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 theory is a branch of mathematics that studies algebraic structures known as groups. Groups are sets equipped with an operation that combines any two elements to form a third element while satisfying four main conditions known as group axioms: closure, associativity, existence of an identity element, and existence of inverse elements.

Basic definitions and concepts

Let's explore the formal definition of a group. A group is a set (G) combined with an operation (*) (often called multiplication or addition) that satisfies the following axioms:

  1. Closure: For every two elements ( a ) and ( b ) in ( G ), the result of the operation ( a * b ) is also in ( G ).
  2. Associativity: For every three elements ( a, b, c ) in ( G ), the equation ( (a * b) * c = a * (b * c) ) is valid.
  3. Identity element: There exists an element ( e ) in ( G ) such that for every element ( a ) in ( G ) the equation ( e * a = a * e = a ) holds. This element ( e ) is called the identity element.
  4. Inverse element: For every element ( a ) in ( G ), there exists an element ( b ) in ( G ) such that ( a * b = b * a = e ), where ( e ) is the identity element. The element ( b ) is called the inverse of ( a ), usually denoted as ( a^{-1} ).

Examples of groups

Examining specific sets and operations can bring clarity to these abstract definitions. Let's look at some examples that illustrate different types of groups.

Integers under addition

Consider the set of integers (mathbb{Z}) with the operation of addition ((+)).

1. Closure: For any ( a, b in mathbb{Z} ), ( a + b ) is also an integer.
2. Associativity: For any ( a, b, c in mathbb{Z} ), ( (a + b) + c = a + (b + c) ).
3. Identity Element: The number 0 acts as an identity because ( 0 + a = a + 0 = a ) for any ( a in mathbb{Z} ).
4. Inverse Elements: Every integer ( a ) has an inverse (-a) such that ( a + (-a) = 0 ).

Symmetric group

The symmetric group ( S_n ) is the group of all permutations of ( n ) elements. The operation is the composition of permutations, and the identity element is the identity permutation, which leaves every element in its original position.

Example: For ( n=3 ), the symmetric group ( S_3 ) contains permutations such as ( {(1), (12), (13), (23), (123), (132)} ).

Visual example: circle group

An intuitive example of a group is the group of rotations around a circle. Consider a circle marked 0 degrees. Consider the operation to be rotation through any angle (theta). This set of all possible rotations forms a group. Here is a visual description.

<svg width="400" height="200"> <circle cx="100" cy="100" r="80" stroke="black" stroke-width="3" fill="none" /> <line x1="100" y1="100" x2="180" y2="100" stroke="red" stroke-width="2" /> <line x1="100" y1="100" x2="140" y2="60" stroke="blue" stroke-width="2" /> <line x1="100" y1="100" x2="120" y2="180" stroke="green" stroke-width="2" /> </svg>
<svg width="400" height="200"> <circle cx="100" cy="100" r="80" stroke="black" stroke-width="3" fill="none" /> <line x1="100" y1="100" x2="180" y2="100" stroke="red" stroke-width="2" /> <line x1="100" y1="100" x2="140" y2="60" stroke="blue" stroke-width="2" /> <line x1="100" y1="100" x2="120" y2="180" stroke="green" stroke-width="2" /> </svg>

Properties of groups

Groups exhibit fascinating properties that contribute to their rich structure and functionality in mathematics. Below, we will discuss some of these fundamental properties in depth.

Order of the group

The order of a group is the number of elements it contains, often denoted as (|G|). If the number of elements in a group is finite, then it is called a finite group. Otherwise, it is infinite.

Subgroups

A subset ( H ) of a group ( G ) which is also a group with the same operations as ( G ) is called a subgroup of ( G ).

Example: In (mathbb{Z}), the group of even integers forms a subgroup because it satisfies all the group axioms.

Cyclic groups

A group is cyclic if it can be generated by a single element. This means that every element in the group can be written as powers (or multiples in additive notation) of this element. If ( g ) is a generator of ( G ), then:

( g = { g^n : n in mathbb{Z} } ).

Cayley theorem and Lagrange theorem

Cayley's theorem

Cayley's theorem states that every group (G) is isomorphic to a subgroup of the symmetric group that acts on (G). This implies that even though groups may differ in terms of elements and operations, they can be represented through permutations of objects, which connects group theory to geometry and combinatorics.

Lagrange's theorem

Lagrange's theorem plays an important role in understanding the relationship between a group and its subgroups. It states that the order of any subgroup (H) of a finite group (G) divides the order of the group (|G|). This theorem is helpful in analysing the structure and constraints of various groups.

Applications of group theory

Group theory is not just an abstract field, but it is used in a wide range of subjects. Below are some of the areas where group theory is used:

  • Physics: Group theory is used extensively in theoretical physics, especially in the study of symmetries in quantum mechanics and particle physics.
  • Chemistry: In chemistry, group theory aids in the study of molecular symmetry and helps predict spectral properties.
  • Cryptography: Modern cryptographic systems such as RSA use the properties of algebraic groups to secure data transmission.
  • Computer science: Algorithms and data structures often use group-like features to improve efficiency and security.

Conclusion

Group theory is a foundational concept in algebra that provides insight into how symmetries and transformations operate in a variety of mathematical and scientific fields. Understanding groups facilitates a deeper understanding of the underlying structure and organization in many mathematical systems. Although its principles are abstract, the applications of group theory span diverse and practical fields of inquiry, making it an invaluable tool in both theoretical research and practical problem-solving.


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Group Theory

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