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 ↓


Cosets and Lagrange's Theorem


Group theory is a branch of algebra that studies algebraic structures known as groups. One of the fundamental concepts within group theory is the coset and the Lagrange theorem. These concepts are important for understanding the structure of groups and their subgroups.

Understanding groups and subgroups

A group G is a set equipped with a binary operation (often called multiplication) that satisfies certain axioms: closure, associativity, the existence of an identity element, and the existence of inverse elements. Mathematically, if g, h are elements of G, then:

1. Completion: ( g cdot h in G )
2. Associative: ( (g cdot h) cdot k = g cdot (h cdot k) )
3. Identity: There exists an element ( e in G ) such that ( g cdot e = e cdot g = g )
4. Inverse: For every ( g in G ) there exists an element ( g^{-1} in G ) such that ( g cdot g^{-1} = g^{-1} cdot g = e )

A subgroup H of a group G is a subset of G which is itself a group under an operation defined on G

What are cosets?

In the context of groups, a coset is a subgroup formed by multiplying (or adding, in the case of additive groups) all the elements of a group by a fixed element of the group. Specifically, if H is a subgroup of G, and g is an element of G, then the left coset of H with respect to g is:

( gH = { g cdot h mid h in H } )

Similarly, the right coset of H with respect to g is:

( HG = { H cdot G mid H in H } )

Visualize this with a simple SVG example. Let's say ( o ) is the identity element, and H is a subgroup containing the elements ( { o, a, b } ). If g is an element from G that is not in H, and we multiply each element in H by g, we generate a new set of elements:

Hey A B

Here, gH = { g cdot o, g cdot a, g cdot b } creates a new set in G that denotes the left coset of H by g.

Properties of cosets

An interesting aspect of cosets is that they partition the group into disjoint subgroups of equal size. Below are some fundamental properties:

  • The cosets of a subgroup H in G are either equal or disjoint.
  • All cosets of a subgroup H have the same cardinality (size), which is equal to the cardinality of H

Let us understand this property with a text example:

For a group G = { e, a, b, c } and a subgroup H = { e, a }, compute the left coset:

eH = { e cdot e, e cdot a } = { e, a }
bH = { b cdot e, b cdot a } = { b, c }
cH = { c cdot e, c cdot a } = { c, b }

Note that bH and cH result in the same set, which shows that these two cosets are equal. Also, every element of G belongs to exactly one coset, which shows how cosets form a partition of the group.

Lagrange's theorem

The Lagrange theorem is a fundamental result in group theory, providing insight into the relationship between a group and its subgroups. It states:

If G is a finite group and H is a subgroup of G, then the order (number of elements) of H divides the order of G

Expressed mathematically:

|g| = n times |h|

where |G| is the order of the group, |H| is the order of the subgroup, and n is the number of distinct cosets of H in G (also called the index of H in G).

Example of Lagrange's Theorem

Consider the symmetric group S_3, which consists of all possible permutations of a three-element set. It has the following elements:

S_3 = {e, (12), (13), (23), (123), (132) }

The order of S_3 is 6. Let us take H = { e, (12) } as a subgroup of S_3:

|h| = 2

According to Lagrange's theorem, the cosets of H must partition S_3. The possible cosets can be calculated as follows:

eH = { e cdot e, e cdot (12) } = { e, (12) }
(13)H = { (13) cdot E, (13) cdot (12) } = { (13), (132) }
(23)H = { (23) cdot E, (23) cdot (12) } = { (23), (123) }

The three cosets eH, (13)H, and (23)H are clearly distinct and cover all elements of S_3, which shows the truth of Lagrange's theorem.

Conclusion

The concepts of cosets and the Lagrange theorem play an important role in understanding the structure of groups in algebra. Cosets provide a way to divide groups into uniform and structured subgroups, and the Lagrange theorem provides a powerful tool for analyzing the relationship between a group and its subgroups. These ideas provide the basis for more advanced topics in algebra and have wide applications in physics, cryptography, and other fields.


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