MAGMA Computational Algebra System

Introduction

This online help node and the nodes below it present the functions designed for computing with finitely-presented groups (fp-groups for short). The name of the corresponding Magma category is GrpFP. The functions considered here are designed for doing what is sometimes referred to as combinatorial group theory.

Subsections

• Overview of Facilities
• The Construction of Finitely Presented Groups

Overview of Facilities

The facilities provided for fp-groups fall into a number of natural groupings:

The construction of fp-groups in terms of generators and relations;

The construction of particular types of quotient groups: abelian quotient, p-quotient, nilpotent quotient and soluble quotient;

Index determination and subgroup building based on the Todd-Coxeter procedure;

Calculations with subgroups having finite index in a group, where the subgroups are represented by coset tables;

The construction of all subgroups having index less than some (small) specified bound;

The construction of representations of an fp-group corresponding to actions on coset spaces and elementary abelian sections;

The use of a rewriting process for constructing presentations of subgroups;

The simplification of words with respect to a given set of relations.

The Construction of Finitely Presented Groups

The construction of fp-groups utilises the fact that every group is a quotient of some free group. Thus, two general fp-group constructors are provided: FreeGroup(n) which constructs a free group of rank n, and quo< F | R > which constructs the quotient of group F by the normal subgroup defined by the relations R.

The naming of generators presents special difficulties since they are not always used in a consistent manner in the mathematical literature. A generator name is used in two distinct ways. Firstly, it plays the role of a variable having as its value a designated generator of G. Secondly, it appears as the symbol designating the specified generator whenever elements of the group are output. These two uses are separated in the Magma semantics.

In Magma, a standard indexing notation is provided for referencing the generators of any fp-group G. Thus, G.i denotes the i-th generator of G. However, users may give individual names to the generators by means of the generator-assignment. Suppose that the group G is defined on r generators. Then if the right hand side of the following statement creates a group, the special assignment

> G< v_1, ..., v_r> := construction;

> G := construction;
>    v_1 := G.1;
>    ...
>    v_r := G.r;