Module: sage.groups.perm_gps.permgroup_morphism
Permutation group homomorphisms
Author: - David Joyner (2006-03-21): first version - David Joyner (2008-06): fixed kernel and image to return a group, instead of a string.
sage: G = CyclicPermutationGroup(4) sage: gens = G.gens() sage: H = DihedralGroup(4) sage: g = G([(1,2,3,4)]) sage: phi = PermutationGroupMorphism_im_gens( G, H, gens, gens) sage: phi.image(G) Permutation Group with generators [(1,2,3,4)] sage: phi.kernel() Permutation Group with generators [()] sage: phi.image(g) (1,2,3,4) sage: phi(g) (1,2,3,4) sage: phi.range() Dihedral group of order 8 as a permutation group sage: phi.codomain() Dihedral group of order 8 as a permutation group sage: phi.domain() Cyclic group of order 4 as a permutation group
Module-level Functions
x) |
Put a permutation in Gap format, as a string.
f) |
Class: PermutationGroupMap
self, parent) |
Special Functions: __init__,
_repr_type
Class: PermutationGroupMorphism
sage: G = CyclicPermutationGroup(4) sage: gens = G.gens() sage: H = DihedralGroup(4) sage: g = G([(1,3),(2,4)]); g (1,3)(2,4) sage: phi = PermutationGroupMorphism_im_gens( G, H, gens, gens) sage: phi Homomorphism : Cyclic group of order 4 as a permutation group --> Dihedral group of order 8 as a permutation group sage: phi(g) (1,3)(2,4) sage: gens1 = G.gens() sage: gens2 = ((4,3,2,1),) sage: phi = PermutationGroupMorphism_im_gens( G, G, gens1, gens2) sage: g = G([(1,2,3,4)]); g (1,2,3,4) sage: phi(g) (1,4,3,2)
Author: David Joyner (2-2006)
self, G, H, gensG, imgsH) |
Functions: codomain,
domain,
image,
kernel,
range
self, J) |
J must be a subgroup of G. Computes the subgroup of H which is the image of J.
sage: G = CyclicPermutationGroup(4) sage: gens = G.gens() sage: H = DihedralGroup(4) sage: g = G([(1,2,3,4)]) sage: phi = PermutationGroupMorphism_im_gens( G, H, gens, gens) sage: phi.image(G) Permutation Group with generators [(1,2,3,4)] sage: phi.image(g) (1,2,3,4)
Special Functions: __call__,
__init__,
__repr__,
__str__,
_latex_
self, g) |
Some python code for wrapping GAP's Images function but only for permutation groups. Returns an error if g is not in G.
sage: G = CyclicPermutationGroup(4) sage: gens = G.gens() sage: H = DihedralGroup(4) sage: phi = PermutationGroupMorphism_im_gens( G, H, gens, gens) sage: g = G([(1,3),(2,4)]); g (1,3)(2,4) sage: phi(g) (1,3)(2,4)
Class: PermutationGroupMorphism_from_gap
Basic syntax:
PermutationGroupMorphism_from_gap(domain_group, range_group,'phi:=gap_hom_command;','phi') And don't forget the line: from sage.groups.perm_gps.permgroup_morphism import PermutationGroupMorphism_from_gap in your program.
sage: from sage.groups.perm_gps.permgroup_morphism import PermutationGroupMorphism_from_gap sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]]) sage: H = G.subgroup([G([(1,2,3,4)])]) sage: PermutationGroupMorphism_from_gap(H, G, 'phi:=Identity','phi') Homomorphism : Subgroup of Permutation Group with generators [(1,2)(3,4), (1,2,3,4)] generated by [(1,2,3,4)] --> Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]
self, G, H, gap_hom_str, [name=phi]) |
Functions: codomain,
domain,
image,
kernel,
range
self, J) |
J must be an element or a subgroup of G. Computes the subgroup of H which is the image of J.
sage: G = PSL(2,7) sage: D = G.direct_product(G) sage: H = D[0] sage: pr1 = D[3] sage: pr1.image(G) Permutation Group with generators [(3,7,5)(4,8,6), (1,2,6)(3,4,8)] sage: G.is_isomorphic(pr1.image(G)) True
self) |
Computes the subgroup of the domain group which is the kernel of self.
sage: G = PSL(2,7) sage: D = G.direct_product(G) sage: H = D[0] sage: pr1 = D[3] sage: G.is_isomorphic(pr1.kernel()) True
Special Functions: __call__,
__init__,
__repr__,
__str__,
_latex_
self, g) |
Some python code for wrapping GAP's Images function but only for permutation groups. Returns an error if g is not in G.
Class: PermutationGroupMorphism_id
self, X) |
Special Functions: __init__,
_repr_defn
Class: PermutationGroupMorphism_im_gens
sage: G = CyclicPermutationGroup(4) sage: gens = G.gens() sage: H = DihedralGroup(4) sage: g = G([(1,3),(2,4)]); g (1,3)(2,4) sage: phi = PermutationGroupMorphism_im_gens( G, H, gens, gens) sage: phi Homomorphism : Cyclic group of order 4 as a permutation group --> Dihedral group of order 8 as a permutation group sage: phi(g) (1,3)(2,4) sage: gens1 = G.gens() sage: gens2 = ((4,3,2,1),) sage: phi = PermutationGroupMorphism_im_gens( G, G, gens1, gens2) sage: g = G([(1,2,3,4)]); g (1,2,3,4) sage: phi(g) (1,4,3,2)
Author: David Joyner (2-2006)
self, G, H, gensG, imgsH) |
Functions: codomain,
domain,
image,
kernel,
range
self, J) |
J must be a subgroup of G. Computes the subgroup of H which is the image of J.
sage: G = CyclicPermutationGroup(4) sage: gens = G.gens() sage: H = DihedralGroup(4) sage: g = G([(1,2,3,4)]) sage: phi = PermutationGroupMorphism_im_gens( G, H, gens, gens) sage: phi.image(G) Permutation Group with generators [(1,2,3,4)] sage: phi.image(g) (1,2,3,4)
Special Functions: __call__,
__init__,
__repr__,
__str__,
_latex_
self, g) |
Some python code for wrapping GAP's Images function but only for permutation groups. Returns an error if g is not in G.
sage: G = CyclicPermutationGroup(4) sage: gens = G.gens() sage: H = DihedralGroup(4) sage: phi = PermutationGroupMorphism_im_gens( G, H, gens, gens) sage: g = G([(1,3),(2,4)]); g (1,3)(2,4) sage: phi(g) (1,3)(2,4)
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