Elements

Contents

Construction of Elements

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

A ! [a1, ... ,an] : GrpAb, [RngIntElt] -> GrpAbElt
A ! [a1, ... ,an] : GrpAbGen, [RngIntElt] -> GrpAbGenElt
Given an abelian group A with generators e1, ..., er and a sequence Q = [a1, ..., ar] of integers, construct the element a1 e1 + ... + ar er 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, ..., en and suppose g is an element of A, where g = a1 e1 + ... + an en. These functions return the sequence Q of n integers defined by Q[i] = (ai), for i = 1, ..., n. Moreover, each ai, 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, ..., un and suppose g is an element of A, where g = a1 u1 + ... + an un. This function returns the sequence Q of n integers defined by Q[i] = (ai), for i = 1, ..., n. Moreover, each ai, 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, ..., sm] be any sequence of elements of A. Assume g is an element of A such that b g = a1 s1 + ... + am sm. This function returns as its first value the sequence Q of m integers defined by Q[i] = (ai), for i = 1, ..., m. The second value returned is the coefficient b of g. Note that b might not be 1.

Example GrpAb_ElementCreationAndRep (H75E7)

We use the quadratic forms example considered above to illustrate these functions. 
> 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

Arithmetic with Elements

If the generic abelian group A has been constructed with the flag UseRepresentation set true, then arithmetic with elements of A is trivial.

u + v : GrpAbElt, GrpAbElt -> GrpAbElt
Given elements u and v belonging to the same abelian group A, return the sum of u and v.
- u : GrpAbElt -> GrpAbElt
The inverse of element u.
u - v : GrpAbElt, GrpAbElt -> GrpAbElt
Given elements u and v belonging to the same abelian group A, return the sum of u and the inverse of v.
m * u : RngIntElt, GrpAbElt-> GrpAbElt
u * m : GrpAb, RngIntElt -> GrpAbElt
Given an integer m, return the element w + w + ... w (|m| summands), where w = u, if m is positive and w = - u if m is negative.
V2.28, 13 July 2023