Permutation group homomorphisms#

AUTHORS:

David Joyner (2006-03-21): first version
David Joyner (2008-06): fixed kernel and image to return a group, instead of a string.

EXAMPLES:

Sage

sage: G = CyclicPermutationGroup(4)
sage: H = DihedralGroup(4)
sage: g = G([(1,2,3,4)])
sage: phi = PermutationGroupMorphism_im_gens(G, H, map(H, G.gens()))
sage: phi.image(G)
Subgroup generated by [(1,2,3,4)] of
 (Dihedral group of order 8 as a permutation group)
sage: phi.kernel()
Subgroup generated by [()] of (Cyclic group of order 4 as a permutation group)
sage: phi.image(g)
(1,2,3,4)
sage: phi(g)
(1,2,3,4)
sage: phi.codomain()
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

Python

>>> from sage.all import *
>>> G = CyclicPermutationGroup(Integer(4))
>>> H = DihedralGroup(Integer(4))
>>> g = G([(Integer(1),Integer(2),Integer(3),Integer(4))])
>>> phi = PermutationGroupMorphism_im_gens(G, H, map(H, G.gens()))
>>> phi.image(G)
Subgroup generated by [(1,2,3,4)] of
 (Dihedral group of order 8 as a permutation group)
>>> phi.kernel()
Subgroup generated by [()] of (Cyclic group of order 4 as a permutation group)
>>> phi.image(g)
(1,2,3,4)
>>> phi(g)
(1,2,3,4)
>>> phi.codomain()
Dihedral group of order 8 as a permutation group
>>> phi.codomain()
Dihedral group of order 8 as a permutation group
>>> phi.domain()
Cyclic group of order 4 as a permutation group

class sage.groups.perm_gps.permgroup_morphism.PermutationGroupMorphism[source]#

Bases: Morphism

A set-theoretic map between PermutationGroups.

image(J)[source]#

Compute the subgroup of the codomain \(H\) which is the image of \(J\).

\(J\) must be a subgroup of the domain \(G\).

EXAMPLES:

Sage

sage: G = CyclicPermutationGroup(4)
sage: H = DihedralGroup(4)
sage: g = G([(1,2,3,4)])
sage: phi = PermutationGroupMorphism_im_gens(G, H, map(H, G.gens()))
sage: phi.image(G)
Subgroup generated by [(1,2,3,4)] of
 (Dihedral group of order 8 as a permutation group)
sage: phi.image(g)
(1,2,3,4)

Python

>>> from sage.all import *
>>> G = CyclicPermutationGroup(Integer(4))
>>> H = DihedralGroup(Integer(4))
>>> g = G([(Integer(1),Integer(2),Integer(3),Integer(4))])
>>> phi = PermutationGroupMorphism_im_gens(G, H, map(H, G.gens()))
>>> phi.image(G)
Subgroup generated by [(1,2,3,4)] of
 (Dihedral group of order 8 as a permutation group)
>>> phi.image(g)
(1,2,3,4)

Sage

sage: G = PSL(2,7)
sage: D = G.direct_product(G)
sage: H = D[0]
sage: pr1 = D[3]
sage: pr1.image(G)
Subgroup generated by [(3,7,5)(4,8,6), (1,2,6)(3,4,8)] of
 (The projective special linear group of degree 2 over Finite Field of size 7)
sage: G.is_isomorphic(pr1.image(G))
True

Python

>>> from sage.all import *
>>> G = PSL(Integer(2),Integer(7))
>>> D = G.direct_product(G)
>>> H = D[Integer(0)]
>>> pr1 = D[Integer(3)]
>>> pr1.image(G)
Subgroup generated by [(3,7,5)(4,8,6), (1,2,6)(3,4,8)] of
 (The projective special linear group of degree 2 over Finite Field of size 7)
>>> G.is_isomorphic(pr1.image(G))
True

Check that Issue #28324 is fixed:

Sage

sage: # needs sage.rings.number_field
sage: R.<x> = QQ[]
sage: f = x^4 + x^2 - 3
sage: L.<a> = f.splitting_field()
sage: G = L.galois_group()
sage: D4 = DihedralGroup(4)
sage: h = D4.isomorphism_to(G)
sage: h.image(D4).is_isomorphic(G)
True
sage: all(h.image(g) in G for g in D4.gens())
True

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> f = x**Integer(4) + x**Integer(2) - Integer(3)
>>> L = f.splitting_field(names=('a',)); (a,) = L._first_ngens(1)
>>> G = L.galois_group()
>>> D4 = DihedralGroup(Integer(4))
>>> h = D4.isomorphism_to(G)
>>> h.image(D4).is_isomorphic(G)
True
>>> all(h.image(g) in G for g in D4.gens())
True

kernel()[source]#

Return the kernel of this homomorphism as a permutation group.

EXAMPLES:

Sage

sage: G = CyclicPermutationGroup(4)
sage: H = DihedralGroup(4)
sage: g = G([(1,2,3,4)])
sage: phi = PermutationGroupMorphism_im_gens(G, H, [1])
sage: phi.kernel()
Subgroup generated by [(1,2,3,4)] of
 (Cyclic group of order 4 as a permutation group)

Python

>>> from sage.all import *
>>> G = CyclicPermutationGroup(Integer(4))
>>> H = DihedralGroup(Integer(4))
>>> g = G([(Integer(1),Integer(2),Integer(3),Integer(4))])
>>> phi = PermutationGroupMorphism_im_gens(G, H, [Integer(1)])
>>> phi.kernel()
Subgroup generated by [(1,2,3,4)] of
 (Cyclic group of order 4 as a permutation group)

Sage

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

Python

>>> from sage.all import *
>>> G = PSL(Integer(2),Integer(7))
>>> D = G.direct_product(G)
>>> H = D[Integer(0)]
>>> pr1 = D[Integer(3)]
>>> G.is_isomorphic(pr1.kernel())
True

class sage.groups.perm_gps.permgroup_morphism.PermutationGroupMorphism_from_gap(G, H, gap_hom)[source]#

Bases: PermutationGroupMorphism

This is a Python trick to allow Sage programmers to create a group homomorphism using GAP using very general constructions. An example of its usage is in the direct_product instance method of the PermutationGroup_generic class in permgroup.py.

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.

EXAMPLES:

Sage

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, gap.Identity)
Permutation group morphism:
  From: Subgroup generated by [(1,2,3,4)] of
        (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)])
  To:   Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]
  Defn: Identity

Python

>>> from sage.all import *
>>> from sage.groups.perm_gps.permgroup_morphism import PermutationGroupMorphism_from_gap
>>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3),Integer(4))]])
>>> H = G.subgroup([G([(Integer(1),Integer(2),Integer(3),Integer(4))])])
>>> PermutationGroupMorphism_from_gap(H, G, gap.Identity)
Permutation group morphism:
  From: Subgroup generated by [(1,2,3,4)] of
        (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)])
  To:   Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]
  Defn: Identity

class sage.groups.perm_gps.permgroup_morphism.PermutationGroupMorphism_id[source]#: Bases: PermutationGroupMorphism

class sage.groups.perm_gps.permgroup_morphism.PermutationGroupMorphism_im_gens(G, H, gens=None)[source]#

Bases: PermutationGroupMorphism

Some python code for wrapping GAP’s GroupHomomorphismByImages function but only for permutation groups. Can be expensive if G is large. This returns “fail” if gens does not generate self or if the map does not extend to a group homomorphism, self - other.

EXAMPLES:

Sage

sage: G = CyclicPermutationGroup(4)
sage: H = DihedralGroup(4)
sage: phi = PermutationGroupMorphism_im_gens(G, H, map(H, G.gens())); phi
Permutation group morphism:
  From: Cyclic group of order 4 as a permutation group
  To:   Dihedral group of order 8 as a permutation group
  Defn: [(1,2,3,4)] -> [(1,2,3,4)]
sage: g = G([(1,3),(2,4)]); g
(1,3)(2,4)
sage: phi(g)
(1,3)(2,4)
sage: images = ((4,3,2,1),)
sage: phi = PermutationGroupMorphism_im_gens(G, G, images)
sage: g = G([(1,2,3,4)]); g
(1,2,3,4)
sage: phi(g)
(1,4,3,2)

Python

>>> from sage.all import *
>>> G = CyclicPermutationGroup(Integer(4))
>>> H = DihedralGroup(Integer(4))
>>> phi = PermutationGroupMorphism_im_gens(G, H, map(H, G.gens())); phi
Permutation group morphism:
  From: Cyclic group of order 4 as a permutation group
  To:   Dihedral group of order 8 as a permutation group
  Defn: [(1,2,3,4)] -> [(1,2,3,4)]
>>> g = G([(Integer(1),Integer(3)),(Integer(2),Integer(4))]); g
(1,3)(2,4)
>>> phi(g)
(1,3)(2,4)
>>> images = ((Integer(4),Integer(3),Integer(2),Integer(1)),)
>>> phi = PermutationGroupMorphism_im_gens(G, G, images)
>>> g = G([(Integer(1),Integer(2),Integer(3),Integer(4))]); g
(1,2,3,4)
>>> phi(g)
(1,4,3,2)

AUTHORS:

David Joyner (2006-02)

sage.groups.perm_gps.permgroup_morphism.is_PermutationGroupMorphism(f)[source]#

Return True if the argument f is a PermutationGroupMorphism.

EXAMPLES:

Sage

sage: from sage.groups.perm_gps.permgroup_morphism import is_PermutationGroupMorphism
sage: G = CyclicPermutationGroup(4)
sage: H = DihedralGroup(4)
sage: phi = PermutationGroupMorphism_im_gens(G, H, map(H, G.gens()))
sage: is_PermutationGroupMorphism(phi)
doctest:warning...
DeprecationWarning: The function is_PermutationGroupMorphism is deprecated;
use 'isinstance(..., PermutationGroupMorphism)' instead.
See https://github.com/sagemath/sage/issues/38103 for details.
True

Python

>>> from sage.all import *
>>> from sage.groups.perm_gps.permgroup_morphism import is_PermutationGroupMorphism
>>> G = CyclicPermutationGroup(Integer(4))
>>> H = DihedralGroup(Integer(4))
>>> phi = PermutationGroupMorphism_im_gens(G, H, map(H, G.gens()))
>>> is_PermutationGroupMorphism(phi)
doctest:warning...
DeprecationWarning: The function is_PermutationGroupMorphism is deprecated;
use 'isinstance(..., PermutationGroupMorphism)' instead.
See https://github.com/sagemath/sage/issues/38103 for details.
True