We start with a brief introduction to Free Groups, thereby appreciating Nielsen’s approach to the
Subgroup theorem. Beautiful results of J. H. C. Whitehead, J. Nielsen, E. S. Rapaport, Higgins and
Lyndon, and J. McCool form our building block. We study different automorphisms of a finitely generated
free group as well as a finite set of automorphisms which Whitehead used to deduce that if two elements
of a finitely generated free group are equivalent under an automorphism of the group, then they are
equivalent under such automorphisms. We write program aimed at appreciating Whitehead’s theorem,
starting with programs for appreciating Whitehead automorphisms to programs for determining whether
two elements of a finitely generated free group are equivalent or not. We conclude by classifying all
minimal words of lengths 2, 3, 4, 5 and 6 in F n (for some n ∈ [2, 6]) up to equivalence.
Table of Contents
1 Introduction to Free Groups
1.1 Definition, Existence and Uniqueness Theorem
1.2 Basic Properties of Free Groups
1.3 Subgroups of Free Groups
2 Automorphisms of Free Groups
2.1 Special Automorphisms of FX
2.2 Automorphisms of F¯n
2.3 Nielsen Automorphisms
2.4 The Whitehead Automorphisms
2.5 Equivalence of Elements under Aut(Fn)
2.6 A Presentation for Aut(Fn)
3 The Whitehead Algorithm
3.1 Whitehead Theorem Revisited
3.2 Algorithm Developments
3.3 GAP Exhibitions
4 Concluding Remarks
4.1 Summary
4.2 Further Research Questions
4.3 Conclusions
A All reduced words and minimal words of lengths l in Fn
A.1 A program for finding all reduced words of lengths l in Fn
A.2 A program for finding all minimal words of lengths l in Fn
B Equivalence classes of minimal words of certain lengths in F2 and F3
B.1 Equivalence classes of minimal words of lengths 2, 3, 4, 5 and 6 in F2
B.2 Equivalence classes of minimal words of lengths 2, 3, 4 and 5 in F3
Research Objectives and Key Topics
The primary objective of this work is to explore the theory of automorphisms of finitely generated free groups, specifically focusing on the Whitehead algorithm as a powerful tool for determining the equivalence of elements. The research aims to implement these theoretical concepts through computational methods using the GAP (Groups, Algorithms, and Programming) software, ultimately classifying minimal words of various lengths within these groups up to equivalence.
- Theoretical foundations of Free Groups and their automorphisms.
- Detailed analysis of Whitehead automorphisms, including Type 1 and Type 2.
- Algorithmic development for testing word minimality and equivalence.
- Implementation of GAP programs to classify minimal words and investigate equivalence classes.
- Mathematical characterization and classification of minimal words of lengths 2 through 6.
Excerpt from the Book
2.4 The Whitehead Automorphisms
Here, we introduce a set of automorphisms which Whitehead ([Whi36a], [Whi36b]) used to deduce that if two elements of a finitely generated free group are equivalent under any automorphism of the group, then they are equivalent under a finite sequence of automorphisms which we shall refer to in the sequel as “Whitehead automorphisms”. We start by defining some terms and notations that will be used.
2.4.1 Definition. • For w1, w2 ∈ Fn, we denote equivalence by ∼, and write w1 ∼α w2 if α ∈ Aut(Fn) such that α(w1) = w2.
• In FX (where | X |= n < ∞), the set Ln of generators and inverses of Fn is defined as: Ln = {x1, x2, ··· , xn, x¯1, x¯2, ··· , x¯n} with x¯i = x−1 i for 1 ≤ i ≤ n.
• A word w ∈ Fn is said to be Minimal if |w|≤|α(w)| ∀ α ∈ Aut(Fn).
• A Cyclic word is a set of all cyclic permutations of a given cyclically reduced word.
• The Length of a cyclic word w ∈ Fn is denoted by |w|, and defined as the number of letters in the cyclically ordered set.
Chapter Summaries
1 Introduction to Free Groups: This chapter provides fundamental definitions and properties of free groups, including the existence and uniqueness theorem and an introduction to their subgroups.
2 Automorphisms of Free Groups: This chapter examines various types of automorphisms, including Special, Nielsen, and Whitehead automorphisms, and investigates their roles in group theory and equivalence.
3 The Whitehead Algorithm: This chapter focuses on the design and implementation of algorithms to identify minimal words and determine the equivalence of elements within finitely generated free groups using GAP.
4 Concluding Remarks: This chapter summarizes the findings regarding minimal words, presents conjectures for lengths 4 and 5, and suggests directions for future research.
Keywords
Free Groups, Automorphisms, Whitehead Algorithm, Nielsen Automorphisms, Whitehead Automorphisms, Word Minimality, Word Equivalence, Group Theory, GAP, Minimal Words, Finitely Generated Groups, Computational Group Theory, Cyclic Words, Character Pairs, Abelianization.
Frequently Asked Questions
What is the core focus of this research?
The work primarily focuses on the theory of automorphisms in finitely generated free groups and the application of the Whitehead algorithm to determine if two elements are equivalent.
What are the central themes discussed in the book?
The book covers the structural properties of free groups, the classification of different types of automorphisms, the computational approach to word minimality, and the resulting classification of equivalence classes of minimal words.
What is the main research question or goal?
The main goal is to effectively implement the Whitehead algorithm to classify minimal words of certain lengths in free groups and to provide a systematic computational approach to study their equivalence.
What scientific methods are utilized?
The research combines algebraic group theory proofs with algorithmic developments, utilizing the GAP (Groups, Algorithms, and Programming) software for computational verification and implementation.
What is covered in the main body of the work?
The main body covers the mathematical theory of free group automorphisms, the formal definition and properties of Whitehead automorphisms, the development of specific algorithms for word reduction, and detailed GAP program implementations.
Which keywords best describe this study?
Key terms include Free Groups, Whitehead Algorithm, Automorphisms, Word Minimality, Word Equivalence, and Computational Group Theory.
What are "Type 1" and "Type 2" Whitehead automorphisms?
Type 1 automorphisms are permutations of generators and inverses that do not change word length, while Type 2 automorphisms are length-reducing operations crucial for reaching word minimality.
How does the author characterize minimal words of lengths 2 and 3?
The author proves that every minimal word of length 2 or 3 is "exact," meaning it is the only reduced word within its cyclic structure, and identifies a single equivalence class for these lengths.
- Arbeit zitieren
- Chimere Anabanti (Autor:in), 2013, The Whitehead Algorithm for free groups, München, GRIN Verlag, https://www.grin.com/document/294023
