Elements

Construction of Elements

Unless otherwise stated, the operations in this section apply to fp-abelian groups and generic abelian groups.

A ! [a₁, ... ,a_n] : GrpAb, [RngIntElt] -> GrpAbElt

A ! [a₁, ... ,a_n] : GrpAbGen, [RngIntElt] -> GrpAbGenElt

Given an abelian group A with generators e₁, ..., e_r and a sequence Q = [a₁, ..., a_r] of integers, construct the element a₁ e₁ + ... + a_r e_r of A.

A ! e : GrpAbGen, Elt -> GrpAbGenElt

Given a generic abelian group A and an element e of the domain over which it is defined, return e as an element of A. If A is a proper subset of its underlying domain, then e must be a linear combination of the generators (which may be user-supplied) of A.

A ! g : GrpAbGen, GrpAbGenElt -> GrpAbGenElt

Given a generic abelian group A and an element g of the underlying set X of A, return g as an element of A.

A ! n : GrpAb, RngIntElt -> GrpAbElt

Given an abelian group A with exactly one generator x, construct the element n x.

Random(A) : GrpAbGen -> GrpAbGenElt

Given either a finite fp-abelian group or a generic abelian group A, return a random element of A.

Identity(A) : GrpAb -> GrpAbElt

Id(A) : GrpAb -> GrpAbElt

A ! 0 : GrpAb, RngIntElt -> GrpAbElt

Construct the identity element (empty word) for the abelian group A.

Let A be a generic abelian group defined in the universe U of A. If g is an element of A, then U!g is an element of U.

Representation of an Element

An element g of an abelian group A can be represented as a linear combination with respect to a given generating sequence. The coefficients appearing in this linear combination provide an alternative representation for g. If A is a fp-group, the generating set will be the one on which the group was defined. In the case of a generic group, the generating set can either be that obtained when constructing a presentation for A or a user-supplied generating set.

Representation(g) : GrpAbGenElt -> [RngIntElt]

ElementToSequence(g) : GrpAbGenElt -> [RngIntElt]

Eltseq(g) : GrpAbGenElt -> [RngIntElt]

Let A be an abelian group with generating set e₁, ..., e_n and suppose g is an element of A, where g = a₁ e₁ + ... + a_n e_n. These functions return the sequence Q of n integers defined by Q[i] = (a_i), for i = 1, ..., n. Moreover, each a_i, i = 1, ..., n, is the integer residue modulus the order of the ith generator.

UserRepresentation(g) : GrpAbGenElt -> [RngIntElt]

Let A be a generic abelian group with a user-supplied set of generators u₁, ..., u_n and suppose g is an element of A, where g = a₁ u₁ + ... + a_n u_n. This function returns the sequence Q of n integers defined by Q[i] = (a_i), for i = 1, ..., n. Moreover, each a_i, i = 1, ..., n, is the integer residue modulus the order of the ith generator.

Representation(S, g) : SeqEnum, GrpAbGenElt -> [RngIntElt], RngIntElt

Let A be a generic abelian group and let S = [s₁, ..., s_m] be any sequence of elements of A. Assume g is an element of A such that b g = a₁ s₁ + ... + a_m s_m. This function returns as its first value the sequence Q of m integers defined by Q[i] = (a_i), for i = 1, ..., m. The second value returned is the coefficient b of g. Note that b might not be 1.

> Generators(QF);
[ <2,2,500001>, <206,-102,4867> ]
> g := QF ! [5, 6];
> g;
<837,-766,1370>
> Representation(g);
[ 1, 6 ]
> 
> g := Random(QF);
> Representation(g);
[ 1, 270 ]
> 
> UserRepresentation(g);
[ 377, 0, 515, 0, 0, 0, 0, 0, 0, 0 ]
>
> S := [];
> for i in [1..3] do
>     d := Random(QF);
>     Include(~S, d);
> end for;
> seq, coeff := Representation(S, g);
> seq; coeff;
[ -170, -3, 0 ]
1