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 github issue #26750)
Sebastian Oehms (2019-02): Implemented gap() properly (github 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)]

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

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

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

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)

class sage.groups.perm_gps.permgroup_element.PermutationGroupElement#

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

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

cycle_string(singletons=False)#

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)'

cycle_tuples(singletons=False)#

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,)]

EXAMPLES:

sage: S = SymmetricGroup(4)
sage: S.gen(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')]

cycle_type(singletons=True, as_list=False)#

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]

cycles()#

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)

dict()#

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}

domain()#

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]

gap()#

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()

has_descent(i, side='right', positive=False)#

INPUT:

i – an element of the index set
side – "left" or "right" (default: "right")
positive – a boolean (default: False)

Returns 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).

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()

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()

inverse()#

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)

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

matrix()#

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]

multiplicative_order()#

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

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

sage: s.order()
6

orbit(n, sorted=True)#

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]

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

sign()#

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

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()#

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')

word_problem(words, display=True, as_list=False)#

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

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]]

class sage.groups.perm_gps.permgroup_element.SymmetricGroupElement#

Bases: PermutationGroupElement

An element of the symmetric group.

absolute_length()#

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.