def PermutationGroupElement(g, parent=None, check=True):
r"""
Builds a permutation from ``g``.
INPUT:
- ``g`` -- either
- a list of images
- a tuple describing a single cycle
- a list of tuples describing the cycle decomposition
- a string describing the cycle decomposition
- ``parent`` -- (optional) an ambient permutation group for the result;
it is mandatory if you want a permutation on a domain different
from `\{1, \ldots, n\}`
- ``check`` -- (default: ``True``) whether additional check are performed;
setting it to ``False`` is likely to result in faster code
EXAMPLES:
Initialization as a list of images::
sage: p = PermutationGroupElement([1,4,2,3])
sage: p
(2,4,3)
sage: p.parent()
Symmetric group of order 4! as a permutation group
Initialization as a list of cycles::
sage: p = PermutationGroupElement([(3,5),(4,6,9)])
sage: p
(3,5)(4,6,9)
sage: p.parent()
Symmetric group of order 9! as a permutation group
Initialization as a string representing a cycle decomposition::
sage: p = PermutationGroupElement('(2,4)(3,5)')
sage: p
(2,4)(3,5)
sage: p.parent()
Symmetric group of order 5! as a permutation group
By default the constructor assumes that the domain is `\{1, \dots, n\}`
but it can be set to anything via its second ``parent`` argument::
sage: S = SymmetricGroup(['a', 'b', 'c', 'd', 'e'])
sage: PermutationGroupElement(['e', 'c', 'b', 'a', 'd'], S)
('a','e','d')('b','c')
sage: PermutationGroupElement(('a', 'b', 'c'), S)
('a','b','c')
sage: PermutationGroupElement([('a', 'c'), ('b', 'e')], S)
('a','c')('b','e')
sage: PermutationGroupElement("('a','b','e')('c','d')", S)
('a','b','e')('c','d')
But in this situation, you might want to use the more direct::
sage: S(['e', 'c', 'b', 'a', 'd'])
('a','e','d')('b','c')
sage: S(('a', 'b', 'c'))
('a','b','c')
sage: S([('a', 'c'), ('b', 'e')])
('a','c')('b','e')
sage: S("('a','b','e')('c','d')")
('a','b','e')('c','d')
"""
if isinstance(g, permgroup_element.PermutationGroupElement):
if parent is None or g.parent() is parent:
return g
if parent is None:
from sage.groups.perm_gps.permgroup_named import SymmetricGroup
try:
v = standardize_generator(g, None)
except KeyError:
raise ValueError("invalid permutation vector: %s" % g)
parent = SymmetricGroup(len(v))
# We have constructed the parent from the element and already checked
# that it is a valid permutation
check = False
return parent.element_class(g, parent, check)
