# 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: 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
```
class sage.groups.perm_gps.permgroup_morphism.PermutationGroupMorphism

A set-theoretic map between PermutationGroups.

image(J)

J must be a subgroup of G. Computes the subgroup of H which is the image of J.

EXAMPLES:

```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)
```
```sage: G = PSL(2,7)
sage: D = G.direct_product(G)
sage: H = D
sage: pr1 = D
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
```

Check that trac ticket #28324 is fixed:

```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)
Subgroup generated by [(1,2)(3,4)(5,7)(6,8), (1,6,4,7)(2,5,3,8)] of (Galois group 8T4 (2) with order 8 of x^8 + 4*x^7 + 12*x^6 + 22*x^5 + 23*x^4 + 14*x^3 + 28*x^2 + 24*x + 16)
sage: r, s = D4.gens()
sage: h.image(r)
(1,6,4,7)(2,5,3,8)
```
kernel()

Returns the kernel of this homomorphism as a permutation group.

EXAMPLES:

```sage: G = CyclicPermutationGroup(4)
sage: H = DihedralGroup(4)
sage: g = G([(1,2,3,4)])
sage: phi = PermutationGroupMorphism_im_gens(G, H, )
sage: phi.kernel()
Subgroup generated by [(1,2,3,4)] of (Cyclic group of order 4 as a permutation group)
```
```sage: G = PSL(2,7)
sage: D = G.direct_product(G)
sage: H = D
sage: pr1 = D
sage: G.is_isomorphic(pr1.kernel())
True
```
class sage.groups.perm_gps.permgroup_morphism.PermutationGroupMorphism_from_gap(G, H, gap_hom)

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: 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
```
class sage.groups.perm_gps.permgroup_morphism.PermutationGroupMorphism_id
class sage.groups.perm_gps.permgroup_morphism.PermutationGroupMorphism_im_gens(G, H, gens=None)

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

EXAMPLES:

```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)
```

AUTHORS:

• David Joyner (2006-02)

sage.groups.perm_gps.permgroup_morphism.is_PermutationGroupMorphism(f)

Returns True if the argument `f` is a PermutationGroupMorphism.

EXAMPLES:

```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)
True
```