prevent caching GAP fails
        if gap.eval('%s = fail' % C.name()) == 'true':
            return None
        return C

    @cached_method
    def has_regular_subgroup(self, return_group = False):
        r"""
        Return whether the group contains a regular subgroup.

        INPUT:

        - ``return_group`` (boolean) -- If ``return_group = True``, a regular
          subgroup is returned if there is one, and ``None`` if there isn't.
          When ``return_group = False`` (default), only a boolean indicating
          whether such a group exists is returned instead.

        EXAMPLES:

        The symmetric group on 4 elements has a regular subgroup::

            sage: S4 = groups.permutation.Symmetric(4)
            sage: S4.has_regular_subgroup()
            True
            sage: S4.has_regular_subgroup(return_group = True) # random
            Subgroup of (Symmetric group of order 4! as a permutation group) generated by [(1,3)(2,4), (1,4)(2,3)]

        But the automorphism group of Petersen's graph does not::

            sage: G = graphs.PetersenGraph().automorphism_group()
            sage: G.has_regular_subgroup()
            False

        """
        b = False
        G = None
        if self.order() % self.degree() == 0:
            if self.order() == len(self.domain()):
                b = self.is_transitive()
                if b:
                    G = self
            else:
                C = self._regular_subgroup_gap()
                b = (C is not None)
                if b and return_group:
                    G = self.subgroup(gap_group=C.Representative())
        if return_group:
            return G
        else:
            return b

    def blocks_all(self, representatives = True):
        r"""
        Returns the list of block systems of imprimitivity.

        For more information on primitivity, see the :wikipedia:`Wikipedia
        article on primitive group actions <Primitive_permutation_group>`.

        INPUT:

        - ``representative`` (boolean) -- whether to return all possible block
          systems of imprimitivity or only one of their representatives (the
          block can be obtained from its representative set `S` by computing the
          orbit of `S` under ``self``).

          This parameter is set to ``True`` by default (as it is GAP's default
          behaviour).

        OUTPUT:

        This method returns a description of *all* block systems. Hence, the
        output is a "list of lists of lists" or a "list of lists" depending on
        the value of ``representatives``. A bit more clearly, output is :

        * A list of length (#number of different block systems) of

           * block systems, each of them being defined as

               * If ``representatives = True`` : a list of representatives of
                 each set of the block system

               * If ``representatives = False`` : a partition of the elements
                 defining an imprimitivity block.

        .. SEEALSO::

            - :meth:`~PermutationGroup_generic.is_primitive`

        EXAMPLES:

        Picking an interesting group::

            sage: g = graphs.DodecahedralGraph()
            sage: g.is_vertex_transitive()
            True
            sage: ag = g.automorphism_group()
            sage: ag.is_primitive()
            False

        Computing its blocks representatives::

            sage: ag.blocks_all()
            [[0, 15]]

        Now the full block::

            sage: sorted(ag.blocks_all(representatives = False)[0])
            [[0, 15], [1, 16], [2, 12], [3, 13], [4, 9], [5, 10], [6, 11], [7, 18], [8, 17], [14, 19]]

        TESTS::

            sage: g = PermutationGroup([("a","b","c","d")])
            sage: g.blocks_all()
            [['a', 'c']]
            sage: g.blocks_all(False)
            [[['a', 'c'], ['b', 'd']]]
        """
        ag = self._gap_()
        if representatives:
            return [[self._domain_from_gap[x] for x in rep]
                    for rep in ag.AllBlocks().sage()]
        else:
            return [[[self._domain_from_gap[x] for x in cl] for cl in ag.Orbit(rep,"OnSets").sage()]
                    for rep in ag.AllBlocks()]

    def cosets(self, S, side='right'):
        r"""
        Returns a list of the cosets of ``S`` in ``self``.

        INPUT:

        - ``S`` - a subgroup of ``self``.  An error is raised
          if ``S`` is not a subgroup.

        - ``side`` - default: 'right' - determines if right cosets or
          left cosets are returned.  ``side`` refers to where the
          representative is placed in the products forming the cosets
          and thus allowable values are only 'right' and 'left'.

        OUTPUT:

        A list of lists.  Each inner list is a coset of the subgroup
        in the group.  The first element of each coset is the smallest
        element (based on the ordering of the elements of ``self``)
        of all the group elements that have not yet appeared in a
        previous coset. The elements of each coset are in the same
        order as the subgroup elements used to build the coset's
        elements.

        As a consequence, the subgroup itself is the first coset,
        and its first element is the identity element.  For each coset,
        the first element listed is the element used as a representative
        to build the coset.  These representatives form an increasing
        sequence across the list of cosets, and within a coset the
        representative is the smallest element of its coset (both
        orderings are based on of the ordering of elements of ``self``).

        In the case of a normal subgroup, left and right cosets should
        appear in the same order as part of the outer list.  However,
        the list of the elements of a particular coset may be in a
        different order for the right coset versus the order in the
        left coset. So, if you check to see if a subgroup is normal,
        it is necessary to sort each individual coset first (but not
        the list of cosets, due to the ordering of the representatives).
        See below for examples of this.

        .. note::

            This is a naive implementation intended for instructional
            purposes, and hence is slow for larger groups.  Sage and GAP
            provide more sophisticated functions for working quickly with
            cosets of larger groups.

        EXAMPLES:

        The default is to build right cosets. This example works with
        the symmetry group of an 8-gon and a normal subgroup.
        Notice that a straight check on the equality of the output
        is not sufficient to check normality, while sorting the
        individual cosets is sufficient to then simply test equality of
        the list of lists.  Study the second coset in each list to understand the
        need for sorting the elements of the cosets.  ::

            sage: G = DihedralGroup(8)
            sage: quarter_turn = G('(1,3,5,7)(2,4,6,8)'); quarter_turn
            (1,3,5,7)(2,4,6,8)
            sage: S = G.subgroup([quarter_turn])
            sage: rc = G.cosets(S); rc
            [[(), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8), (1,7,5,3)(2,8,6,4)],
             [(2,8)(3,7)(4,6), (1,7)(2,6)(3,5), (1,5)(2,4)(6,8), (1,3)(4,8)(5,7)],
             [(1,2)(3,8)(4,7)(5,6), (1,8)(2,7)(3,6)(4,5), (1,6)(2,5)(3,4)(7,8), (1,4)(2,3)(5,8)(6,7)],
             [(1,2,3,4,5,6,7,8), (1,4,7,2,5,8,3,6), (1,6,3,8,5,2,7,4), (1,8,7,6,5,4,3,2)]]
            sage: lc = G.cosets(S, side='left'); lc
            [[(), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8), (1,7,5,3)(2,8,6,4)],
             [(2,8)(3,7)(4,6), (1,3)(4,8)(5,7), (1,5)(2,4)(6,8), (1,7)(2,6)(3,5)],
             [(1,2)(3,8)(4,7)(5,6), (1,4)(2,3)(5,8)(6,7), (1,6)(2,5)(3,4)(7,8), (1,8)(2,7)(3,6)(4,5)],
             [(1,2,3,4,5,6,7,8), (1,4,7,2,5,8,3,6), (1,6,3,8,5,2,7,4), (1,8,7,6,5,4,3,2)]]

            sage: S.is_normal(G)
            True
            sage: rc == lc
            False
            sage: rc_sorted = [sorted(c) for c in rc]
            sage: lc_sorted = [sorted(c) for c in lc]
            sage: rc_sorted == lc_sorted
            True

        An example with the symmetry group of a regular
        tetrahedron and a subgroup that is not normal.
        Thus, the right and left cosets are different
        (and so are the representatives). With each
        individual coset sorted, a naive test of normality
        is possible.  ::

            sage: A = AlternatingGroup(4)
            sage: face_turn = A('(1,2,3)'); face_turn
            (1,2,3)
            sage: stabilizer = A.subgroup([face_turn])
            sage: rc = A.cosets(stabilizer, side='right'); rc
            [[(), (1,2,3), (1,3,2)],
             [(2,3,4), (1,3)(2,4), (1,4,2)],
             [(2,4,3), (1,4,3), (1,2)(3,4)],
             [(1,2,4), (1,4)(2,3), (1,3,4)]]
            sage: lc = A.cosets(stabilizer, side='left'); lc
            [[(), (1,2,3), (1,3,2)],
             [(2,3,4), (1,2)(3,4), (1,3,4)],
             [(2,4,3), (1,2,4), (1,3)(2,4)],
             [(1,4,2), (1,4,3), (1,4)(2,3)]]

            sage: stabilizer.is_normal(A)
            False
            sage: rc_sorted = [sorted(c) for c in rc]
            sage: lc_sorted = [sorted(c) for c in lc]
            sage: rc_sorted == lc_sorted
            False

        TESTS:

        The keyword ``side`` is checked for the two possible values. ::

            sage: G = SymmetricGroup(3)
            sage: S = G.subgroup([G("(1,2)")])
            sage: G.cosets(S, side='junk')
            Traceback (most recent call last):
            ...
            ValueError: side should be 'right' or 'left', not junk

        The subgroup argument is checked to see if it is a permutation group.
        Even a legitimate GAP object can be rejected. ::

            sage: G=SymmetricGroup(3)
            sage: G.cosets(gap(3))
            Traceback (most recent call last):
            ...
            TypeError: 3 is not a permutation group

        The subgroup is verified as a subgroup of ``self``. ::

            sage: A = AlternatingGroup(3)
            sage: G = SymmetricGroup(3)
            sage: S = G.subgroup([G("(1,2)")])
            sage: A.cosets(S)
            Traceback (most recent call last):
            ...
            ValueError: Subgroup of (Symmetric group of order 3! as a permutation group) generated by [(1,2)] is not a subgroup of Alternating group of order 3!/2 as a permutation group

        AUTHOR:

        - Rob Beezer (2011-01-31)
        """
        from copy import copy
        from sage.categories.finite_permutation_groups import FinitePermutationGroups

        if not side in ['right','left']:
            raise ValueError("side should be 'right' or 'left', not %s" % side)
        if not S in FinitePermutationGroups():
            raise TypeError("%s is not a permutation group" % S)
        if not S.is_subgroup(self):
            raise ValueError("%s is not a subgroup of %s" % (S, self))

        group = sorted(copy(self.list()))
        subgroup = sorted([self(s) for s in S.list()])
        decomposition = []
        while group:
            rep = group[0]
            if side == 'right':
                coset = [e*rep for e in subgroup]
            if side == 'left':
                coset = [rep*e for e in subgroup]
            for e in coset:
                group.remove(e)
            decomposition.append(coset)
        return decomposition

    def minimal_generating_set(self):
        r"""
        Return a minimal generating set

        EXAMPLES::

            sage: g = graphs.CompleteGraph(4)
            sage: g.relabel(['a','b','c','d'])
            sage: mgs = g.automorphism_group().minimal_generating_set(); len(mgs)
            2
            sage: mgs # random
            [('b','d','c'), ('a','c','b','d')]


        TESTS::

            sage: PermutationGroup(["(1,2,3)(4,5,6)","(1,2,3,4,5,6)"]).minimal_generating_set()
            [(2,5)(3,6), (1,5,3,4,2,6)]
        """
        return from_gap_list(self,str(self._gap_().MinimalGeneratingSet()))

    def normalizer(self, g):
        """
        Returns the normalizer of ``g`` in ``self``.

        EXAMPLES::

            sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
            sage: g = G([(1,3)])
            sage: G.normalizer(g)
            Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(2,4), (1,3)]
            sage: g = G([(1,2,3,4)])
            sage: G.normalizer(g)
            Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(2,4), (1,2,3,4), (1,3)(2,4)]
            sage: H = G.subgroup([G([(1,2,3,4)])])
            sage: G.normalizer(H)
            Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(2,4), (1,2,3,4), (1,3)(2,4)]
        """
        return self.subgroup(gap_group=self._gap_().Normalizer(g))

    def centralizer(self, g):
        """
        Returns the centralizer of ``g`` in ``self``.

        EXAMPLES::

            sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
            sage: g = G([(1,3)])
            sage: G.centralizer(g)
            Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(2,4), (1,3)]
            sage: g = G([(1,2,3,4)])
            sage: G.centralizer(g)
            Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(1,2,3,4)]
            sage: H = G.subgroup([G([(1,2,3,4)])])
            sage: G.centralizer(H)
            Subgroup of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) generated by [(1,2,3,4)]
        """
        return self.subgroup(gap_group=self._gap_().Centralizer(g))

    def isomorphism_type_info_simple_group(self):
        """
        If the group is simple, then this returns the name of the group.

        EXAMPLES::

            sage: G = CyclicPermutationGroup(5)
            sage: G.isomorphism_type_info_simple_group()
            rec(
              name := "Z(5)",
              parameter := 5,
              series := "Z" )

        TESTS:

        This shows that the issue at :trac:`7360` is fixed::

            sage: G = KleinFourGroup()
            sage: G.is_simple()
            False
            sage: G.isomorphism_type_info_simple_group()
            Traceback (most recent call last):
            ...
            TypeError: Group must be simple.
        """
        if self.is_simple():
            info = self._gap_().IsomorphismTypeInfoFiniteSimpleGroup()
            return info
        else:
            raise TypeError("Group must be simple.")

    ######################  Boolean tests #####################

    def is_abelian(self):
        """
        Return ``True`` if this group is abelian.

        EXAMPLES::

            sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
            sage: G.is_abelian()
            False
            sage: G = PermutationGroup(['(1,2,3)(4,5)'])
            sage: G.is_abelian()
            True
        """
        return self._gap_().IsAbelian().bool()

    def is_commutative(self):
        """
        Return ``True`` if this group is commutative.

        EXAMPLES::

            sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
            sage: G.is_commutative()
            False
            sage: G = PermutationGroup(['(1,2,3)(4,5)'])
            sage: G.is_commutative()
            True
        """
        return self.is_abelian()

    def is_cyclic(self):
        """
        Return ``True`` if this group is cyclic.

        EXAMPLES::

            sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
            sage: G.is_cyclic()
            False
            sage: G = PermutationGroup(['(1,2,3)(4,5)'])
            sage: G.is_cyclic()
            True
        """
        return self._gap_().IsCyclic().bool()

    def is_elementary_abelian(self):
        """
        Return ``True`` if this group is elementary abelian. An elementary
        abelian group is a finite abelian group, where every nontrivial
        element has order `p`, where `p` is a prime.

        EXAMPLES::

            sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
            sage: G.is_elementary_abelian()
            False
            sage: G = PermutationGroup(['(1,2,3)','(4,5,6)'])
            sage: G.is_elementary_abelian()
            True
        """
        return self._gap_().IsElementaryAbelian().bool()

    def isomorphism_to(self, right):
        """
        Return an isomorphism from ``self`` to ``right`` if the groups
        are isomorphic, otherwise ``None``.

        INPUT:


        -  ``self`` - this group

        -  ``right`` - a permutation group


        OUTPUT:

        - ``None`` or a morphism of permutation groups.

        EXAMPLES::

            sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
            sage: H = PermutationGroup(['(1,2,3)(4,5)'])
            sage: G.isomorphism_to(H) is None
            True
            sage: G = PermutationGroup([(1,2,3), (2,3)])
            sage: H = PermutationGroup([(1,2,4), (1,4)])
            sage: G.isomorphism_to(H)  # not tested, see below
            Permutation group morphism:
              From: Permutation Group with generators [(2,3), (1,2,3)]
              To:   Permutation Group with generators [(1,2,4), (1,4)]
              Defn: [(2,3), (1,2,3)] -> [(2,4), (1,2,4)]

        TESTS:

        Partial check that the output makes some sense::

            sage: G.isomorphism_to(H)
            Permutation group morphism:
              From: Permutation Group with generators [(2,3), (1,2,3)]
              To:   Permutation Group with generators [(1,2,4), (1,4)]
              Defn: [(2,3), (1,2,3)] -> [...]
        """
        current_randstate().set_seed_gap()
        if not isinstance(right, PermutationGroup_generic):
            raise TypeError("right must be a permutation group")

        iso = self._gap_().IsomorphismGroups(right)
        if str(iso) == "fail":
            return None

        dsts = [right(iso.Image(x), check=False) for x in self.gens()]

        from .permgroup_morphism import PermutationGroupMorphism_im_gens
        return PermutationGroupMorphism_im_gens(self, right, dsts)

    def is_isomorphic(self, right):
        """
        Return ``True`` if the groups are isomorphic.

        INPUT:


        -  ``self`` - this group

        -  ``right`` - a permutation group


        OUTPUT:

        - boolean; ``True`` if ``self`` and ``right`` are isomorphic groups;
          ``False`` otherwise.

        EXAMPLES::

            sage: v = ['(1,2,3)(4,5)', '(1,2,3,4,5)']
            sage: G = PermutationGroup(v)
            sage: H = PermutationGroup(['(1,2,3)(4,5)'])
            sage: G.is_isomorphic(H)
            False
            sage: G.is_isomorphic(G)
            True
            sage: G.is_isomorphic(PermutationGroup(list(reversed(v))))
            True
        """
        if not isinstance(right, PermutationGroup_generic):
            raise TypeError("right must be a permutation group")
        iso = self._gap_().IsomorphismGroups(right)
        return str(iso) != 'fail'

    def is_monomial(self):
        """
        Returns ``True`` if the group is monomial. A finite group is monomial
        if every irreducible complex character is induced from a linear
        character of a subgroup.

        EXAMPLES::

            sage: G = PermutationGroup(['(1,2,3)(4,5)'])
            sage: G.is_monomial()
            True
        """
        return self._gap_().IsMonomialGroup().bool()

    def is_nilpotent(self):
        """
        Return ``True`` if this group is nilpotent.

        EXAMPLES::

            sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
            sage: G.is_nilpotent()
            False
            sage: G = PermutationGroup(['(1,2,3)(4,5)'])
            sage: G.is_nilpotent()
            True
        """
        return self._gap_().IsNilpotent().bool()

    def is_normal(self, other):
        """
        Return ``True`` if this group is a normal subgroup of ``other``.

        EXAMPLES::

            sage: AlternatingGroup(4).is_normal(SymmetricGroup(4))
            True
            sage: H = PermutationGroup(['(1,2,3)(4,5)'])
            sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
            sage: H.is_normal(G)
            False
        """
        if not(self.is_subgroup(other)):
            raise TypeError("%s must be a subgroup of %s"%(self, other))
        return other._gap_().IsNormal(self).bool()

    def is_perfect(self):
        """
        Return ``True`` if this group is perfect. A group is perfect if it
        equals its derived subgroup.

        EXAMPLES::

            sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
            sage: G.is_perfect()
            False
            sage: G = PermutationGroup(['(1,2,3)(4,5)'])
            sage: G.is_perfect()
            False
        """
        return self._gap_().IsPerfectGroup().bool()

    def is_pgroup(self):
        """
        Returns ``True`` if this group is a `p`-group. A finite group is
        a `p`-group if its order is of the form `p^n` for a prime integer
        `p` and a nonnegative integer `n`.

        EXAMPLES::

            sage: G = PermutationGroup(['(1,2,3,4,5)'])
            sage: G.is_pgroup()
            True
        """
        return self._gap_().IsPGroup().bool()

    def is_polycyclic(self):
        r"""
        Return ``True`` if this group is polycyclic. A group is polycyclic if
        it has a subnormal series with cyclic factors. (For finite groups,
        this is the same as if the group is solvable - see
        ``is_solvable``.)

        EXAMPLES::

            sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
            sage: G.is_polycyclic()
            False
            sage: G = PermutationGroup(['(1,2,3)(4,5)'])
            sage: G.is_polycyclic()
            True
        """
        return self._gap_().IsPolycyclicGroup().bool()

    def is_simple(self):
        """
        Returns ``True`` if the group is simple. A group is simple if it has no
        proper normal subgroups.

        EXAMPLES::

            sage: G = PermutationGroup(['(1,2,3)(4,5)'])
            sage: G.is_simple()
            False
        """
        return self._gap_().IsSimpleGroup().bool()

    def is_solvable(self):
        """
        Returns ``True`` if the group is solvable.

        EXAMPLES::

            sage: G = PermutationGroup(['(1,2,3)(4,5)'])
            sage: G.is_solvable()
            True
        """
        return self._gap_().IsSolvableGroup().bool()

    def is_subgroup(self, other):
        """
        Returns ``True`` if ``self`` is a subgroup of ``other``.

        EXAMPLES::

            sage: G = AlternatingGroup(5)
            sage: H = SymmetricGroup(5)
            sage: G.is_subgroup(H)
            True
        """
        return all((x in other) for x in self.gens())

    def is_supersolvable(self):
        """
        Returns ``True`` if the group is supersolvable. A finite group is
        supersolvable if it has a normal series with cyclic factors.

        EXAMPLES::

            sage: G = PermutationGroup(['(1,2,3)(4,5)'])
            sage: G.is_supersolvable()
            True
        """
        return self._gap_().IsSupersolvableGroup().bool()

    def non_fixed_points(self):
        r"""
        Return the list of points not fixed by ``self``, i.e., the subset
        of ``self.domain()`` moved by some element of ``self``.

        EXAMPLES::

            sage: G = PermutationGroup([[(3,4,5)],[(7,10)]])
            sage: G.non_fixed_points()
            [3, 4, 5, 7, 10]
            sage: G = PermutationGroup([[(2,3,6)],[(9,)]]) # note: 9 is fixed
            sage: G.non_fixed_points()
            [2, 3, 6]
        """
        pnts = set()
        for gens in self.gens():
            for cycles in gens.cycle_tuples():
                for thispnt in cycles:
                    if thispnt not in pnts:
                        pnts.add(thispnt)
        return sorted(pnts)

    def fixed_points(self):
        r"""
        Return the list of points fixed by ``self``, i.e., the subset
        of ``.domain()`` not moved by any element of ``self``.

        EXAMPLES::

            sage: G = PermutationGroup([(1,2,3)])
            sage: G.fixed_points()
            []
            sage: G = PermutationGroup([(1,2,3),(5,6)])
            sage: G.fixed_points()
            [4]
            sage: G = PermutationGroup([[(1,4,7)],[(4,3),(6,7)]])
            sage: G.fixed_points()
            [2, 5]
        """
        non_fixed_points = self.non_fixed_points()
        return [i for i in self.domain() if i not in non_fixed_points]

    def is_transitive(self, domain=None):
        r"""
        Returns ``True`` if ``self`` acts transitively on ``domain``.
        A group $G$ acts transitively on set $S$ if for all `x,y\in S`
        there is some `g\in G` such that `x^g=y`.

        EXAMPLES::

            sage: G = SymmetricGroup(5)
            sage: G.is_transitive()
            True
            sage: G = PermutationGroup(['(1,2)(3,4)(5,6)'])
            sage: G.is_transitive()
            False

        ::

            sage: G = PermutationGroup([[(1,2,3,4,5)],[(1,2)]]) #S_5 on [1..5]
            sage: G.is_transitive([1,4,5])
            True
            sage: G.is_transitive([2..6])
            False
            sage: G.is_transitive(G.non_fixed_points())
            True
            sage: H = PermutationGroup([[(1,2,3)],[(4,5,6)]])
            sage: H.is_transitive(H.non_fixed_points())
            False

        Note that this differs from the definition in GAP, where
        ``IsTransitive`` returns whether the group is transitive on the
        set of points moved by the group.

        ::

            sage: G = PermutationGroup([(2,3)])
            sage: G.is_transitive()
            False
            sage: gap(G).IsTransitive()
            true
        """
        #If the domain is not a subset of self.domain(), then the
        #action isn't transitive.
        try:
            domain = self._domain_gap(domain)
        except ValueError:
            return False

        return self._gap_().IsTransitive(domain).bool()

    def is_primitive(self, domain=None):
        r"""
        Returns ``True`` if ``self`` acts primitively on ``domain``.
        A group $G$ acts primitively on a set $S$ if

        1. $G$ acts transitively on $S$ and

        2. the action induces no non-trivial block system on $S$.

        INPUT:

        - ``domain`` (optional)

        .. SEEALSO::

            - :meth:`~PermutationGroup_generic.blocks_all`

        EXAMPLES:

        By default, test for primitivity of ``self`` on its domain::

            sage: G = PermutationGroup([[(1,2,3,4)],[(1,2)]])
            sage: G.is_primitive()
            True
            sage: G = PermutationGroup([[(1,2,3,4)],[(2,4)]])
            sage: G.is_primitive()
            False

        You can specify a domain on which to test primitivity::

            sage: G = PermutationGroup([[(1,2,3,4)],[(2,4)]])
            sage: G.is_primitive([1..4])
            False
            sage: G.is_primitive([1,2,3])
            True
            sage: G = PermutationGroup([[(3,4,5,6)],[(3,4)]]) #S_4 on [3..6]
            sage: G.is_primitive(G.non_fixed_points())
            True

        """
        #If the domain is not a subset of self.domain(), then the
        #action isn't primitive.
        try:
            domain = self._domain_gap(domain)
        except ValueError:
            return False

        return self._gap_().IsPrimitive(domain).bool()

    def is_semi_regular(self, domain=None):
        r"""
        Returns ``True`` if ``self`` acts semi-regularly on ``domain``.
        A group $G$ acts semi-regularly on a set $S$ if the point
        stabilizers of $S$ in $G$ are trivial.

        ``domain`` is optional and may take several forms. See examples.

        EXAMPLES::

            sage: G = PermutationGroup([[(1,2,3,4)]])
            sage: G.is_semi_regular()
            True
            sage: G = PermutationGroup([[(1,2,3,4)],[(5,6)]])
            sage: G.is_semi_regular()
            False

        You can pass in a domain to test semi-regularity::

            sage: G = PermutationGroup([[(1,2,3,4)],[(5,6)]])
            sage: G.is_semi_regular([1..4])
            True
            sage: G.is_semi_regular(G.non_fixed_points())
            False

        """
        try:
            domain = self._domain_gap(domain)
        except ValueError:
            return False
        return self._gap_().IsSemiRegular(domain).bool()

    def is_regular(self, domain=None):
        r"""
        Returns ``True`` if ``self`` acts regularly on ``domain``.
        A group $G$ acts regularly on a set $S$ if

        1. $G$ acts transitively on $S$ and
        2. $G$ acts semi-regularly on $S$.

        EXAMPLES::

            sage: G = PermutationGroup([[(1,2,3,4)]])
            sage: G.is_regular()
            True
            sage: G = PermutationGroup([[(1,2,3,4)],[(5,6)]])
            sage: G.is_regular()
            False

        You can pass in a domain on which to test regularity::

            sage: G = PermutationGroup([[(1,2,3,4)],[(5,6)]])
            sage: G.is_regular([1..4])
            True
            sage: G.is_regular(G.non_fixed_points())
            False

        """
        try:
            domain = self._domain_gap(domain)
        except ValueError:
            return False
        return self._gap_().IsRegular(domain).bool()


    def normalizes(self, other):
        r"""
        Returns ``True`` if the group ``other`` is normalized by ``self``.
        Wraps GAP's ``IsNormal`` function.

        A group `G` normalizes a group `U` if and only if for every
        `g \in G` and `u \in U` the element `u^g`
        is a member of `U`. Note that `U` need not be a subgroup of `G`.

        EXAMPLES::

            sage: G = PermutationGroup(['(1,2,3)(4,5)'])
            sage: H = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
            sage: H.normalizes(G)
            False
            sage: G = SymmetricGroup(3)
            sage: H = PermutationGroup( [ (4,5,6) ] )
            sage: G.normalizes(H)
            True
            sage: H.normalizes(G)
            True

        In the last example, `G` and `H` are disjoint, so each normalizes the
        other.
        """
        return self._gap_().IsNormal(other).bool()

    ############## Series ######################

    def composition_series(self):
        """
        Return the composition series of this group as a list of
        permutation groups.

        EXAMPLES:

        These computations use pseudo-random numbers, so we set
        the seed for reproducible testing.

        ::

            sage: set_random_seed(0)
            sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
            sage: G.composition_series()  # random output
            [Permutation Group with generators [(1,2,3)(4,5), (3,4)], Permutation Group with generators [(1,5)(3,4), (1,5)(2,3), (1,5,4)], Permutation Group with generators [()]]
            sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]])
            sage: CS = G.composition_series()
            sage: CS[3]
            Subgroup of (Permutation Group with generators [(1,2), (1,2,3)(4,5)]) generated by [()]
        """
        current_randstate().set_seed_gap()
        CS = self._gap_().CompositionSeries()
        return [self.subgroup(gap_group=group) for group in CS]

    def derived_series(self):
        """
        Return the derived series of this group as a list of permutation
        groups.

        EXAMPLES:

        These computations use pseudo-random numbers, so we set
        the seed for reproducible testing.

        ::

            sage: set_random_seed(0)
            sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
            sage: G.derived_series()  # random output
            [Permutation Group with generators [(1,2,3)(4,5), (3,4)], Permutation Group with generators [(1,5)(3,4), (1,5)(2,4), (2,4)(3,5)]]
        """
        current_randstate().set_seed_gap()
        DS = self._gap_().DerivedSeries()
        return [self.subgroup(gap_group=group) for group in DS]

    def lower_central_series(self):
        """
        Return the lower central series of this group as a list of
        permutation groups.

        EXAMPLES:

        These computations use pseudo-random numbers, so we set
        the seed for reproducible testing.

        ::

            sage: set_random_seed(0)
            sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
            sage: G.lower_central_series()  # random output
            [Permutation Group with generators [(1,2,3)(4,5), (3,4)], Permutation Group with generators [(1,5)(3,4), (1,5)(2,3), (1,3)(2,4)]]
        """
        current_randstate().set_seed_gap()
        LCS = self._gap_().LowerCentralSeriesOfGroup()
        return [self.subgroup(gap_group=group) for group in LCS]

    def molien_series(self):
        r"""
        Return the Molien series of a permutation group. The
        function

        .. MATH::

                     M(x) = (1/|G|)\sum_{g\in G} \det(1-x*g)^{-1}

        is sometimes called the "Molien series" of `G`. GAP's
        ``MolienSeries`` is associated to a character of a
        group `G`. How are these related? A group `G`, given as a permutation
        group on `n` points, has a "natural" representation of dimension `n`,
        given by permutation matrices. The Molien series of `G` is the one
        associated to that permutation representation of `G` using the above
        formula. Character values then count fixed points of the
        corresponding permutations.

        EXAMPLES::

            sage: G = SymmetricGroup(5)
            sage: G.molien_series()
            1/(-x^15 + x^14 + x^13 - x^10 - x^9 - x^8 + x^7 + x^6 + x^5 - x^2 - x + 1)
            sage: G = SymmetricGroup(3)
            sage: G.molien_series()
            1/(-x^6 + x^5 + x^4 - x^2 - x + 1)

        Some further tests (after :trac:`15817`)::

            sage: G = PermutationGroup([[(1,2,3,4)]])
            sage: S4ms = SymmetricGroup(4).molien_series()
            sage: G.molien_series() / S4ms
            x^5 + 2*x^4 + x^3 + x^2 + 1

        This works for not-transitive groups::

            sage: G = PermutationGroup([[(1,2)],[(3,4)]])
            sage: G.molien_series() / S4ms
            x^4 + x^3 + 2*x^2 + x + 1

        This works for groups with fixed points::

            sage: G = PermutationGroup([[(2,)]])
            sage: G.molien_series()
            1/(x^2 - 2*x + 1)
        """
        pi = self._gap_().NaturalCharacter()
        # because NaturalCharacter forgets about fixed points :
        pi += self._gap_().TrivialCharacter() * len(self.fixed_points())

        M = pi.MolienSeries()

        R = QQ['x']
        nn = M.NumeratorOfRationalFunction()
        dd = M.DenominatorOfRationalFunction()
        return (R(str(nn).replace("_1", "")) /
                R(str(dd).replace("_1", "")))

    def normal_subgroups(self):
        """
        Return the normal subgroups of this group as a (sorted in
        increasing order) list of permutation groups.

        The normal subgroups of `H = PSL(2,7) \\times PSL(2,7)` are
        `1`, two copies of `PSL(2,7)` and `H`
        itself, as the following example shows.

        EXAMPLES::

            sage: G = PSL(2,7)
            sage: D = G.direct_product(G)
            sage: H = D[0]
            sage: NH = H.normal_subgroups()
            sage: len(NH)
            4
            sage: NH[1].is_isomorphic(G)
            True
            sage: NH[2].is_isomorphic(G)
            True
        """
        NS = self._gap_().NormalSubgroups()
        return [self.subgroup(gap_group=group) for group in NS]

    def poincare_series(self, p=2, n=10):
        r"""
        Return the Poincaré series of `G \mod p` (`p \geq 2` must be a prime),
        for `n` large.

        In other words, if you input a finite group `G`, a prime `p`, and a
        positive integer `n`, it returns a quotient of polynomials `f(x) = P(x)
        / Q(x)` whose coefficient of `x^k` equals the rank of the vector space
        `H_k(G, \ZZ / p \ZZ)`, for all `k` in the range `1 \leq k \leq n`.

        REQUIRES: GAP package HAP (in gap_packages-\*.spkg).

        EXAMPLES::

            sage: G = SymmetricGroup(5)
            sage: G.poincare_series(2,10)                              # optional - gap_packages
            (x^2 + 1)/(x^4 - x^3 - x + 1)
            sage: G = SymmetricGroup(3)
            sage: G.poincare_series(2,10)                              # optional - gap_packages
            1/(-x + 1)

        AUTHORS:

        - David Joyner and Graham Ellis

        """
        load_hap()
        from sage.arith.all import is_prime
        if not (p == 0 or is_prime(p)):
            raise ValueError("p must be 0 or prime")

        ff = self._gap_().PoincareSeriesPrimePart(p, n)
        R = QQ['x']
        nn = ff.NumeratorOfRationalFunction()
        dd = ff.DenominatorOfRationalFunction()
        return (R(str(nn).replace('x_1', 'x')) /
                R(str(dd).replace('x_1', 'x')))


    def sylow_subgroup(self, p):
        """
        Returns a Sylow `p`-subgroup of the finite group `G`, where `p` is a
        prime. This is a `p`-subgroup of `G` whose index in `G` is coprime to
        `p`.

        Wraps the GAP function ``SylowSubgroup``.

        EXAMPLES::

            sage: G = PermutationGroup(['(1,2,3)', '(2,3)'])
            sage: G.sylow_subgroup(2)
            Subgroup of (Permutation Group with generators [(2,3), (1,2,3)]) generated by [(2,3)]
            sage: G.sylow_subgroup(5)
            Subgroup of (Permutation Group with generators [(2,3), (1,2,3)]) generated by [()]

        TESTS:

        Implementation details should not prevent us from computing
        large subgroups (:trac:`5491`)::

            sage: PSL(10,2).sylow_subgroup(7)
            Subgroup of...

        """
        return self.subgroup(gap_group=self._gap_().SylowSubgroup(p))

    def upper_central_series(self):
        """
        Return the upper central series of this group as a list of
        permutation groups.

        EXAMPLES:

        These computations use pseudo-random numbers, so we set
        the seed for reproducible testing::

            sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
            sage: G.upper_central_series()
            [Subgroup of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]) generated by [()]]
        """
        current_randstate().set_seed_gap()
        UCS = self._gap_().UpperCentralSeriesOfGroup()
        return [self.subgroup(gap_group=group) for group in UCS]

    from sage.groups.generic import structure_description



class PermutationGroup_subgroup(PermutationGroup_generic):
    """
    Subgroup subclass of ``PermutationGroup_generic``, so instance methods
    are inherited.

    EXAMPLES::

        sage: G = CyclicPermutationGroup(4)
        sage: gens = G.gens()
        sage: H = DihedralGroup(4)
        sage: H.subgroup(gens)
        Subgroup of (Dihedral group of order 8 as a permutation group) generated by [(1,2,3,4)]
        sage: K = H.subgroup(gens)
        sage: K.list()
        [(), (1,2,3,4), (1,3)(2,4), (1,4,3,2)]
        sage: K.ambient_group()
        Dihedral group of order 8 as a permutation group
        sage: K.gens()
        [(1,2,3,4)]
    """
    def __init__(self, ambient, gens=None, gap_group=None, domain=None,
                 category=None, canonicalize=True, check=True):
        r"""
        Initialization method for the
        ``PermutationGroup_subgroup`` class.

        INPUT:

        - ``ambient`` - the ambient group from which to construct this
          subgroup

        - ``gens`` - the generators of the subgroup

        - ``from_group`` - ``True``: subgroup is generated from a Gap
          string representation of the generators (default: ``False``)

        - ``check`` - ``True``: checks if ``gens`` are indeed elements of the
          ambient group

        - ``canonicalize`` - boolean (default: ``True``); if ``True``, sort
          generators and remove duplicates

        EXAMPLES:

        An example involving the dihedral group on four elements.
        `D_8` contains a cyclic subgroup or order four::

            sage: G = DihedralGroup(4)
            sage: H = CyclicPermutationGroup(4)
            sage: gens = H.gens(); gens
            [(1,2,3,4)]
            sage: S = G.subgroup(gens)
            sage: S
            Subgroup of (Dihedral group of order 8 as a permutation group) generated by [(1,2,3,4)]
            sage: S.list()
            [(), (1,2,3,4), (1,3)(2,4), (1,4,3,2)]
            sage: S.ambient_group()
            Dihedral group of order 8 as a permutation group

        However, `D_8` does not contain a cyclic subgroup of order
        three::

            sage: G = DihedralGroup(4)
            sage: H = CyclicPermutationGroup(3)
            sage: gens = H.gens()
            sage: S = PermutationGroup_subgroup(G,list(gens))
            Traceback (most recent call last):
            ...
            TypeError: each generator must be in the ambient group
        """
        if not isinstance(ambient, PermutationGroup_generic):
            raise TypeError("ambient (=%s) must be perm group"%ambient)
        if domain is None:
            domain = ambient.domain()
        if category is None:
            category = ambient.category()
        PermutationGroup_generic.__init__(self, gens=gens, gap_group=gap_group, domain=domain,
                                          category=category, canonicalize=canonicalize)

        self._ambient_group = ambient
        if check:
            from sage.groups.perm_gps.permgroup_named import SymmetricGroup
            if not isinstance(ambient, SymmetricGroup):
                ambient_gap_group = ambient._gap_()
                for g in self.gens():
                    if g._gap_() not in ambient_gap_group:
                        raise TypeError("each generator must be in the ambient group")

    def __richcmp__(self, other, op):
        r"""
        Compare ``self`` and ``other``.

        First, ``self`` and ``other`` are compared as permutation
        groups, see :meth:`PermutationGroup_generic.__richcmp__`.

        Second, if both are equal, the ambient groups are compared,
        where (if necessary) ``other`` is considered a subgroup of
        itself.

        EXAMPLES::

            sage: G = SymmetricGroup(6)
            sage: G1 = G.subgroup([G((1,2,3,4,5)),G((1,2))])
            sage: G2 = G.subgroup([G((1,2,3,4)),G((1,2))])
            sage: K = G2.subgroup([G2((1,2,3))])
            sage: H = G1.subgroup([G1(())])

        ``H`` is subgroup of ``K``, and therefore we have::

            sage: H < K
            True
            sage: K < H
            False

        In the next example, both ``H`` and ``H2`` are trivial
        groups, but the ambient group of ``H2`` is strictly contained
        in the ambient group of ``H``, and thus we obtain::

            sage: H2 = G2.subgroup([G2(())])
            sage: H2 < H
            True

        Of course, ``G`` as a permutation group and ``G`` considered
        as sub-group of itself are different objects. But they
        compare equal, both when considered as permutation groups
        or permutation subgroups::

            sage: G3 = G.subgroup([G((1,2,3,4,5,6)),G((1,2))])
            sage: G
            Symmetric group of order 6! as a permutation group
            sage: G3
            Subgroup of (Symmetric group of order 6! as a permutation group) generated by [(1,2), (1,2,3,4,5,6)]
            sage: G is G3
            False
            sage: G == G3 # as permutation groups
            True
            sage: G3 == G # as permutation subgroups
            True

        TESTS::

            sage: G = SymmetricGroup(4)
            sage: G.subgroup([G((1,2,3))]) == G.subgroup([G((2,3,1))])
            True
            sage: G.subgroup([G((1,2,3))]) == G.subgroup([G((1,3,2))])
            True

        """
        if self is other:
            return rich_to_bool(op, 0)

        if not isinstance(other, PermutationGroup_generic):
            return NotImplemented

        c = PermutationGroup_generic.__richcmp__(self, other, op_EQ)
        if not c:
            return PermutationGroup_generic.__richcmp__(self, other, op)

        if hasattr(other, 'ambient_group'):
            o_ambient = other.ambient_group()
        else:
            o_ambient = other
        return richcmp(self.ambient_group(), o_ambient, op)

    def _repr_(self):
        r"""
        Returns a string representation / description of the permutation
        subgroup.

        EXAMPLES:

        An example involving the dihedral group on four elements,
        `D_8`::

            sage: G = DihedralGroup(4)
            sage: H = CyclicPermutationGroup(4)
            sage: gens = H.gens()
            sage: S = PermutationGroup_subgroup(G, list(gens))
            sage: S
            Subgroup of (Dihedral group of order 8 as a permutation group) generated by [(1,2,3,4)]
            sage: S._repr_()
            'Subgroup of (Dihedral group of order 8 as a permutation group) generated by [(1,2,3,4)]'
        """
        s = "Subgroup of (%s) generated by %s"%(self.ambient_group(), self.gens())
        return s

    def _latex_(self):
        r"""
        Return LaTeX representation of this group.

        EXAMPLES:

        An example involving the dihedral group on four elements,
        `D_8`::

            sage: G = DihedralGroup(4)
            sage: H = CyclicPermutationGroup(4)
            sage: gens = H.gens()
            sage: S = PermutationGroup_subgroup(G, list(gens))
            sage: latex(S)
            \hbox{Subgroup } \langle (1,2,3,4) \rangle \hbox{ of } \langle (1,2,3,4), (1,4)(2,3) \rangle
            sage: S._latex_()
            '\\hbox{Subgroup } \\langle (1,2,3,4) \\rangle \\hbox{ of } \\langle (1,2,3,4), (1,4)(2,3) \\rangle'
        """
        gens = '\\langle ' + \
               ', '.join([x._latex_() for x in self.gens()]) + ' \\rangle'
        return '\\hbox{Subgroup } %s \\hbox{ of } %s' % \
                 (gens, self.ambient_group()._latex_())

    def ambient_group(self):
        """
        Return the ambient group related to ``self``.

        EXAMPLES:

        An example involving the dihedral group on four elements,
        `D_8`::

            sage: G = DihedralGroup(4)
            sage: H = CyclicPermutationGroup(4)
            sage: gens = H.gens()
            sage: S = PermutationGroup_subgroup(G, list(gens))
            sage: S.ambient_group()
            Dihedral group of order 8 as a permutation group
            sage: S.ambient_group() == G
            True
        """
        return self._ambient_group


    def is_normal(self, other=None):
        """
        Return ``True`` if this group is a normal subgroup of
        ``other``.  If ``other`` is not specified, then it is assumed
        to be the ambient group.

        EXAMPLES::

           sage: S = SymmetricGroup(['a','b','c'])
           sage: H = S.subgroup([('a', 'b', 'c')]); H
           Subgroup of (Symmetric group of order 3! as a permutation group) generated by [('a','b','c')]
           sage: H.is_normal()
           True

        """
        if other is None:
            other = self.ambient_group()
        return PermutationGroup_generic.is_normal(self, other)

from sage.misc.rest_index_of_methods import gen_rest_table_index
__doc__ = __doc__.format(METHODS_OF_PermutationGroup_generic=gen_rest_table_index(PermutationGroup_generic))