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 Homomorphisms


In the vast field of algebra, the field of abstract structures known as group theory plays an important role. It provides a framework for studying symmetry and its generalizations. One of the fundamental concepts within this framework is the notion of group isomorphism.

What is a group?

Before diving into homomorphisms, let's first understand what a group is in the mathematical sense. A group is a collection of elements equipped with binary operations that satisfy four fundamental properties: closure, associativity, identity, and invertibility.

  • Closure: If a and b are elements of a group G, then the result of the operation a * b will also be in G
  • Associativity: If a, b, c are elements of G, then (a * b) * c = a * (b * c)
  • Identity element: there exists an element e in G such that e * a = a * e = a for every element a in G
  • 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.

Defining group homomorphisms

A group isomorphism is a function between two groups that preserves the group structure. Consider two groups (G, *) and (H, •). The function f: G → H is an isomorphism if for every pair of elements a, b in G, the following condition holds:

f(a * b) = f(a) • f(b)

This equation essentially tells us that the function f respects the group operation.

Examples of group isomorphisms

Example 1: Identity isomorphism

Consider any group (G, *). The function id: G → G defined by id(a) = a for all a in G is a homomorphism. This is because for any elements a and b in G, we have:

id(a * b) = a * b = id(a) * id(b)

Example 2: Zero homomorphism

Consider an isomorphism between two groups (G, *) and (H, •), where e_H is the identity element of H. The function f: G → H defined by f(a) = e_H for all a in G is an isomorphism, usually called the zero isomorphism.

Evidence:

f(a * b) = e_H = e_H • e_H = f(a) • f(b)

Example 3: Real numbers under addition

Consider the set of real numbers under addition, (ℝ, +) and the set of positive real numbers under multiplication, (ℝ^+, ×). The logarithmic function log: ℝ^+ → ℝ is an isomorphism because:

log(xy) = log(x) + log(y)

This equation shows that log transforms the multiplication operation into addition, thereby preserving the group structure with respect to the addition.

Visual example

To understand the concept visually, imagine two groups represented as circles with elements inside them. A homomorphism is like an arrow that maps elements from one circle to another while maintaining the integrity of their operations.

Yes A B H f(a) f(b) F

Properties of group isomorphisms

Group homomorphisms have a number of useful properties that reflect the structure of the groups they relate. Below are some of these properties:

Image and kernel

  • Image: The image of a homomorphism f: G → H is the set of all elements h in H for which there exists a g in G such that f(g) = h. It is denoted by Im(f) or f(G).
  • Kernel: The kernel of a homomorphism is the set of all elements in G that coincide with the identity element in H. It is denoted by Ker(f).

A homomorphism is called injective or a monomorphism if its kernel contains only the identity element of G

Isomorphism

A homomorphism that is both injective and surjective (onto) is called an isomorphism. Two groups are considered isomorphic if there exists an isomorphism between them. Isomorphic groups share the same structural properties, even though their elements may differ.

Examples demonstrating properties

Example 4: Kernel of a homeomorphism

Consider the isomorphism f: ℤ → ℤ_6 defined by f(n) = n mod 6, where ℤ is the group of integers under addition, and ℤ_6 is the group of integers modulo 6. The kernel of this isomorphism is the group of all integers n such that n mod 6 = 0 Thus:

Ker(f) = { ..., -12, -6, 0, 6, 12, ... }

Example 5: Isomorphic groups

Consider a cyclic group C_4 consisting of the elements {0, 1, 2, 3} under modulo addition of 4, and another group H with the elements {1, i, -1, -i} under multiplication (where i is the imaginary unit). The function f: C_4 → H is defined by:

  • f(0) = 1
  • f(1) = i
  • f(2) = -1
  • f(3) = -i

There is a symmetry. This can be verified by checking both the symmetry property and the double bonding.

Importance and applications

Group symmetries are important because they allow mathematicians to transfer structure and properties from one group to another. They serve as a bridge to compare and understand different algebraic structures. Applications of group theory and symmetries span many disciplines, including physics, cryptography, and music theory. In physics, symmetries and conservation laws are deeply rooted in group theory. In cryptography, secure communication algorithms often involve groups and symmetries. This further demonstrates that these mathematical concepts play important roles beyond abstract theory.

Group symmetries also contribute to the classification of groups by identifying simple groups and constructing more complex groups from them via extensions, by dealing with modules and representation theory, and by understanding the underlying symmetries of mathematical objects.

Conclusion

Group symmetries serve as fundamental connectives between algebraic structures, helping to map and preserve group properties across different domains. Through a combination of practical examples and rigorous proofs, we have not only defined what a group symmetry is, but also explored the various layers of complexities and properties that arise from it. Understanding these contributors to the overarching structure of group theory allows one to appreciate the beauty and versatility of algebra in solving real-world problems.


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Group Homomorphisms

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