Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Algebra → Abstract Algebra ↓


Homomorphisms


In abstract algebra, a branch of mathematics dealing with algebraic structures, homomorphisms play a key role in understanding how these structures relate to each other. In simple terms, homomorphisms are functions between algebraic structures that preserve structure. This means that if you understand homomorphisms, you have discovered a way to map elements between different sets that respect the operations of the structures you are working with.

To fully understand homeomorphisms, let's take a deeper look at some key topics:

  • Definition of homeomorphism
  • Examples of homeomorphisms
  • Properties of homeomorphisms
  • Kernel and image of a homeomorphism
  • Homeomorphism and isomorphism
  • Different types of homeomorphisms

Definition of homeomorphism

A homomorphism is a map between two algebraic structures of the same type, such as groups, rings, or vector spaces, that preserves the operations they define. Let's see what this means in context:

Mathematical definition

Consider two algebraic structures (A, *) and (B, •), where * and • denote binary operations in each structure, respectively. A function f: A → B is called a homomorphism if for all elements x, y in A, the following is true:

f(x * y) = f(x) • f(y)

This condition ensures that the structure of A is preserved when mapped onto B

Let A and B be groups and let *, • denote the group operations. Then f: A → B is a group isomorphism if for all x, y in A:

f(x * y) = f(x) • f(y).

Examples of homeomorphisms

Examples are very helpful in understanding the concept of homomorphism better. Let's look at some common examples:

Example 1: Real numbers under addition

Consider the set of real numbers (ℝ, +), where the operation is standard addition. Define a function f: ℝ → ℝ by f(x) = 2x. Then, for any real numbers a and b:

f(a + b) = 2(a + b) = 2a + 2b = f(a) + f(b)

This proves that f is an isomorphism from (ℝ, +) to (ℝ, +).

Example 2: Qualitative isomorphism

Consider the set ℝ{0} of non-zero real numbers under multiplication. Define a function g: ℝ{0} → ℝ{0} as g(x) = x^2. Verify that for all x in ℝ{0} x, y:

g(xy) = (xy)^2 = x^2y^2 = g(x)g(y)

Thus, g is an isomorphism from (ℝ{0}, times) to (ℝ{0}, times).

A B F : A → B

Properties of homeomorphisms

Homeomorphisms have several important properties that make them useful in the study of algebraic structures. Let's explore some of these:

Identity protection

A homomorphism preserves the identity element of a structure. If e_A is the identity element in A, then for any homomorphism f: A → B, f(e_A) is the identity element in B

Consider the groups (ℤ, +) and (ℝ, +). A homomorphism f: ℤ → ℝ must map the identity 0 ∈ ℤ to the identity 0 ∈ ℝ.

Inverse conservation

A homeomorphism also preserves inverse relations of elements. For an element x in a domain structure with a homeomorphism f and inverse x -1, it is valid that:

f(x -1) = [f(x)] -1

Kernel and image of a homeomorphism

Two important concepts associated with homeomorphisms are the kernel and the image.

Kernels

The kernel of a homomorphism f: A → B is the set of all elements in A that are mapped to the identity element of B Formally, it is defined as:

Ker(f) = {x ∈ A | f(x) = e_B}

Image

The image of a homomorphism is the set of all elements in B that are mapped to elements in A It is defined as:

{displaystyle f(x)|xin A}
For the isomorphism f: ℝ → ℝ defined by f(x) = 2x as before, the kernel is: 
ker(f) = {0}
and the image is: 
ℝ = ℝ

Homeomorphism and isomorphism

Isomorphism is a special type of homomorphism that is binary and preserves structure. If an isomorphism exists between two algebraic structures, then the structures are said to be isomorphic, meaning that they are essentially the same structurally.

Example of isomorphism

Consider (ℤ, +) and (2ℤ, +), where 2ℤ denotes the even integers. The function h: ℤ → 2ℤ defined by h(x) = 2x is an isomorphism. It is bijective because every integer maps uniquely to an even integer, and this is a homomorphism:

h(x + y) = 2(x + y) = 2x + 2y = h(x) + h(y)

Thus, (ℤ, +) and (2ℤ, +) are isomorphic.

Different types of homeomorphisms

Depending on the algebraic structure under consideration, there are different types of homeomorphisms:

  • Group homomorphism : A function between two groups that respects the group operation.
  • Ring homomorphism : a function between two rings that preserves both addition and multiplication.
  • Linear transformation : An isomorphism of vector spaces, also known as a linear map.
  • Module homomorphism : a structure-preserving map between two modules over a ring.

Understanding homeomorphisms in each of these contexts highlights their versatility and importance in various algebraic systems.

Conclusion

Homomorphisms are fundamental concepts in the study of algebraic structures and provide a powerful tool for understanding how different mathematical systems are related. By preserving operations between structures, they reveal insights into the structure and properties of algebraic groups, rings, vector spaces, and much more. Through their kernels and images, we gain important information that can help us build a deeper understanding of the underlying algebraic systems.


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