Module: sage.groups.matrix_gps.matrix_group_morphism
Homomorphisms Between Matrix Groups
Author Log:
Class: MatrixGroupMap
self, parent) |
Special Functions: __init__,
_repr_type
Class: MatrixGroupMorphism
Class: MatrixGroupMorphism_im_gens
TODO: what does it mean to return fail? It's a constructor for a class.
sage: F = GF(5); MS = MatrixSpace(F,2,2) sage: G = MatrixGroup([MS([1,1,0,1])]) sage: H = MatrixGroup([MS([1,0,1,1])]) sage: phi = G.hom(H.gens()) sage: phi Homomorphism : Matrix group over Finite Field of size 5 with 1 generators: [[[1, 1], [0, 1]]] --> Matrix group over Finite Field of size 5 with 1 generators: [[[1, 0], [1, 1]]] sage: phi(MS([1,1,0,1])) [1 0] [1 1] sage: F = GF(7); MS = MatrixSpace(F,2,2) sage: F.multiplicative_generator() 3 sage: G = MatrixGroup([MS([3,0,0,1])]) sage: a = G.gens()[0]^2 sage: phi = G.hom([a])
self, homset, imgsH, [check=True]) |
Functions: image,
kernel
self, J) |
J must be a subgroup of G. Computes the subgroup of H which is the image of J.
sage: F = GF(7); MS = MatrixSpace(F,2,2) sage: F.multiplicative_generator() 3 sage: G = MatrixGroup([MS([3,0,0,1])]) sage: a = G.gens()[0]^2 sage: phi = G.hom([a]) sage: phi.image(G.gens()[0]) '[ [ Z(7)^2, 0*Z(7) ], [ 0*Z(7), Z(7)^0 ] ]' sage: H = MatrixGroup([MS(a.list())]) sage: H Matrix group over Finite Field of size 7 with 1 generators: [[[2, 0], [0, 1]]] sage: phi.image(H) 'Group([ [ [ Z(7)^4, 0*Z(7) ], [ 0*Z(7), Z(7)^0 ] ] ])'
self) |
sage: F = GF(7); MS = MatrixSpace(F,2,2) sage: F.multiplicative_generator() 3 sage: G = MatrixGroup([MS([3,0,0,1])]) sage: a = G.gens()[0]^2 sage: phi = G.hom([a]) sage: phi.kernel() 'Group([ [ [ Z(7)^3, 0*Z(7) ], [ 0*Z(7), Z(7)^0 ] ] ])' sage: phi.image(G.gens()[0]) '[ [ Z(7)^2, 0*Z(7) ], [ 0*Z(7), Z(7)^0 ] ]'
Special Functions: __call__,
__init__,
_gap_init_,
_latex_,
_repr_
self, g) |
Some python code for wrapping GAP's Images function for a matrix group G. Returns an error if g is not in G.
sage: F = GF(5); MS = MatrixSpace(F,2,2) sage: g = MS([1,1,0,1]) sage: G = MatrixGroup([g]) sage: phi = G.hom(G.gens()) sage: phi(G.0) [1 1] [0 1] sage: phi(G(g^2)) [1 2] [0 1]
sage: F = GF(5); MS = MatrixSpace(F,2,2) sage: gens = [MS([1,2, -1,1]),MS([1,1, 0,1])] sage: G = MatrixGroup(gens) sage: phi = G.hom(G.gens()) sage: phi(G.0) [1 2] [4 1] sage: phi(G.1) [1 1] [0 1]
self) |
sage: F = GF(5); MS = MatrixSpace(F,2,2) sage: G = MatrixGroup([MS([1,1,0,1])]) sage: phi = G.hom(G.gens()) sage: print latex(phi) \left\langle \left(\begin{array}{rr} 1 \& 1 \\ 0 \& 1 \end{array}\right) \right\rangle \rightarrow{} \left\langle \left(\begin{array}{rr} 1 \& 1 \\ 0 \& 1 \end{array}\right) \right\rangle
self) |
sage: F = GF(5); MS = MatrixSpace(F,2,2) sage: G = MatrixGroup([MS([1,1,0,1])]) sage: H = MatrixGroup([MS([1,0,1,1])]) sage: phi = G.hom(H.gens()) sage: phi Homomorphism : Matrix group over Finite Field of size 5 with 1 generators: [[[1, 1], [0, 1]]] --> Matrix group over Finite Field of size 5 with 1 generators: [[[1, 0], [1, 1]]] sage: phi(MS([1,1,0,1])) [1 0] [1 1]
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