def __init__(self, homset, imgsH, check=True):
"""
Group morphism specified by images of generators.
Some Python code for wrapping GAP's GroupHomomorphismByImages
function but only for matrix groups. Can be expensive if G is
large.
EXAMPLES::
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: G = MatrixGroup([MS([1,1,0,1])])
sage: H = MatrixGroup([MS([1,0,1,1])])
sage: phi = G.hom(H.gens())
sage: phi
Homomorphism : Matrix group over Finite Field of size 5 with 1 generators (
[1 1]
[0 1]
) --> Matrix group over Finite Field of size 5 with 1 generators (
[1 0]
[1 1]
)
sage: phi(MS([1,1,0,1]))
[1 0]
[1 1]
sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: F.multiplicative_generator()
3
sage: G = MatrixGroup([MS([3,0,0,1])])
sage: a = G.gens()[0]^2
sage: phi = G.hom([a])
TESTS:
Check that :trac:`19406` is fixed::
sage: G = GL(2, GF(3))
sage: H = GL(3, GF(2))
sage: mat1 = H([[-1,0,0],[0,0,-1],[0,-1,0]])
sage: mat2 = H([[1,1,1],[0,0,-1],[-1,0,0]])
sage: phi = G.hom([mat1, mat2])
Traceback (most recent call last):
...
TypeError: images do not define a group homomorphism
"""
MatrixGroupMorphism.__init__(self, homset) # sets the parent
from sage.libs.gap.libgap import libgap
G = homset.domain()
H = homset.codomain()
gens = [x.gap() for x in G.gens()]
imgs = [to_libgap(x) for x in imgsH]
self._phi = phi = libgap.GroupHomomorphismByImages(
G.gap(), H.gap(), gens, imgs)
if not phi.IsGroupHomomorphism():
raise ValueError('the map {}-->{} is not a homomorphism'.format(
G.gens(), imgsH))
def _element_constructor_(self, x, check=True, **options):
r"""
Handle conversions and coercions.
EXAMPLES::
sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap
sage: A = AbelianGroupGap([2,4])
sage: H = A.Hom(A)
sage: H([g^2 for g in A.gens()])
Group endomorphism of Abelian group with gap, generator orders (2, 4)
sage: H(2)
Traceback (most recent call last):
...
TypeError: unable to convert 2 to an element of ...
A group homomorphism between a finitely presented group and a subgroup of a permutation group::
sage: PG = PGU(6,2)
sage: g, h = PG.gens()
sage: p1 = h^-3*(h^-1*g^-1)^2*h*g*h^2*g^-1*h^2*g*h^-5*g^-1
sage: p2 = g*(g*h)^2*g*h^-4*(g*h)^2*(h^2*g*h^-2*g)^2*h^-2*g*h^-2*g^-1*h^-1*g*h*g*h^-1*g
sage: p3 = h^-3*g^-1*h*g*h^4*g^-1*h^-1*g*h*(h^2*g^-1)^2*h^-4*g*h^2*g^-1*h^-7*g^-2*h^-2*g*h^-2*g^-1*h^-1*(g*h)^2*h^3
sage: p4 = h*(h^3*g)^2*h*g*h^-1*g*h^2*g^-1*h^-2*g*h^4*g^-1*h^3*g*h^-2*g*h^-1*g^-1*h^2*g*h*g^-1*h^-2*g*h*g^-1*h^2*g*h^2*g^-1
sage: p5 = h^2*g*h^2*g^-1*h*g*h^-1*g*h*g^-1*h^2*g*h^-2*g*h^2*g*h^-2*(h^-1*g)^2*h^4*(g*h^-1)^2*g^-1
sage: UPG = PG.subgroup([p1, p2, p3, p4, p5], canonicalize=False)
sage: B6 = BraidGroup(6)
sage: reprB6 = B6.hom(UPG.gens())
sage: b1, b2, b3, b4, b5 = B6.gens()
sage: reprB6(b1*b2*b3*b4*b5) == p1*p2*p3*p4*p5
True
TESTS:
The following input does not define a valid group homomorphism::
sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap
sage: A = AbelianGroupGap([2])
sage: G = MatrixGroup([Matrix(ZZ, 2, [0,1,-1,0])])
sage: A.hom([G.gen(0)])
Traceback (most recent call last):
...
ValueError: images do not define a group homomorphism
"""
if isinstance(x, (tuple, list)):
# there should be a better way
from sage.libs.gap.libgap import libgap
dom = self.domain()
codom = self.codomain()
gens = dom.gap().GeneratorsOfGroup()
imgs = [codom(g).gap() for g in x]
if check:
if not len(gens) == len(imgs):
raise ValueError("provide an image for each generator")
phi = libgap.GroupHomomorphismByImages(dom.gap(), codom.gap(), gens, imgs)
# if it is not a group homomorphism, then
# self._phi is the gap boolean fail
if phi.is_bool(): # check we did not fail
raise ValueError("images do not define a group homomorphism")
else:
ByImagesNC = libgap.function_factory("GroupHomomorphismByImagesNC")
phi = ByImagesNC(dom.gap(), codom.gap(), gens, imgs)
return self.element_class(self, phi, check=check, **options)
if isinstance(x, GapElement):
try:
return self.element_class(self, x, check=True, **options)
except ValueError:
pass
return super(GroupHomset_libgap, self)._element_constructor_(x, check=check, **options)