def as_permutation(self):
r"""
Return the element of the permutation group G (isomorphic to the
abelian group A) associated to a in A.
EXAMPLES::
sage: G = AbelianGroup(3,[2,3,4],names="abc"); G
Multiplicative Abelian group isomorphic to C2 x C3 x C4
sage: a,b,c=G.gens()
sage: Gp = G.permutation_group(); Gp
Permutation Group with generators [(6,7,8,9), (3,4,5), (1,2)]
sage: a.as_permutation()
(1,2)
sage: ap = a.as_permutation(); ap
(1,2)
sage: ap in Gp
True
"""
from sage.libs.gap.libgap import libgap
G = self.parent()
A = libgap.AbelianGroup(G.gens_orders())
phi = libgap.IsomorphismPermGroup(A)
gens = libgap.GeneratorsOfGroup(A)
L2 = libgap.Product([geni**Li for geni, Li in zip(gens, self.list())])
pg = libgap.Image(phi, L2)
return G.permutation_group()(pg)
def SymmetricPresentation(n):
r"""
Build the Symmetric group of order `n!` as a finitely presented group.
INPUT:
- ``n`` -- The size of the underlying set of arbitrary symbols being acted
on by the Symmetric group of order `n!`.
OUTPUT:
Symmetric group as a finite presentation, implementation uses GAP to find an
isomorphism from a permutation representation to a finitely presented group
representation. Due to this fact, the exact output presentation may not be
the same for every method call on a constant ``n``.
EXAMPLES::
sage: S4 = groups.presentation.Symmetric(4)
sage: S4.as_permutation_group().is_isomorphic(SymmetricGroup(4))
True
TESTS::
sage: S = [groups.presentation.Symmetric(i) for i in range(1,4)]; S[0].order()
1
sage: S[1].order(), S[2].as_permutation_group().is_isomorphic(DihedralGroup(3))
(2, True)
sage: S5 = groups.presentation.Symmetric(5)
sage: perm_S5 = S5.as_permutation_group(); perm_S5.is_isomorphic(SymmetricGroup(5))
True
sage: groups.presentation.Symmetric(8).order()
40320
"""
from sage.groups.perm_gps.permgroup_named import SymmetricGroup
from sage.groups.free_group import _lexi_gen
n = Integer(n)
perm_rep = SymmetricGroup(n)
GAP_fp_rep = libgap.Image(
libgap.IsomorphismFpGroupByGenerators(perm_rep, perm_rep.gens()))
image_gens = GAP_fp_rep.FreeGeneratorsOfFpGroup()
name_itr = _lexi_gen() # Python generator object for variable names
F = FreeGroup([next(name_itr) for x in perm_rep.gens()])
ret_rls = tuple([
F(rel_word.TietzeWordAbstractWord(image_gens).sage())
for rel_word in GAP_fp_rep.RelatorsOfFpGroup()
])
return FinitelyPresentedGroup(F, ret_rls)
def AlternatingPresentation(n):
r"""
Build the Alternating group of order `n!/2` as a finitely presented group.
INPUT:
- ``n`` -- The size of the underlying set of arbitrary symbols being acted
on by the Alternating group of order `n!/2`.
OUTPUT:
Alternating group as a finite presentation, implementation uses GAP to find an
isomorphism from a permutation representation to a finitely presented group
representation. Due to this fact, the exact output presentation may not be
the same for every method call on a constant ``n``.
EXAMPLES::
sage: A6 = groups.presentation.Alternating(6)
sage: A6.as_permutation_group().is_isomorphic(AlternatingGroup(6)), A6.order()
(True, 360)
TESTS::
sage: #even permutation test..
sage: A1 = groups.presentation.Alternating(1); A2 = groups.presentation.Alternating(2)
sage: A1.is_isomorphic(A2), A1.order()
(True, 1)
sage: A3 = groups.presentation.Alternating(3); A3.order(), A3.as_permutation_group().is_cyclic()
(3, True)
sage: A8 = groups.presentation.Alternating(8); A8.order()
20160
"""
from sage.groups.perm_gps.permgroup_named import AlternatingGroup
from sage.groups.free_group import _lexi_gen
n = Integer(n)
perm_rep = AlternatingGroup(n)
GAP_fp_rep = libgap.Image(
libgap.IsomorphismFpGroupByGenerators(perm_rep, perm_rep.gens()))
image_gens = GAP_fp_rep.FreeGeneratorsOfFpGroup()
name_itr = _lexi_gen() # Python generator object for variable names
F = FreeGroup([next(name_itr) for x in perm_rep.gens()])
ret_rls = tuple([
F(rel_word.TietzeWordAbstractWord(image_gens).sage())
for rel_word in GAP_fp_rep.RelatorsOfFpGroup()
])
return FinitelyPresentedGroup(F, ret_rls)