6 - The Reidemeister-Schreier procedure

Summary

In Proposition 3.5 of Chapter 4, we proved that subgroups of finite index in finitely generated groups are finitely generated. In this chapter we shall give a constructive proof that subgroups of finite index in finitely presented groups are finitely presented. The algorithm developed as part of the proof is the last of the major tools available for studying subgroups of finitely presented groups. This algorithm is usually referred to as the Reidemeister-Schreier procedure.

Let G be a group given by a presentation and let H be a subgroup of G. In order to obtain a presentation for H, we need to determine some additional information about H and the way H is embedded in G. Section 6.1 describes the data needed, demonstrates how to get the presentation from that data, and shows how to derive the data in a special case. Section 6.2 gives some examples showing how to derive the data in the general case using coset enumeration. Section 6.3 formalizes the procedure. The initial presentations of subgroups are often unpleasant. Section 6.4 discusses ways of simplifying these presentations.

Presentations of subgroups

For the time being we shall work with monoid presentations. Let G = Mon 〈X | S〉 and assume that G is a group. The image in G of a word U in X* will be denoted [U], and ≡ will be the congruence on X* generated by S. Let H be a subgroup of G. Our goal is to find a set Y and a subset of Y* × Y* such that H is isomorphic to Mon〈Y |〉.