Book contents
- Frontmatter
- Contents
- Preface
- 1 Free Groups
- 2 Schreier's Method
- 3 Nielsen's Method
- 4 Free Presentations of Groups
- 5 Some Popular Groups
- 6 Finitely-generated Abelian Groups
- 7 Finite Groups with few Relations
- 8 Coset Enumeration
- 9 Presentations of Subgroups
- 10 Presentations of Group Extensions
- 11 Relation Modules
- 12 An Algorithm for N/N′
- 13 Finite p-groups
- 14 The Nilpotent Quotient Algorithm
- 15 The Golod-Shafarevich Theorem
- 16 Proving some Groups Infinite
- Guide to the literature and references
- Index
- Dramatis personae
4 - Free Presentations of Groups
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- 1 Free Groups
- 2 Schreier's Method
- 3 Nielsen's Method
- 4 Free Presentations of Groups
- 5 Some Popular Groups
- 6 Finitely-generated Abelian Groups
- 7 Finite Groups with few Relations
- 8 Coset Enumeration
- 9 Presentations of Subgroups
- 10 Presentations of Group Extensions
- 11 Relation Modules
- 12 An Algorithm for N/N′
- 13 Finite p-groups
- 14 The Nilpotent Quotient Algorithm
- 15 The Golod-Shafarevich Theorem
- 16 Proving some Groups Infinite
- Guide to the literature and references
- Index
- Dramatis personae
Summary
Basic concepts
Suppose that
X is a set,
F = F(X) is the free group on X,
R is a subset of F,
N = R is the normal closure of R in F, and
G is the factor group F/N.
Definition 1. With this notation, we write G = <X|R> and call this a free presentation, or simply a presentation of G. The elements of X are called generators and those of R defining relators. A group G is called finitely presented if it has a presentation with both X and R finite sets.
Remarks. 1. This makes precise the notion that the x ∈ X generate G, that the r ∈ R are equal to e in G, and that G is the “largest” group with these properties. Note the abuse of notation in referring to x and r as elements of G. This is done for convenience. It is always clear from the context to which group (F or G) a given word in X± belongs.
2. It is sometimes convenient to replace R in < X | R > by the set of equations R = e, that is, [r = e|r ∈ R}, called defining relations for G. A defining relation may even take the form “u = v”, where u,v ∈ F(X), corresponding to the defining relator uv-1.
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- Information
- Presentations of Groups , pp. 41 - 57Publisher: Cambridge University PressPrint publication year: 1997