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 ↓


Group Actions


Group actions are a way of describing symmetries in a more structured way. In the field of group theory, which is a branch of mathematics that deals with algebraic structures known as groups, understanding how groups can act on other structures provides deep insight into the nature of groups and the structures they are acting on.

Understanding groups

Before diving into group actions, it is important to understand what a group is. A group is a set equipped with an operation that satisfies four fundamental properties: closure, associativity, identity element, and existence of an inverse.

1. Closure: For a set G with an operation *, if a and b are in G, then a * b is also in G. 2. Associativity: For any a, b, and c in G, (a * b) * c = a * (b * c). 3. Identity Element: There exists an element e in G such that for any element a in G, e * a = a * e = a. 4. Inverses: For each a in G, there is an element b in G such that a * b = b * a = e.

What is group action?

A group action is a formal way of interpreting group elements as symmetries or transformations of another set. Formally, we say that a group G acts on a set X if there is a function:

G × X → X

satisfying two properties:

1. Identity: For every element x in X, e * x = x, where e is the identity element of G. 2. Compatibility: For every pair of elements g, h in G, and every x in X, (g * h) * x = g * (h * x).

Visualizing group actions

Let's use a simple example. Consider the group C3, which represents the symmetries of an equilateral triangle. This group has three elements: the identity rotation (e), the rotation by 120 degrees (r), and the rotation by 240 degrees (r²).

ABC

When C3 acts on the triangle, each group element changes the positions of the vertices (A, B, C) of the triangle. For example:

- Identity (e): (A, B, C) → (A, B, C) - Rotation by 120 degrees (r): (A, B, C) → (B, C, A) - Rotation by 240 degrees (r²): (A, B, C) → (C, A, B)

This example shows how group actions allow us to talk about the symmetries (rotations) of a triangle in a structured way using the group C3.

Examples of group actions

Example 1: Permutation group

Consider the symmetric group S3, which consists of all permutations of three objects. Suppose that S3 acts on the set {1, 2, 3}.

- Identity (e): (1, 2, 3) → (1, 2, 3) - Swap the first two elements ((1 2)): (1, 2, 3) → (2, 1, 3) - Cycle (1 2 3): (1, 2, 3) → (2, 3, 1) - Cycle (1 3 2): (1, 2, 3) → (3, 1, 2) - Swap the last two elements ((2 3)): (1, 2, 3) → (1, 3, 2) - Cycle (1 3): (1, 2, 3) → (3, 2, 1)

Here, every permutation of the set S is a group action by3 onto {1, 2, 3}.

Example 2: Matrix group

Consider the group of 2x2 invertible matrices, known as GL(2, R), which acts on the vector space R². A matrix A from GL(2, R) acts on a vector v in R² by multiplying A by v.

If A = [[a, b], [c, d]] and v = [x, y], then A • v = [[a, b], [c, d]] • [x, y] = [ax + by, cx + dy]

Properties of group actions

Group functions are more than just abstract operations. They have important properties that allow mathematicians to investigate deep relationships within structures.

Types of group activities

Group actions may be classified based on their behaviour:

- Faithful: If for any two distinct elements of G, there is a point x in X such that the actions differ on x. - Transitive: If for any x, y in X, there exists ag in G such that g * x = y. - Free: If the stabilizer of every element in X is trivial (only the identity element in G fixes x).

Visual example of a transitive verb

XYJade

In a transitive action, you can move from any point in the set to any other point using a group element.

Applications of group actions

Understanding group actions is important in several areas:

  • Crystallography: Describing symmetries in crystal structures.
  • Coding theory: Code structuring using group symmetry for error correction.
  • Physics: Describing symmetric properties of systems and particles.
  • Geometry: Study of transformations that preserve certain properties of places.

Conclusion

Group actions form a key concept in both pure and applied mathematics. They enable us to understand symmetries and transformations in a structured way, providing insights into algebra, geometry, and beyond. Whether it is symmetrical motion in a dance or molecular vibrations in a crystal, group actions provide a powerful lens through which we can analyze and predict the behavior of complex systems.


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Group Actions

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