Book contents
- Frontmatter
- Contents
- Preface
- Chapter I Free groups and free presentations
- Chapter II Examples of presentations
- Chapter III Groups with few relations
- Chapter IV Presentations of subgroups
- Chapter V The triangle groups
- Chapter VI Extensions of groups
- Chapter VII Small cancellation groups
- Chapter VIII Groups from topology
- Guide to the literature and references
- Index of notation
- Index
Chapter IV - Presentations of subgroups
Published online by Cambridge University Press: 28 January 2010
- Frontmatter
- Contents
- Preface
- Chapter I Free groups and free presentations
- Chapter II Examples of presentations
- Chapter III Groups with few relations
- Chapter IV Presentations of subgroups
- Chapter V The triangle groups
- Chapter VI Extensions of groups
- Chapter VII Small cancellation groups
- Chapter VIII Groups from topology
- Guide to the literature and references
- Index of notation
- Index
Summary
Write the vision, and make it plain upon tables, that he may run that readeth it.
(Habakkuk)Suppose we are given a presentation <X|R> for a group G. As might be expected, the derivation of a presentation for a specified factor group of G is easy enough (Theorem 3.3). By contrast, the corresponding problem for a subgroup H of G is no simple matter, and is in general very undecidable. As usual we dodge the pathology by confining ourselves to propitious cases, and describe the general method in §12. The whole thing hinges on the derivation of a certain Schreier transversal U, in terms of which we obtain free generators for H (as in Lemma 2.3). A simple trick gives relators for H in terms of the generators X of G and these must be rewritten as words in the free generators of H. If necessary, we can then perform Tietze transformations (Theorem 4.3) on the resulting presentation to reduce it to a more suitable form.
The method of calculating U varies according to what H is and how it is specified. For example, the method of §6 (see Example 6.2) contains an algorithm for computing U in the case where G is finitely generated and H ⊇ G'. The best general method, however, is that invented by J.A. Todd and H.S.M. Coxeter in 1936 and known as coset enumeration. It works in any specific situation when H is the subgroup generated by a set Y of words in X, provided only that |X|, |R|, |Y| and |G:H| are all finite. We describe it in §§10, 11.
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- Topics in the Theory of Group Presentations , pp. 85 - 129Publisher: Cambridge University PressPrint publication year: 1980