Submodules and Quotient Modules#

Construction#

sub<M | L>: ModRng, List -> ModRng#

Given an \(A\)-module \(M\), construct the submodule \(N\) generated by the elements of \(M\) specified by the list \(L\). Each term \(L_i\) 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 \(L_i\) together with the stored generators for submodules specified by terms of \(L_i\). 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 \(f : N \rightarrow M\).

ImageWithBasis(X, M): ModMatRngElt, ModRng -> ModRng#

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 \(X\).

Morphism(M, N): ModRng, ModRng -> ModMatRngElt#: If the \(A\)-module \(M\) was created as a submodule of the module \(N\), return the inclusion homomorphism \(\phi : M \rightarrow N\) as an element of \({\operatorname{Hom}}_A(M, N)\). Thus, \(\phi\) gives the correspondence between elements of \(M\) (represented with respect to the standard basis of \(M\)) and elements for \(N\).

Example: Submodule (ex-260f18)#

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]

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Note that as a \({\bf F}_{2}\)-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)
]

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Membership and Equality#

The operators described below refer to the underlying vector space.

u in M: ModRngElt, ModRng -> BoolElt#: Returns true if the element \(u\) lies in the \(A\)-module \(M\).

N subset M: ModRng, ModRng -> BoolElt#: Returns true if the \(A\)-module \(N\) is contained in the \(A\)-module \(M\).

N eq M: ModRng, ModRng -> BoolElt#: Returns true if the \(A\)-modules \(N\) and \(M\) are equal, where \(N\) and \(M\) are contained in a common \(A\)-module.

Operations on Submodules#

M + N: ModRng, ModRng -> ModRng#: Sum of the submodules \(M\) and \(N\), where \(M\) and \(N\) belong to a a common \(A\)-module.

M meet N: ModRng, ModRng -> ModRng#: Intersection of the submodules \(M\) and \(N\), where \(M\) and \(N\) belong to a common \(A\)-module.

Quotient Modules#

quo<M | L>: ModRng, List -> ModRng#

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 \(L_i\) 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 \(L_i\) together with the stored generators for submodules specified by terms of \(L_i\). The constructor returns the quotient module \(P\) as an \(A\)-module together with the natural homomorphism \(f : M \rightarrow P\).

Morphism(M, N): ModRng, ModRng -> ModMatRngElt#: If the \(A\)-module \(N\) was created as a quotient module of the module \(M\), return the natural homomorphism \(\phi : M \rightarrow N\) as an element of \({\operatorname{Hom}}_R(M, N)\). Thus \(\phi\) gives the correspondence between elements of \(M\) and elements of \(N\) (represented with respect to the standard basis for \(N\)).

Example: Quotient Module (ex-26899c)#

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)

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We locate the kernel of the epimorphism \(\phi\) and check that it is the same as \(N\).

> K :=  Kernel(phi);
GModule Ker of dimension 9 over GF(2)
> K eq N;
true

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Given an element \(x\) in the codomain \(Q\) of the epimorphism \(\phi\), 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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