Permutation group elements#

AUTHORS:

  • David Joyner (2006-02)

  • David Joyner (2006-03): word problem method and reorganization

  • Robert Bradshaw (2007-11): convert to Cython

  • Sebastian Oehms (2018-11): Added gap() as synonym to _gap_() (compatibility to libgap framework, see Issue #26750)

  • Sebastian Oehms (2019-02): Implemented gap() properly (Issue #27234)

There are several ways to define a permutation group element:

  • Define a permutation group \(G\), then use G.gens() and multiplication * to construct elements.

  • Define a permutation group \(G\), then use, e.g., G([(1,2),(3,4,5)]) to construct an element of the group. You could also use G('(1,2)(3,4,5)')

  • Use, e.g., PermutationGroupElement([(1,2),(3,4,5)]) or PermutationGroupElement('(1,2)(3,4,5)') to make a permutation group element with parent \(S_5\).

EXAMPLES:

We illustrate construction of permutation using several different methods.

First we construct elements by multiplying together generators for a group:

sage: G = PermutationGroup(['(1,2)(3,4)', '(3,4,5,6)'], canonicalize=False)
sage: s = G.gens()
sage: s[0]
(1,2)(3,4)
sage: s[1]
(3,4,5,6)
sage: s[0]*s[1]
(1,2)(3,5,6)
sage: (s[0]*s[1]).parent()
Permutation Group with generators [(1,2)(3,4), (3,4,5,6)]

>>> from sage.all import *
>>> G = PermutationGroup(['(1,2)(3,4)', '(3,4,5,6)'], canonicalize=False)
>>> s = G.gens()
>>> s[Integer(0)]
(1,2)(3,4)
>>> s[Integer(1)]
(3,4,5,6)
>>> s[Integer(0)]*s[Integer(1)]
(1,2)(3,5,6)
>>> (s[Integer(0)]*s[Integer(1)]).parent()
Permutation Group with generators [(1,2)(3,4), (3,4,5,6)]

Next we illustrate creation of a permutation using coercion into an already-created group:

sage: g = G([(1,2),(3,5,6)])
sage: g
(1,2)(3,5,6)
sage: g.parent()
Permutation Group with generators [(1,2)(3,4), (3,4,5,6)]
sage: g == s[0]*s[1]
True

>>> from sage.all import *
>>> g = G([(Integer(1),Integer(2)),(Integer(3),Integer(5),Integer(6))])
>>> g
(1,2)(3,5,6)
>>> g.parent()
Permutation Group with generators [(1,2)(3,4), (3,4,5,6)]
>>> g == s[Integer(0)]*s[Integer(1)]
True

We can also use a string or one-line notation to specify the permutation:

sage: h = G('(1,2)(3,5,6)')
sage: i = G([2,1,5,4,6,3])
sage: g == h == i
True

>>> from sage.all import *
>>> h = G('(1,2)(3,5,6)')
>>> i = G([Integer(2),Integer(1),Integer(5),Integer(4),Integer(6),Integer(3)])
>>> g == h == i
True

The Rubik’s cube group:

sage: f = [(17,19,24,22),(18,21,23,20),( 6,25,43,16),( 7,28,42,13),( 8,30,41,11)]
sage: b = [(33,35,40,38),(34,37,39,36),( 3, 9,46,32),( 2,12,47,29),( 1,14,48,27)]
sage: l = [( 9,11,16,14),(10,13,15,12),( 1,17,41,40),( 4,20,44,37),( 6,22,46,35)]
sage: r = [(25,27,32,30),(26,29,31,28),( 3,38,43,19),( 5,36,45,21),( 8,33,48,24)]
sage: u = [( 1, 3, 8, 6),( 2, 5, 7, 4),( 9,33,25,17),(10,34,26,18),(11,35,27,19)]
sage: d = [(41,43,48,46),(42,45,47,44),(14,22,30,38),(15,23,31,39),(16,24,32,40)]
sage: cube = PermutationGroup([f, b, l, r, u, d])
sage: F, B, L, R, U, D = cube.gens()
sage: cube.order()
43252003274489856000
sage: F.order()
4

>>> from sage.all import *
>>> f = [(Integer(17),Integer(19),Integer(24),Integer(22)),(Integer(18),Integer(21),Integer(23),Integer(20)),( Integer(6),Integer(25),Integer(43),Integer(16)),( Integer(7),Integer(28),Integer(42),Integer(13)),( Integer(8),Integer(30),Integer(41),Integer(11))]
>>> b = [(Integer(33),Integer(35),Integer(40),Integer(38)),(Integer(34),Integer(37),Integer(39),Integer(36)),( Integer(3), Integer(9),Integer(46),Integer(32)),( Integer(2),Integer(12),Integer(47),Integer(29)),( Integer(1),Integer(14),Integer(48),Integer(27))]
>>> l = [( Integer(9),Integer(11),Integer(16),Integer(14)),(Integer(10),Integer(13),Integer(15),Integer(12)),( Integer(1),Integer(17),Integer(41),Integer(40)),( Integer(4),Integer(20),Integer(44),Integer(37)),( Integer(6),Integer(22),Integer(46),Integer(35))]
>>> r = [(Integer(25),Integer(27),Integer(32),Integer(30)),(Integer(26),Integer(29),Integer(31),Integer(28)),( Integer(3),Integer(38),Integer(43),Integer(19)),( Integer(5),Integer(36),Integer(45),Integer(21)),( Integer(8),Integer(33),Integer(48),Integer(24))]
>>> u = [( Integer(1), Integer(3), Integer(8), Integer(6)),( Integer(2), Integer(5), Integer(7), Integer(4)),( Integer(9),Integer(33),Integer(25),Integer(17)),(Integer(10),Integer(34),Integer(26),Integer(18)),(Integer(11),Integer(35),Integer(27),Integer(19))]
>>> d = [(Integer(41),Integer(43),Integer(48),Integer(46)),(Integer(42),Integer(45),Integer(47),Integer(44)),(Integer(14),Integer(22),Integer(30),Integer(38)),(Integer(15),Integer(23),Integer(31),Integer(39)),(Integer(16),Integer(24),Integer(32),Integer(40))]
>>> cube = PermutationGroup([f, b, l, r, u, d])
>>> F, B, L, R, U, D = cube.gens()
>>> cube.order()
43252003274489856000
>>> F.order()
4

We create element of a permutation group of large degree:

sage: G = SymmetricGroup(30)
sage: s = G(srange(30,0,-1)); s
(1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)

>>> from sage.all import *
>>> G = SymmetricGroup(Integer(30))
>>> s = G(srange(Integer(30),Integer(0),-Integer(1))); s
(1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)
class sage.groups.perm_gps.permgroup_element.PermutationGroupElement[source]#

Bases: MultiplicativeGroupElement

An element of a permutation group.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)']); G
Permutation Group with generators [(1,2,3)(4,5)]
sage: g = G.random_element()
sage: g in G
True
sage: g = G.gen(0); g
(1,2,3)(4,5)
sage: print(g)
(1,2,3)(4,5)
sage: g*g
(1,3,2)
sage: g**(-1)
(1,3,2)(4,5)
sage: g**2
(1,3,2)
sage: G = PermutationGroup([(1,2,3)])
sage: g = G.gen(0); g
(1,2,3)
sage: g.order()
3

>>> from sage.all import *
>>> G = PermutationGroup(['(1,2,3)(4,5)']); G
Permutation Group with generators [(1,2,3)(4,5)]
>>> g = G.random_element()
>>> g in G
True
>>> g = G.gen(Integer(0)); g
(1,2,3)(4,5)
>>> print(g)
(1,2,3)(4,5)
>>> g*g
(1,3,2)
>>> g**(-Integer(1))
(1,3,2)(4,5)
>>> g**Integer(2)
(1,3,2)
>>> G = PermutationGroup([(Integer(1),Integer(2),Integer(3))])
>>> g = G.gen(Integer(0)); g
(1,2,3)
>>> g.order()
3

This example illustrates how permutations act on multivariate polynomials.

sage: R = PolynomialRing(RationalField(), 5, ["x","y","z","u","v"])
sage: x, y, z, u, v = R.gens()
sage: f = x**2 - y**2 + 3*z**2
sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: sigma = G.gen(0)
sage: f * sigma
3*x^2 + y^2 - z^2

>>> from sage.all import *
>>> R = PolynomialRing(RationalField(), Integer(5), ["x","y","z","u","v"])
>>> x, y, z, u, v = R.gens()
>>> f = x**Integer(2) - y**Integer(2) + Integer(3)*z**Integer(2)
>>> G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
>>> sigma = G.gen(Integer(0))
>>> f * sigma
3*x^2 + y^2 - z^2
cycle_string(singletons=False)[source]#

Return string representation of this permutation.

EXAMPLES:

sage: g = PermutationGroupElement([(1,2,3),(4,5)])
sage: g.cycle_string()
'(1,2,3)(4,5)'

sage: g = PermutationGroupElement([3,2,1])
sage: g.cycle_string(singletons=True)
'(1,3)(2)'

>>> from sage.all import *
>>> g = PermutationGroupElement([(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))])
>>> g.cycle_string()
'(1,2,3)(4,5)'

>>> g = PermutationGroupElement([Integer(3),Integer(2),Integer(1)])
>>> g.cycle_string(singletons=True)
'(1,3)(2)'
cycle_tuples(singletons=False)[source]#

Return self as a list of disjoint cycles, represented as tuples rather than permutation group elements.

INPUT:

  • singletons – boolean (default: False) whether or not consider the cycle that correspond to fixed point

EXAMPLES:

sage: p = PermutationGroupElement('(2,6)(4,5,1)')
sage: p.cycle_tuples()
[(1, 4, 5), (2, 6)]
sage: p.cycle_tuples(singletons=True)
[(1, 4, 5), (2, 6), (3,)]

>>> from sage.all import *
>>> p = PermutationGroupElement('(2,6)(4,5,1)')
>>> p.cycle_tuples()
[(1, 4, 5), (2, 6)]
>>> p.cycle_tuples(singletons=True)
[(1, 4, 5), (2, 6), (3,)]

EXAMPLES:

sage: S = SymmetricGroup(4)
sage: S.gen(0).cycle_tuples()
[(1, 2, 3, 4)]

>>> from sage.all import *
>>> S = SymmetricGroup(Integer(4))
>>> S.gen(Integer(0)).cycle_tuples()
[(1, 2, 3, 4)]

sage: S = SymmetricGroup(['a','b','c','d'])
sage: S.gen(0).cycle_tuples()
[('a', 'b', 'c', 'd')]
sage: S([('a', 'b'), ('c', 'd')]).cycle_tuples()
[('a', 'b'), ('c', 'd')]

>>> from sage.all import *
>>> S = SymmetricGroup(['a','b','c','d'])
>>> S.gen(Integer(0)).cycle_tuples()
[('a', 'b', 'c', 'd')]
>>> S([('a', 'b'), ('c', 'd')]).cycle_tuples()
[('a', 'b'), ('c', 'd')]
cycle_type(singletons=True, as_list=False)[source]#

Return the partition that gives the cycle type of self as an element of its parent.

INPUT:

  • g – an element of the permutation group self.parent()

  • singletons – True or False depending on whether or not trivial cycles should be counted (default: True)

  • as_list – True or False depending on whether the cycle type should be returned as a list or as a Partition (default: False)

OUTPUT: a Partition, or list if is_list is True, giving the cycle type of self

If speed is a concern, then as_list=True should be used.

EXAMPLES:

sage: # needs sage.combinat
sage: G = DihedralGroup(3)
sage: [g.cycle_type() for g in G]
[[1, 1, 1], [3], [3], [2, 1], [2, 1], [2, 1]]
sage: PermutationGroupElement('(1,2,3)(4,5)(6,7,8)').cycle_type()
[3, 3, 2]
sage: G = SymmetricGroup(3); G('(1,2)').cycle_type()
[2, 1]
sage: G = SymmetricGroup(4); G('(1,2)').cycle_type()
[2, 1, 1]
sage: G = SymmetricGroup(4); G('(1,2)').cycle_type(singletons=False)
[2]
sage: G = SymmetricGroup(4); G('(1,2)').cycle_type(as_list=False)
[2, 1, 1]

>>> from sage.all import *
>>> # needs sage.combinat
>>> G = DihedralGroup(Integer(3))
>>> [g.cycle_type() for g in G]
[[1, 1, 1], [3], [3], [2, 1], [2, 1], [2, 1]]
>>> PermutationGroupElement('(1,2,3)(4,5)(6,7,8)').cycle_type()
[3, 3, 2]
>>> G = SymmetricGroup(Integer(3)); G('(1,2)').cycle_type()
[2, 1]
>>> G = SymmetricGroup(Integer(4)); G('(1,2)').cycle_type()
[2, 1, 1]
>>> G = SymmetricGroup(Integer(4)); G('(1,2)').cycle_type(singletons=False)
[2]
>>> G = SymmetricGroup(Integer(4)); G('(1,2)').cycle_type(as_list=False)
[2, 1, 1]
cycles()[source]#

Return self as a list of disjoint cycles.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5,6,7)'])
sage: g = G.0
sage: g.cycles()
[(1,2,3), (4,5,6,7)]
sage: a, b = g.cycles()
sage: a(1), b(1)
(2, 1)

>>> from sage.all import *
>>> G = PermutationGroup(['(1,2,3)(4,5,6,7)'])
>>> g = G.gen(0)
>>> g.cycles()
[(1,2,3), (4,5,6,7)]
>>> a, b = g.cycles()
>>> a(Integer(1)), b(Integer(1))
(2, 1)
dict()[source]#

Return a dictionary associating each element of the domain with its image.

EXAMPLES:

sage: G = SymmetricGroup(4)
sage: g = G((1,2,3,4)); g
(1,2,3,4)
sage: v = g.dict(); v
{1: 2, 2: 3, 3: 4, 4: 1}
sage: type(v[1])
<... 'int'>
sage: x = G([2,1]); x
(1,2)
sage: x.dict()
{1: 2, 2: 1, 3: 3, 4: 4}

>>> from sage.all import *
>>> G = SymmetricGroup(Integer(4))
>>> g = G((Integer(1),Integer(2),Integer(3),Integer(4))); g
(1,2,3,4)
>>> v = g.dict(); v
{1: 2, 2: 3, 3: 4, 4: 1}
>>> type(v[Integer(1)])
<... 'int'>
>>> x = G([Integer(2),Integer(1)]); x
(1,2)
>>> x.dict()
{1: 2, 2: 1, 3: 3, 4: 4}
domain()[source]#

Return the domain of self.

EXAMPLES:

sage: G = SymmetricGroup(4)
sage: x = G([2,1,4,3]); x
(1,2)(3,4)
sage: v = x.domain(); v
[2, 1, 4, 3]
sage: type(v[0])
<... 'int'>
sage: x = G([2,1]); x
(1,2)
sage: x.domain()
[2, 1, 3, 4]

>>> from sage.all import *
>>> G = SymmetricGroup(Integer(4))
>>> x = G([Integer(2),Integer(1),Integer(4),Integer(3)]); x
(1,2)(3,4)
>>> v = x.domain(); v
[2, 1, 4, 3]
>>> type(v[Integer(0)])
<... 'int'>
>>> x = G([Integer(2),Integer(1)]); x
(1,2)
>>> x.domain()
[2, 1, 3, 4]
gap()[source]#

Return self as a libgap element.

EXAMPLES:

sage: S = SymmetricGroup(4)
sage: p = S('(2,4)')
sage: p_libgap = libgap(p)
sage: p_libgap.Order()
2
sage: S(p_libgap) == p
True

sage: # needs sage.rings.finite_rings
sage: P = PGU(8,2)
sage: p, q = P.gens()
sage: p_libgap  = p.gap()

>>> from sage.all import *
>>> S = SymmetricGroup(Integer(4))
>>> p = S('(2,4)')
>>> p_libgap = libgap(p)
>>> p_libgap.Order()
2
>>> S(p_libgap) == p
True

>>> # needs sage.rings.finite_rings
>>> P = PGU(Integer(8),Integer(2))
>>> p, q = P.gens()
>>> p_libgap  = p.gap()
has_descent(i, side='right', positive=False)[source]#

Return whether self has a left (resp. right) descent at position i. If positive is True, then test for a non descent instead.

Beware that, since permutations are acting on the right, the meaning of descents is the reverse of the usual convention. Hence, self has a left descent at position i if self(i) > self(i+1).

INPUT:

  • i – an element of the index set

  • side – "left" or "right" (default: "right")

  • positive – a boolean (default: False)

EXAMPLES:

sage: S = SymmetricGroup([1,2,3])
sage: S.one().has_descent(1)
False
sage: S.one().has_descent(2)
False
sage: s = S.simple_reflections()
sage: x = s[1]*s[2]
sage: x.has_descent(1, side="right")
False
sage: x.has_descent(2, side="right")
True
sage: x.has_descent(1, side="left")
True
sage: x.has_descent(2, side="left")
False
sage: S._test_has_descent()

>>> from sage.all import *
>>> S = SymmetricGroup([Integer(1),Integer(2),Integer(3)])
>>> S.one().has_descent(Integer(1))
False
>>> S.one().has_descent(Integer(2))
False
>>> s = S.simple_reflections()
>>> x = s[Integer(1)]*s[Integer(2)]
>>> x.has_descent(Integer(1), side="right")
False
>>> x.has_descent(Integer(2), side="right")
True
>>> x.has_descent(Integer(1), side="left")
True
>>> x.has_descent(Integer(2), side="left")
False
>>> S._test_has_descent()

The symmetric group acting on a set not of the form \((1,\dots,n)\) is also supported:

sage: S = SymmetricGroup([2,4,1])
sage: s = S.simple_reflections()
sage: x = s[2]*s[4]
sage: x.has_descent(4)
True
sage: S._test_has_descent()

>>> from sage.all import *
>>> S = SymmetricGroup([Integer(2),Integer(4),Integer(1)])
>>> s = S.simple_reflections()
>>> x = s[Integer(2)]*s[Integer(4)]
>>> x.has_descent(Integer(4))
True
>>> S._test_has_descent()
inverse()[source]#

Return the inverse permutation.

OUTPUT:

For an element of a permutation group, this method returns the inverse element, which is both the inverse function and the inverse as an element of a group.

EXAMPLES:

sage: s = PermutationGroupElement("(1,2,3)(4,5)")
sage: s.inverse()
(1,3,2)(4,5)

sage: A = AlternatingGroup(4)
sage: t = A("(1,2,3)")
sage: t.inverse()
(1,3,2)

>>> from sage.all import *
>>> s = PermutationGroupElement("(1,2,3)(4,5)")
>>> s.inverse()
(1,3,2)(4,5)

>>> A = AlternatingGroup(Integer(4))
>>> t = A("(1,2,3)")
>>> t.inverse()
(1,3,2)

There are several ways (syntactically) to get an inverse of a permutation group element.

sage: s = PermutationGroupElement("(1,2,3,4)(6,7,8)")
sage: s.inverse() == s^-1
True
sage: s.inverse() == ~s
True

>>> from sage.all import *
>>> s = PermutationGroupElement("(1,2,3,4)(6,7,8)")
>>> s.inverse() == s**-Integer(1)
True
>>> s.inverse() == ~s
True
matrix()[source]#

Return a deg \(\times\) deg permutation matrix associated to the permutation self.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: g = G.gen(0)
sage: g.matrix()
[0 1 0 0 0]
[0 0 1 0 0]
[1 0 0 0 0]
[0 0 0 0 1]
[0 0 0 1 0]

>>> from sage.all import *
>>> G = PermutationGroup(['(1,2,3)(4,5)'])
>>> g = G.gen(Integer(0))
>>> g.matrix()
[0 1 0 0 0]
[0 0 1 0 0]
[1 0 0 0 0]
[0 0 0 0 1]
[0 0 0 1 0]
multiplicative_order()[source]#

Return the order of this group element, which is the smallest positive integer \(n\) for which \(g^n = 1\).

EXAMPLES:

sage: s = PermutationGroupElement('(1,2)(3,5,6)')
sage: s.multiplicative_order()
6

>>> from sage.all import *
>>> s = PermutationGroupElement('(1,2)(3,5,6)')
>>> s.multiplicative_order()
6

order() is just an alias for multiplicative_order():

sage: s.order()
6

>>> from sage.all import *
>>> s.order()
6
orbit(n, sorted=True)[source]#

Return the orbit of the integer \(n\) under this group element, as a sorted list.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: g = G.gen(0)
sage: g.orbit(4)
[4, 5]
sage: g.orbit(3)
[1, 2, 3]
sage: g.orbit(10)
[10]

>>> from sage.all import *
>>> G = PermutationGroup(['(1,2,3)(4,5)'])
>>> g = G.gen(Integer(0))
>>> g.orbit(Integer(4))
[4, 5]
>>> g.orbit(Integer(3))
[1, 2, 3]
>>> g.orbit(Integer(10))
[10]

sage: s = SymmetricGroup(['a', 'b']).gen(0); s
('a','b')
sage: s.orbit('a')
['a', 'b']

>>> from sage.all import *
>>> s = SymmetricGroup(['a', 'b']).gen(Integer(0)); s
('a','b')
>>> s.orbit('a')
['a', 'b']
sign()[source]#

Return the sign of self, which is \((-1)^{s}\), where \(s\) is the number of swaps.

EXAMPLES:

sage: s = PermutationGroupElement('(1,2)(3,5,6)')
sage: s.sign()
-1

>>> from sage.all import *
>>> s = PermutationGroupElement('(1,2)(3,5,6)')
>>> s.sign()
-1

ALGORITHM: Only even cycles contribute to the sign, thus

\[sign(sigma) = (-1)^{\sum_c len(c)-1}\]

where the sum is over cycles in self.

tuple()[source]#

Return tuple of images of the domain under self.

EXAMPLES:

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

sage: S = SymmetricGroup(['a', 'b'])
sage: S.gen().tuple()
('b', 'a')

>>> from sage.all import *
>>> G = SymmetricGroup(Integer(5))
>>> s = G([Integer(2),Integer(1),Integer(5),Integer(3),Integer(4)])
>>> s.tuple()
(2, 1, 5, 3, 4)

>>> S = SymmetricGroup(['a', 'b'])
>>> S.gen().tuple()
('b', 'a')
word_problem(words, display=True, as_list=False)[source]#

Try to solve the word problem for self.

INPUT:

  • words – a list of elements of the ambient group, generating a subgroup

  • display – boolean (default True) whether to display additional information

  • as_list – boolean (default False) whether to return the result as a list of pairs (generator, exponent)

OUTPUT:

  • a pair of strings, both representing the same word

or

  • a list of pairs representing the word, each pair being (generator as a string, exponent as an integer)

Let \(G\) be the ambient permutation group, containing the given element \(g\). Let \(H\) be the subgroup of \(G\) generated by the list words of elements of \(G\). If \(g\) is in \(H\), this function returns an expression for \(g\) as a word in the elements of words and their inverses.

This function does not solve the word problem in Sage. Rather it pushes it over to GAP, which has optimized algorithms for the word problem. Essentially, this function is a wrapper for the GAP functions EpimorphismFromFreeGroup and PreImagesRepresentative.

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]], canonicalize=False)
sage: g1, g2 = G.gens()
sage: h = g1^2*g2*g1
sage: h.word_problem([g1,g2], False)
('x1^2*x2^-1*x1', '(1,2,3)(4,5)^2*(3,4)^-1*(1,2,3)(4,5)')

sage: h.word_problem([g1,g2])
   x1^2*x2^-1*x1
   [['(1,2,3)(4,5)', 2], ['(3,4)', -1], ['(1,2,3)(4,5)', 1]]
('x1^2*x2^-1*x1', '(1,2,3)(4,5)^2*(3,4)^-1*(1,2,3)(4,5)')

sage: h.word_problem([g1,g2], False, as_list=True)
[['(1,2,3)(4,5)', 2], ['(3,4)', -1], ['(1,2,3)(4,5)', 1]]

>>> from sage.all import *
>>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))],[(Integer(3),Integer(4))]], canonicalize=False)
>>> g1, g2 = G.gens()
>>> h = g1**Integer(2)*g2*g1
>>> h.word_problem([g1,g2], False)
('x1^2*x2^-1*x1', '(1,2,3)(4,5)^2*(3,4)^-1*(1,2,3)(4,5)')

>>> h.word_problem([g1,g2])
   x1^2*x2^-1*x1
   [['(1,2,3)(4,5)', 2], ['(3,4)', -1], ['(1,2,3)(4,5)', 1]]
('x1^2*x2^-1*x1', '(1,2,3)(4,5)^2*(3,4)^-1*(1,2,3)(4,5)')

>>> h.word_problem([g1,g2], False, as_list=True)
[['(1,2,3)(4,5)', 2], ['(3,4)', -1], ['(1,2,3)(4,5)', 1]]
class sage.groups.perm_gps.permgroup_element.SymmetricGroupElement[source]#

Bases: PermutationGroupElement

An element of the symmetric group.

absolute_length()[source]#

Return the absolute length of self.

The absolute length is the size minus the number of its disjoint cycles. Alternatively, it is the length of the shortest expression of the element as a product of reflections.

See also

absolute_le()

EXAMPLES:

sage: S = SymmetricGroup(3)
sage: [x.absolute_length() for x in S]                                      # needs sage.combinat
[0, 2, 2, 1, 1, 1]

>>> from sage.all import *
>>> S = SymmetricGroup(Integer(3))
>>> [x.absolute_length() for x in S]                                      # needs sage.combinat
[0, 2, 2, 1, 1, 1]
has_left_descent(i)[source]#

Return whether \(i\) is a left descent of self.

EXAMPLES:

sage: W = SymmetricGroup(4)
sage: w = W.from_reduced_word([1,3,2,1])
sage: [i for i in W.index_set() if w.has_left_descent(i)]
[1, 3]

>>> from sage.all import *
>>> W = SymmetricGroup(Integer(4))
>>> w = W.from_reduced_word([Integer(1),Integer(3),Integer(2),Integer(1)])
>>> [i for i in W.index_set() if w.has_left_descent(i)]
[1, 3]
sage.groups.perm_gps.permgroup_element.is_PermutationGroupElement(x)[source]#

Return True if x is a PermutationGroupElement.

EXAMPLES:

sage: p = PermutationGroupElement([(1,2),(3,4,5)])
sage: from sage.groups.perm_gps.permgroup_element import is_PermutationGroupElement
sage: is_PermutationGroupElement(p)
doctest:warning...
DeprecationWarning: The function is_PermutationGroupElement is deprecated;
use 'isinstance(..., PermutationGroupElement)' instead.
See https://github.com/sagemath/sage/issues/38184 for details.
True

>>> from sage.all import *
>>> p = PermutationGroupElement([(Integer(1),Integer(2)),(Integer(3),Integer(4),Integer(5))])
>>> from sage.groups.perm_gps.permgroup_element import is_PermutationGroupElement
>>> is_PermutationGroupElement(p)
doctest:warning...
DeprecationWarning: The function is_PermutationGroupElement is deprecated;
use 'isinstance(..., PermutationGroupElement)' instead.
See https://github.com/sagemath/sage/issues/38184 for details.
True
sage.groups.perm_gps.permgroup_element.make_permgroup_element(G, x)[source]#

Return a PermutationGroupElement given the permutation group G and the permutation x in list notation.

This is function is used when unpickling old (pre-domain) versions of permutation groups and their elements. This now does a bit of processing and calls make_permgroup_element_v2() which is used in unpickling the current PermutationGroupElements.

EXAMPLES:

sage: from sage.groups.perm_gps.permgroup_element import make_permgroup_element
sage: S = SymmetricGroup(3)
sage: make_permgroup_element(S, [1,3,2])
(2,3)

>>> from sage.all import *
>>> from sage.groups.perm_gps.permgroup_element import make_permgroup_element
>>> S = SymmetricGroup(Integer(3))
>>> make_permgroup_element(S, [Integer(1),Integer(3),Integer(2)])
(2,3)
sage.groups.perm_gps.permgroup_element.make_permgroup_element_v2(G, x, domain)[source]#

Return a PermutationGroupElement given the permutation group G, the permutation x in list notation, and the domain domain of the permutation group.

This is function is used when unpickling permutation groups and their elements.

EXAMPLES:

sage: from sage.groups.perm_gps.permgroup_element import make_permgroup_element_v2
sage: S = SymmetricGroup(3)
sage: make_permgroup_element_v2(S, [1,3,2], S.domain())
(2,3)

>>> from sage.all import *
>>> from sage.groups.perm_gps.permgroup_element import make_permgroup_element_v2
>>> S = SymmetricGroup(Integer(3))
>>> make_permgroup_element_v2(S, [Integer(1),Integer(3),Integer(2)], S.domain())
(2,3)