Given an A-module M, construct the submodule N generated by the
elements of M specified by the list L. Each term Li of the list
L must be an expression defining an object of one of the following
types:
- (a)
- A sequence of n elements of R defining an element of M;
- (b)
- A set or sequence whose terms are elements of M;
- (c)
- A submodule of M;
- (d)
- A set or sequence whose terms are submodules of M.
The generators stored for N consist of the elements specified by
terms Li together with the stored generators for submodules specified
by terms of Li. Repetitions of an element and occurrences of the
zero element are removed (unless N is trivial).
The constructor returns the submodule N as an A-module
together with the inclusion homomorphism f : N -> M.
Check: BoolElt Default: true
Given a basis matrix X for a A-submodule of the A-module
M, return the submodule N of M such that the morphism of
N into M is X.
If the A-module M was created as a submodule of the module N,
return the inclusion homomorphism φ : M -> N as an element of
HomA(M, N). Thus, φ gives the correspondence between elements of
M (represented with respect to the standard basis of M) and elements
for N.
We construct a submodule of the permutation module for L(3, 4)
in its representation of degree 21.
> G := PSL(3, 4);
> M := PermutationModule(G, GF(2));
> M;
GModule M of dimension 21 over GF(2)
> x := M![0,0,0,1,0,1,0,0,0,1,1,0,0,0,1,0,1,1,0,0,1];
> N := sub< M | x >;
> N:Maximal;
GModule N of dimension 9 over GF(2)
Generators of acting algebra:
[1 0 0 0 1 0 1 0 1]
[0 1 0 1 1 1 0 0 0]
[0 0 1 1 1 1 1 0 1]
[0 0 0 0 0 1 1 0 0]
[0 0 0 1 0 0 1 0 0]
[0 0 0 0 1 0 1 0 0]
[0 0 0 1 1 1 0 0 0]
[0 0 0 0 1 1 0 0 1]
[0 0 0 1 0 1 0 1 1]
[0 0 0 0 0 1 0 1 1]
[1 0 0 0 0 0 0 0 1]
[0 1 1 0 0 1 0 0 1]
[0 0 0 0 0 1 0 0 0]
[0 0 1 0 0 1 0 0 0]
[0 0 1 0 1 1 0 0 1]
[0 0 1 1 0 0 0 0 1]
[0 0 1 0 0 0 0 0 1]
[0 0 0 0 0 0 1 0 0]
Note that as a GF(2)-module V has dimension 1, while
as a K[G]-module it has dimension 9. The submodule N is defined
on a reduced basis so we use Morphism to see N embedded
in M.
> phi := Morphism(N, M);
> [ phi(x) : x in Basis(N) ];
[
M: (1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 0 1),
M: (0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 1 1 1),
M: (0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 0 1 1 0 1 0),
M: (0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 0 1 1 0 1),
M: (0 0 0 0 1 0 0 0 0 1 1 0 1 1 1 0 0 0 0 1 1),
M: (0 0 0 0 0 1 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0),
M: (0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 1 1 1 0 1 0),
M: (0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 1 1 1 0 1),
M: (0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 0 1 1)
]
The operators described below refer to the underlying vector space.
Returns true if the element u lies in the A-module M.
Returns true if the A-module N is contained in the A-module M.
Returns true if the A-modules N and M are equal, where N and M
are contained in a common A-module.
Sum of the submodules M and N, where M and N belong to a
a common A-module.
Intersection of the submodules M and N, where M and N belong
to a common A-module.
Given an A-module M, construct the quotient module P = M/N as
an A-module, where
N is the submodule generated by the elements of M specified by the
list L. Each term Li of the list L must be an expression defining
an object of one of the following types:
- (a)
- A sequence of n elements of R defining an element of M;
- (b)
- A set or sequence whose terms are elements of M;
- (c)
- A submodule of M;
- (d)
- A set or sequence whose terms are submodules of M.
The generators constructed for N consist of the elements specified by
terms Li together with the stored generators for submodules specified
by terms of Li.
The constructor returns the quotient module P as an A-module
together with the natural homomorphism f : M -> P.
If the A-module N was created as a quotient module of the module M,
return the natural homomorphism φ : M -> N as an element of
HomR(M, N). Thus φ gives the correspondence between elements of M
and elements of N (represented with respect to the standard basis for N).
We construct a quotient module of the permutation module for L(3, 4)
considered above.
> G := PSL(3, 4);
> M := PermutationModule(G, GF(2));
> x := M![0,0,0,1,0,1,0,0,0,1,1,0,0,0,1,0,1,1,0,0,1];
> N := sub< M | x >;
> N;
GModule N of dimension 9 over GF(2)
> Q, phi := quo< M | x >;
> Q;
GModule Q of dimension 12 over GF(2)
We locate the kernel of the epimorphism φ and check that it is
the same as N.
> K := Kernel(phi);
GModule Ker of dimension 9 over GF(2)
> K eq N;
true
Given an element x in the codomain Q of the epimorphism φ,
the value returned as the preimage of x is a representative element
of the coset of the kernel that is the actual preimage of x. Since
we are working in a module over a finite field, we can explicitly
construct the full preimage PreIm of x.
> x := Q![0,0,0,1,1,0,0,0,0,1,0,0];
> PreIm := { x@@phi + k : k in K };
> #PreIm;
512
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