Module: sage.groups.abelian_gps.abelian_group_morphism
TODO: * there must be a homspace first * there should be hom and Hom methods in abelian group
Author: David Joyner (2006-03-03): initial version
Module-level Functions
f) |
Class: AbelianGroupMap
self, parent) |
Special Functions: _repr_type
Class: AbelianGroupMorphism
sage: from abelian_group import AbelianGroup 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 abelian_group_morphism import AbelianGroupMorphism sage: phi = AbelianGroupMorphism(H,G,[x,y],[a,b])
AUTHOR: David Joyner (2-2006)
self, G, H, genss, imgss) |
Functions: codomain,
domain,
image,
kernel,
range
self, J) |
Only works for finite groups.
J must be a subgroup of G. Computes the subgroup of H which is the image of J.
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])
self) |
Only works for finite groups.
TODO: not done yet; returns a gap object but should return a SAGE group.
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([ ])'
Special Functions: __call__,
_gap_init_,
_repr_type
self, g) |
Some python code for wrapping GAP's Images function but only for permutation groups. Returns an error if g is not in G.
sage: H = AbelianGroup(3, [2,3,4], names="abc") sage: a,b,c = H.gens() sage: G = AbelianGroup(2, [2,3], names="xy") sage: x,y = G.gens() sage: phi = AbelianGroupMorphism(G,H,[x,y],[a,b]) sage: phi(y*x) a*b sage: phi(y^2) b^2
self) |
Only works for finite groups.
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: phi = AbelianGroupMorphism(H,G,[x,y],[a,b]) sage: phi._gap_init_() 'phi := GroupHomomorphismByImages(G,H,[x, y],[a, b])'
Class: AbelianGroupMorphism_id
self, X) |
Special Functions: _repr_defn
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