Construction of Subgroups and Quotient Groups

The operations in this section apply to both, free abelian groups and arbitrary abelian groups.

Subsections

• Construction of Subgroups
• Construction of Quotient Groups

Construction of Subgroups

sub<A | L> : GrpAb, List -> GrpAb, Map

Construct the subgroup B of the abelian group A generated by the elements specified by the terms of the generator list L. A term L[i] of the generator list may consist of any of the following objects:

(a)

An element liftable to A;

(b)

A sequence of integers representing an element of A;

(c)

A subgroup of A;

(d)

A set or sequence of type (a), (b), or (c).

The collection of words and groups specified by the list must all belong to the group A and the group B will be constructed as a subgroup of A.

Construction of Quotient Groups

quo<F | R> : GrpAb, List -> GrpAb, Hom(GrpAb)

Given an abelian group F, and a set of relations R in the generators of F, construct the quotient A of F by the subgroup of F defined by R. The presentation defining the group A consists of the relations for F (if any), together with the additional relations defined by the list R.

The expression defining F may be either simply the name of a previously constructed group, or an expression defining an abelian group. The possibilities for the relation list R are the same as for the AbelianGroup construction.

The function returns:

(a)

The quotient group A;

(b)

The natural homomorphism phi : F -> A.

A / B : GrpAb, GrpAb -> GrpAb

Given a subgroup B of the abelian group A, construct the quotient of A by B.