Alphabeta MathFree Groups and Presentations
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1 · Prerequisites
Abstract Algebra
Real Analysis
2 · Logical flowchart
3 · Items
Definition
Words in an alphabet with formal inverses, elementary cancellation, and reduced words
Definition
Free group on a set of generators
Theorem
Reduced words form the free group on an alphabet
Definition
Group presentation by generators and relations
Theorem
Free groups on the same set are uniquely isomorphic compatibly with their generators
Theorem
Every group admits a presentation
Proposition
Free equivalence is an equivalence relation and concatenation respects it
Definition
The word-quotient model F_word(X):=W(X)/∼ with multiplication induced by concatenation
Theorem
F_word(X) is a group under [w][v]=[wv]
Lemma
Formal letters act by mutually inverse permutations on the set of reduced words
Theorem
Every class in W(X)/∼ contains exactly one reduced word
Theorem
The word-quotient group W(X)/∼ satisfies the universal property of the free group on X
Corollary
The generator map X→ W(X)/∼ is injective
Corollary
The word-quotient and reduced-word models are uniquely isomorphic compatibly with X
Definition
A free basis of a group
Theorem
Any two finite free bases of the same group have the same cardinality
Definition
The rank of a free group admitting a finite basis
Definition
Relators and relations; finitely generated, finitely related, and finite presentations
Proposition
The normal closure of R is the set of finite products of conjugates of elements of R and their inverses
Proposition
In ⟨ X∣ R⟩, the words u and v represent the same element if and only if u⁻¹v∈⟨⟨ R⟩⟩
Theorem
Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group
Corollary
Every finite group has a finite presentation from its multiplication table
Definition
The abelianisation Gᵃᵇ:=G/[G,G] and its canonical map
Definition
Free abelian group on a set
Theorem
The abelianisation of a free group on X is a free abelian group on X
Definition
Tietze transformations: dictionary generators, redundant relators, renaming, and their inverses
Proposition
Each Tietze transformation preserves the isomorphism type of the presented group
Theorem
Two finite presentations define isomorphic groups if and only if a finite sequence of Tietze transformations and inverses connects them
Definition
Cyclically reduced words
Lemma
Every nonempty reduced word has the form tct⁻¹ with c nonempty and cyclically reduced
Theorem
Free groups are torsion-free
Theorem
Two cyclically reduced words in a free group are conjugate if and only if one is a cyclic permutation of the other
Sources
- McKernan, Presentations and Groups of Small Order, Lecture 12
- John McKernan, Presentations and Groups of Small Order, Lecture 12
- Brittenham, Group Presentations, Class Notes
- M. Brittenham, Group Presentations, Class Notes
- Nicholas Touikan, An Introduction to Combinatorial and Geometric Group Theory, §1.2
- Richard Elman, Lectures on Abstract Algebra, §18
- Nicholas Touikan, An Introduction to Combinatorial and Geometric Group Theory, §1.3
- Encyclopedia of Mathematics, Free group
- D. L. Johnson, Presentations of Groups, Chapter 4
- Encyclopedia of Mathematics, Presentation
- Nicholas Touikan, An Introduction to Combinatorial and Geometric Group Theory, §1.4
- M. Brittenham, Group presentations
- Ashot Minasyan, MATH6138 Geometric Group Theory, §2.2
- Nicholas Touikan, An Introduction to Combinatorial and Geometric Group Theory, §1.6
- Ashot Minasyan, MATH6138 Geometric Group Theory, §2.3
- Encyclopedia of Mathematics, Commutator subgroup
- Nicholas Touikan, An Introduction to Combinatorial and Geometric Group Theory, Theorem 1.6.2
- Alexei Myasnikov and Vladimir Shpilrain, Combinatorics over Free Groups, §2.2.1
- Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial Group Theory
- Alexei Myasnikov and Vladimir Shpilrain, Combinatorics over Free Groups, Proposition 2