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
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class
sage.groups.abelian_gps.abelian_group_morphism.
AbelianGroupMap
(parent)¶ Bases:
sage.categories.morphism.Morphism
A set-theoretic map between AbelianGroups.
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class
sage.groups.abelian_gps.abelian_group_morphism.
AbelianGroupMorphism
(G, H, genss, imgss)¶ Bases:
sage.categories.morphism.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])
AUTHORS:
- David Joyner (2006-02)
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image
(J)¶ Only works for finite groups.
J must be a subgroup of G. Computes the subgroup of H which is the image of J.
EXAMPLES:
sage: G = AbelianGroup(2,[2,3],names="xy") sage: x,y = G.gens() sage: H = AbelianGroup(3,[2,3,4],names="abc") sage: a,b,c = H.gens() sage: phi = AbelianGroupMorphism(G,H,[x,y],[a,b])
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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([ ])'
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class
sage.groups.abelian_gps.abelian_group_morphism.
AbelianGroupMorphism_id
(X)¶ Bases:
sage.groups.abelian_gps.abelian_group_morphism.AbelianGroupMap
Return the identity homomorphism from X to itself.
EXAMPLES:
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sage.groups.abelian_gps.abelian_group_morphism.
is_AbelianGroupMorphism
(f)¶