Presentations

The functions described in this section apply only to finite groups for which a base and strong generating set may be constructed.

Subsections

• Generators and Relations
• Matrices as Words

Generators and Relations

FPGroup(G) : GrpMat :-> GrpFP, Hom(Grp)

Construct a presentation for the matrix group G on the set of defining generators and return the presentation in the form of a finitely presented group F that is isomorphic to G. The presentation is obtained by first computing the regular representation of G and then using the Todd-Coxeter Schreier algorithm to construct a presentation on the strong generators. In this situation the strong generators are identical to the defining generators.

A group homomorphism phi: F -> G, defining G as a matrix representation of F, is also returned.

FPGroupStrong(G) : GrpMat :-> GrpFP, Hom(Grp)

Construct a presentation for the matrix group G on a set of strong generators and return the presentation in the form of a finitely presented group F that is isomorphic to G. In Magma, the Todd-Coxeter Schreier algorithm is used to construct the presentation. If strong generators are not already known for G, they will be constructed. In the case in which strong generators are already known for G, the presentation will be on these strong generators. The group homomorphism f: F -> G, defining G as a matrix representation of F, is also returned.

Matrices as Words

Consider a matrix group G defined on d generators. The word group of G is a free group W of rank d. Then we regard G as a homomorphic image of F with associated homomorphism phi: W -> G. All operations involving words in the generators of G will be performed in W.

WordGroup(G) : GrpMat -> GrpSLP, Map

Given a matrix group G defined on d generators, return (a) a free group W on d generators as an SLP-group, and (b) the homomorphism phi from W to G such that W.i -> G.i, for i = 1, ..., d. The group W associated with G by this function will be referred to as the word group for G.

InverseWordMap(G) : GrpMat -> Map

Given a matrix group G and its associated word group W with canonical homomorphism phi:W -> G, construct the inverse mapping rho. Thus, given a matrix g of G, g@rho returns an element in the preimage of g under phi. If the word group W does not already exist, it will be created.