def automorphism_group(self):
"""
EXAMPLES::
sage: p = PermutationGroupElement((2,3))
sage: S = species.SetSpecies()
sage: F = S * S
sage: a = F.structures([1,2,3,4]).random_element(); a
{1}*{2, 3, 4}
sage: a.automorphism_group()
Permutation Group with generators [(2,3), (2,3,4)]
::
sage: [a.transport(g) for g in a.automorphism_group()]
[{1}*{2, 3, 4},
{1}*{2, 3, 4},
{1}*{2, 3, 4},
{1}*{2, 3, 4},
{1}*{2, 3, 4},
{1}*{2, 3, 4}]
::
sage: a = F.structures([1,2,3,4]).random_element(); a
{2, 3}*{1, 4}
sage: [a.transport(g) for g in a.automorphism_group()]
[{2, 3}*{1, 4}, {2, 3}*{1, 4}, {2, 3}*{1, 4}, {2, 3}*{1, 4}]
"""
from sage.groups.all import PermutationGroupElement, PermutationGroup, SymmetricGroup
from sage.misc.misc import uniq
from sage.combinat.species.misc import change_support
left, right = self._list
n = len(self._labels)
#Get the supports for each of the sides
l_support = self._subset._list
r_support = self._subset.complement()._list
#Get the automorphism group for the left object and
#make it have the correct support. Do the same to the
#right side.
l_aut = change_support(left.automorphism_group(), l_support)
r_aut = change_support(right.automorphism_group(), r_support)
identity = PermutationGroupElement([])
gens = l_aut.gens() + r_aut.gens()
gens = [g for g in gens if g != identity]
gens = uniq(gens) if len(gens) > 0 else [[]]
return PermutationGroup(gens)
def permutation_group_element(self):
"""
Returns this cycle as a permutation group element.
EXAMPLES::
sage: F = species.CycleSpecies()
sage: a = F.structures(["a", "b", "c"]).random_element(); a
('a', 'b', 'c')
sage: a.permutation_group_element()
(1,2,3)
"""
from sage.groups.all import PermutationGroupElement, SymmetricGroup
return PermutationGroupElement(tuple(self._list))
def change_support(perm, support, change_perm=None):
"""
Changes the support of a permutation defined on [1, ..., n] to
support.
EXAMPLES::
sage: from sage.combinat.species.misc import change_support
sage: p = PermutationGroupElement((1,2,3)); p
(1,2,3)
sage: change_support(p, [3,4,5])
(3,4,5)
"""
if change_perm is None:
change_perm = prod([
PermutationGroupElement((i + 1, support[i]))
for i in range(len(support)) if i + 1 != support[i]
], PermutationGroupElement([], SymmetricGroup(support)))
if isinstance(perm, PermutationGroup_generic):
return PermutationGroup(
[change_support(g, support, change_perm) for g in perm.gens()])
return change_perm * perm * ~change_perm