An abelian group with non-trivial relations is constructed as a quotient of an existing abelian group, possibly a free abelian group.

Subsections

• Accessing the Defining Generators and Relations

Given a list X of variables x_1, ..., x_r, and a list of relations R over these generators, first construct the free abelian group F on the generators x_1, ..., x_r and then construct the quotient of F corresponding to the subgroup of F defined by the relations R.

Each term of the list R is either a word, a relation, a relation list or a subgroup of F.

• A relation consists of a pair of words, separated by `='.
• A word w is interpreted as a relator, that is, it is equivalent to the relation w = 0. (See above).
• A relation list consists of a list of words, where each pair of adjacent words is separated by `=': w_1 = w_2 = ... = w_r. This is interpreted as the set of relations w_1 = w_r, ..., w_(r - 1) = w_r. Note that the relation list construct is only meaningful in the context of the quo-constructor.

A subgroup H appearing in the list R contributes its generators to the relation set for A, i.e., each generator of H is interpreted as a relator for A.

The group F may be referred to by the special symbol $ in any word appearing to the right of the `|' symbol in the quo-constructor. Also, in the context of the quo-constructor, the identity element (empty word) may be represented by the digit 0.

The function returns:

• The quotient group A;
• The natural homomorphism phi : F -> A.

> F<a, b, c> := FreeAbelianGroup(3);
> A := quo< F | 7*a + 4*b + c, 8*a + 5*b + 2*c, 9*a + 6*b + 3*c >;
> A;
AbelianGroup isomorphic to Z_3 + Z
Defined on 3 generators
Relations:
    A.1 + A.2 + A.3 = 0
    3 * A.2 + 6 * A.3 = 0

The functions described here provide access to basic information stored for a abelian group A.

A . i : GrpAb, RngIntElt -> GrpAbElt

The i-th defining generator for A.

A set containing the generators for A.

Ngens(A) : GrpAb -> RngIntElt

The number of generators for A.

The parent group A of the word u.

A sequence containing the defining relations for A.