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 ↓


Quotient Groups


In the fascinating world of abstract algebra, group theory plays an important role. An interesting concept within group theory is that of the quotient group. This concept may seem complicated initially, but with clear explanations, visual aids, and examples, it becomes much easier to understand.

Understanding groups

Before we dive deeper into quotient groups, it's important to remember what a group is. In mathematics, a group is a set equipped with an operation that combines any two of its elements to form a third element. This operation must satisfy four fundamental properties:

  • Closure: If a and b are in the group G, then the result of the operation, a * b, must also be in G.
  • Associativity: For any elements a, b, and c in G, the equation (a * b) * c = a * (b * c) must be valid.
  • Identity element: There must exist an element e in G such that the equations e * a = a and a * e = a hold for any element a in G.
  • Inverse element: For every element a in G, there must exist an element b in G such that a * b = e and b * a = e, where e is the identity element.

Normal subgroups

An important concept to understand before quotient groups is that of normal subgroups. A subgroup N of a group G is called a normal subgroup if it is invariant under conjugation; that is, for every element g in G, the element gNg-1 is still in N.

In other words, a subgroup N is normal if for every n in N and g in G, the element g * n * g-1 exists in N. We denote that N is a normal subgroup of G by using N &lhd; G.

Example of a normal subgroup

Consider the symmetric group S3, which is the group of all permutations of three objects. It has six elements:

S3 = {(), (123), (132), (12), (13), (23)}

The subgroup A3 = {(), (123), (132)} is a normal subgroup of S3 since it is invariant under conjugation by any element of S3.

Defining quotient groups

Once we have a normal subgroup N of a group G, we can form the quotient group G/N. The quotient group is the set of all left cosets of N in G. The left coset of N in G is a group of the form gN = { gn | n ∈ N } where g is an element in G.

The main idea is that operations in the quotient group are defined by combinations of cosets. Two elements aN and bN in G/N are multiplied as follows:

(AN) * (BN) = (AB)N

This operation is well-defined because N is a normal subgroup, which allows us to uniquely represent the elements of the group.

Visualization of quotient groups

Consider a group G which can be represented visually as a set of points within a larger circle. Inside this circle, there are smaller circles representing subgroups.

Now, imagine N as a special subgroup in this representation. The quotient group G/N can be thought of as collapsing all the elements of each cogroup into a single point. This creates a simplified structure that still reflects the properties of the group.

G/N

Example of a quotient group

Let's return to the symmetric group S3 and its normal subgroup A3. The quotient group S3/A3 consists of left cosets:

S3 /a3 = {a3, (12)a3, (13)a3}

Each coset represents one of the possible ways in which the structure of S3 can be partitioned, taking A3 as a subset.

Properties of quotient groups

Quotient groups inherit several properties from their parent group, G. These properties retain much of the structure of the parent group:

  • Size: The order of the quotient group G/N is equal to the order of G divided by the order of N. Formally, | G/N | = | G | / | N |.
  • Isomerism: If f: G -> H is an isomorphism and N is in the kernel of f, then there is an induced isomorphism from G/N to H.

Example with homeomorphisms

Suppose we have a group homomorphism from Z (the integers) to Z6 (the modulus of integer 6), defined as:

f(n) = n mod 6

The kernel of this isomorphism is {6k | k ∈ Z}, which allows us to define a quotient group:

{6Z}/{6Z}≅Z6

This shows how the integers mod 6 can be viewed as the quotient group of the set of all integers.

Application

Quotient groups have many applications in mathematics and related fields, including symmetry, number theory, and algebraic structures. They are helpful in constructing other algebraic structures and in studying topological spaces and solutions of polynomial equations.

Examples of applications

In topology, the fundamental group of a space can help describe loops in the space up to deformation, and quotient groups can help understand the relationships between these loops.

In physics, the concept of quotient groups is used in the study of symmetries and conservation laws, especially in the field of particle physics.

Conclusion

The quotient group is a profound concept in group theory that provides insight into the structure and symmetry of a mathematical system. By examining subgroups and their corresponding cogroups, we can infer important properties of the entire group, leading to a better understanding of its overall behavior.


PHD → 1.1.5


U
username
0%
completed in PHD

← Prev (1.1.4)
Normal Subgroups
Next (1.1.6) →
Group Homomorphisms

Comments 



Quotient Groups

https://www.buddymath.com/phd/1.1.5?bmal=en