# 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)

A set-theoretic map between AbelianGroups.

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

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])


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])
sage: H = AbelianGroup(3,[2,3,4],names="abc")
sage: a,b,c = H.gens()
sage: phi = AbelianGroupMorphism(G,H,[x,y],[a,b])
sage: phi.image(subG)
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])
sage: phi.kernel()
'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])
sage: phi.kernel()
'Group([ f1*f2 ])'

sage.groups.abelian_gps.abelian_group_morphism.is_AbelianGroupMorphism(f)