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 intrinsics considered here are designed for computations in the area that is sometimes referred to as combinatorial group theory.

Contents

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 simplification of presentations using Tietze transformations

The Todd-Coxeter procedure for constructing the coset table of a subgroup. This is the basis for many fp-group algorithms

A collection of tools for computing with subgroups having (modest) 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 Reidemeister-Schreier algorithm for constructing presentations of subgroups

The determination of various properties of fp=groups such as perfect, finite/infinite, small cancellation, and large

A search for homomorphisms of an fp-group onto a permutation group or a pc-group

A test for the isomorphism of two fp-groups

The construction of particular types of quotient groups: abelian quotient, p-quotient, nilpotent quotient, soluble quotient, simple group quotient and L2/L3/U3 quotients

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

This chapter contains detailed information about the intrinsics provided for general fp-groups. The much shorter chapter INTRODUCTION TO FP-GROUPS, "Introduction of Finitely Presented Groups", gives a basic description of the more important intrinsics for fp-groups and should serve to get users started. That chapter typically does not list the parameters of an intrinsic and omits the more specialised intrinsics which are mainly useful when generic intrinsics fail.

Free groups are treated in Chapter FREE GROUPS and it should be read in conjunction with this chapter. Automatic groups and hyperbolic groups are discussed in Chapter AUTOMATIC AND HYPERBOLIC GROUPS.

For a description of the fundamental algorithms for finitely presented groups, the reader should consult Sims [Sim94] or Holt [HEO05].

V2.28, 13 July 2023