PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Algebra → Group Theory ↓


Free Groups


The concept of free groups in algebra is crucial to understanding the rich structure of group theory. Free groups are the most basic examples of groups and serve as the building blocks for more complex groups. They are not burdened by additional relations, which makes them freely generated by a set of elements. Essentially, in a free group, you can use the generating elements and their inverses to form any term, subject only to the rules of combination (i.e., how group elements combine) and inversion (i.e., the principle that combining an element with its inverse gives the identity element).

Intuitive understanding of free groups

Consider a set of symbols, say {a, b, c}. A free group generated by these symbols, denoted as F(a, b, c), consists of all possible strings (or "words") that can be formed using these symbols and their inverses {a⁻¹, b⁻¹, c⁻¹}. The only rule is that adjacent pairs of an element and its inverse can be removed from the word, leading to simplification.

Examples of words in the free group F(a, b, c):
- Aaaa
- a b c a ⁻¹ b ⁻¹ c
- abcabc⁻¹b⁻¹a⁻¹

Simplifying a term means to collapse pairs of elements and their inverses until no further simplification is possible. For example: abca⁻¹b⁻¹c → abc⁰b⁻¹c → abc⁰b⁰c → acc.

Formal definition of free groups

Mathematically, a free group with a set of generators S is a group where each element can be uniquely expressed as a product of the elements of S and their inverses. Formally, if S is a set, the free group F(S) is defined as:

F(S) = { the set of all finite words from the alphabet S ∪ S⁻¹ with cancellation rule }

In this definition, the cancellation rule means that any pair of consecutive elements ss⁻¹ or s⁻¹s (where s ∈ S) can be cancelled to obtain simpler words.

Visualizing free groups

To understand how open groups work, imagine a tree where each node represents the partial words being constructed.

E A B Come Now B. A BB

Here, the root node ε represents the identity element, while the branches represent adding elements a or b. The cancellation law would mean two nodes like a and a⁻¹ “disappear” in ε.

Properties of free groups

Free groups have a number of interesting properties due to their definition:

  • Universal property: If G is a group and S is a set, then any function F(S) from S to G can be uniquely extended by a group homomorphism from G This universality gives free groups a special place in category theory.
  • Word problem: In a free group, we can decide whether two terms are equivalent (i.e., simplify to a single term), which is an important aspect of computational group theory.
  • Free product: Any two free groups can combine (or take a "free product") to form another free group, and retain a structure consisting entirely of unique, non-redundant relationships.

Creation of an independent group: an example

Let us consider a practical example for constructing a free group. Suppose we have generators {x, y}.

The free group F(x, y) consists of all possible words made up of x, y and their inverses {x⁻¹, y⁻¹}. Examples of elements include:

- X
-y⁻x
- xyx⁻¹y⁻¹
- xy⁻¹xyx⁻¹

One operation on these elements is concatenation, which simply means adding strings. Consider concatenating xy and y⁻¹x:

xy • y⁻¹x = xyy⁻¹x = xx (after canceling yy⁻¹)

Real-world applications of free groups

Free groups appear in a variety of areas beyond pure mathematics. In computational group theory, they represent structures without any additional constraints beyond the group axioms, which are necessary for algorithms dealing with simplified group structures. In topology, free groups help describe the fundamental group of some topological space.

For example, in topology, the fundamental group of a bouquet of two circles (which looks like a "figure-eight" or infinity symbol drawn by the two loops) is generated independently by the loops going around each circle once in a prescribed path.

Conclusion

Free groups serve as fundamental structures in algebra, providing insight into the construction and analysis of more complex group structures. Their importance spans many mathematical and computational fields, highlighting their versatility and foundational role in mathematics. A strong understanding of free groups not only enriches the understanding of group theory, but also provides tools for exploring any mathematical structure relying on the concept of groups and their symmetries.


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Free Groups

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