MAGMA Computational Algebra System

Homomorphisms

For a general description of homomorphisms, we refer to chapter MAPPINGS. This section describes some special aspects of homomorphisms whose domain or codomain is an automatic group.

Subsections

• General remarks
• Construction of Homomorphisms

General remarks

Groups in the category GrpAtc currently are accepted as codomains only in some special situations. The most important cases in which an automatic group can be used as a codomain are group homomorphisms whose domain is in one of the categories GrpFP, GrpGPC, GrpRWS or GrpAtc.

Construction of Homomorphisms

hom< A -> G | S > : Struct , Struct -> Map

Returns the homomorphism from the automatic group A to the group G defined by the expression S which can be the one of the following:

(i)

A list, sequence or indexed set containing the images of the n generators A.1, ..., A.n of A. Here, the i-th element of S is interpreted as the image of A.i, i.e. the order of the elements in S is important.

(ii)

A list, sequence, enumerated set or indexed set, containing n tuples <x_i, y_i> or arrow pairs x_i - > y_i, where x_i is a generator of A and y_i∈G (i=1, ..., n) and the set {x₁, ..., x_n} is the full set of generators of A. In this case, y_i is assigned as the image of x_i, hence the order of the elements in S is not important.

It is the user's responsibility to ensure that the provided generator images actually give rise to a well-defined homomorphism. No checking is performed by the constructor.

Note that it is currently not possible to define a homomorphism by assigning images to the elements of an arbitrary generating set of A.