# Homomorphisms of abelian groups#

Todo

• there must be a homspace first

• there should be hom and Hom methods in abelian group

AUTHORS:

• David Joyner (2006-03-03): initial version

class sage.groups.abelian_gps.abelian_group_morphism.AbelianGroupMap(parent)#

Bases: `Morphism`

A set-theoretic map between AbelianGroups.

class sage.groups.abelian_gps.abelian_group_morphism.AbelianGroupMorphism(G, H, genss, imgss)#

Bases: `Morphism`

Some python code for wrapping GAP’s GroupHomomorphismByImages function for abelian groups. Returns “fail” if gens does not generate self or if the map does not extend to a group homomorphism, self - other.

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: H = AbelianGroup(2,[2,3],names="xy"); H
Multiplicative Abelian group isomorphic to C2 x C3
sage: x,y = H.gens()

sage: from sage.groups.abelian_gps.abelian_group_morphism import AbelianGroupMorphism
sage: phi = AbelianGroupMorphism(H,G,[x,y],[a,b])  # optional - gap_package_polycyclic
```

AUTHORS:

• David Joyner (2006-02)

image(S)#

Return the image of the subgroup `S` by the morphism.

This only works for finite groups.

INPUT:

• `S` – a subgroup of the domain group `G`

EXAMPLES:

```sage: G = AbelianGroup(2,[2,3],names="xy")
sage: x,y = G.gens()
sage: subG = G.subgroup([x])                       # optional - gap_package_polycyclic
sage: H = AbelianGroup(3,[2,3,4],names="abc")
sage: a,b,c = H.gens()
sage: phi = AbelianGroupMorphism(G,H,[x,y],[a,b])  # optional - gap_package_polycyclic
sage: phi.image(subG)                              # optional - gap_package_polycyclic
Multiplicative Abelian subgroup isomorphic to C2 generated by {a}
```
kernel()#

Only works for finite groups.

Todo

not done yet; returns a gap object but should return a Sage group.

EXAMPLES:

```sage: H = AbelianGroup(3,[2,3,4],names="abc"); H
Multiplicative Abelian group isomorphic to C2 x C3 x C4
sage: a,b,c = H.gens()
sage: G = AbelianGroup(2,[2,3],names="xy"); G
Multiplicative Abelian group isomorphic to C2 x C3
sage: x,y = G.gens()
sage: phi = AbelianGroupMorphism(G,H,[x,y],[a,b])  # optional - gap_package_polycyclic
sage: phi.kernel()                                 # optional - gap_package_polycyclic
Group([  ])

sage: H = AbelianGroup(3,[2,2,2],names="abc")
sage: a,b,c = H.gens()
sage: G = AbelianGroup(2,[2,2],names="x")
sage: x,y = G.gens()
sage: phi = AbelianGroupMorphism(G,H,[x,y],[a,a])  # optional - gap_package_polycyclic
sage: phi.kernel()                                 # optional - gap_package_polycyclic
Group([ f1*f2 ])
```
sage.groups.abelian_gps.abelian_group_morphism.is_AbelianGroupMorphism(f)#