Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Abstract Algebra → Groups ↓


Normal Subgroups


In abstract algebra, group theory is a major area of study. It brings a rigorous approach to understanding symmetry and structure in mathematical systems. At the core of this discussion about groups lies the concept of subgroups, and there is a special kind of subgroup called a "normal subgroup". This is a detailed exploration of normal subgroups, showing their theoretical basis and conceptual importance.

Basic concepts of group theory

Before diving into normal subgroups, let's recap some basic concepts of groups. A group (G) is a set equipped with a binary operation ((cdot)) that satisfies four fundamental properties:

1. Closure: For every ( a, b in G ), the result of the operation ( a cdot b ) is also in ( G ). 
2. Associativity: For every ( a, b, c in G ), ((a cdot b) cdot c = a cdot (b cdot c)). 
3. Identity Element: There exists an element ( e in G ) such that for every element ( a in G ), the equation ( e cdot a = a cdot e = a ) holds. 
4. Inverse Element: For each ( a in G ), there exists an element ( b in G ) such that ( a cdot b = b cdot a = e ), where ( e ) is the identity element.

Subgroups

A subgroup is a subgroup of a group itself that is closed under the group operation and satisfies all group properties. Consider a group ( (G, cdot) ). A subgroup ( H subseteq G ) is called a subgroup if:

1. Closure: For every ( a, b in H ), the result ( a cdot b in H ). 
2. Identity: The identity element ( e ) of ( G ) is in ( H ).
3. Inverses: For every ( a in H ), the inverse ( a^{-1} in H ).

Normal subgroups

Normal subgroups are a special type of subgroup where the left and right cosets of the subgroup in the group are equal. Formally, a subgroup ( N ) of a group ( G ) is normal if it satisfies:

[ gN = Ng text{ for all } g in G ]

This means that for each element of the group, the elements of the subgroup interact symmetrically while forming a coset. If ( N ) is a normal subgroup of ( G ), then we denote it by ( N trianglelefteq G ).

Visualization of cosets and normal subgroups

Let us consider the group (S_3), the symmetric group on three elements, consisting of all permutations of the set {1, 2, 3}.

S_3 = { e, (1 2), (1 3), (2 3), (1 2 3), (1 3 2) }

S_3 has many subgroups. Consider the subgroup ( H = { e, (1 2) } ). We verify ( H trianglelefteq S_3 ) by comparing the left and right cosets:

Left cosets of H: quad gH eH = { e, (1 2) } \ 
(1 2)H = { (1 2), e } quad text{(same as eH)} \ 
(1 3)H = { (1 3), (1 2 3) } \ 
(2 3)H = { (2 3), (1 3 2) } \ 
(1 2 3)H = { (1 2 3), (1 3) } \ 
(1 3 2)H = { (1 3 2), (2 3) } 
Right cosets of H: quad Hg He = { e, (1 2) } \ 
H(1 2) = { (1 2), e } quad text{(same as He)} \ 
H(1 3) = { (1 3), (1 2 3) } \ 
H(2 3) = { (2 3), (1 3 2) } \ 
H(1 2 3) = { (1 2 3), (1 3) } \ 
H(1 3 2) = { (1 3 2), (2 3) }

We see that every left coset ( gH ) has a corresponding right coset ( Hg ). Therefore, ( H ) is a normal subgroup of ( S_3 ).

Algebraic properties and significance

One of the main reasons why normal subgroups are important is that they allow us to create quotient groups. If ( N trianglelefteq G ), then we can define the group of cosets, ( G/N ), using the operation:

(aN) cdot (bN) = (ab)N

The set of all such cosets ( {aN mid a in G} ) forms a group under this operation, the quotient group ( G/N ).

Example of a quotient group

Consider ( G = mathbb{Z} ), the group of all integers under addition, and the subgroup ( N = 2mathbb{Z} ) (all even integers).

Left cosets of ( N ) in ( G ): eg, ( 0 + 2mathbb{Z} = { ldots, -4, -2, 0, 2, 4, ldots } = N ) \ 
( 1 + 2mathbb{Z} = { ldots, -3, -1, 1, 3, 5, ldots } ) \ 
These cosets represent the elements of the quotient group ( mathbb{Z}/2mathbb{Z} ). The operation is addition modulo 2, and this group is isomorphic to ( mathbb{Z}_2 ), the integers modulo 2.

Common subgroup tests

To check whether a subgroup ( N ) of a group ( G ) is normal, you can use the normal subgroup test:

A subgroup ( N ) of ( G ) is normal in ( G ) if and only if for every element ( g in G ) and ( n in N ), the element ( gng^{-1} ) is in ( N ).

gng^{-1} in N quad forall g in G, n in N

Why are normal subgroups important?

Normal subgroups play an important role in the overall structure theory of groups. They allow us to build new groups (quotient groups) from existing groups, making it possible to investigate the properties and characteristics of these groups in depth:

  • Simplification and analysis: By studying quotient groups, we can often break down complex groups into simpler components that are easier to analyze and understand.

  • Symmetry and invariance: Normal subgroups capture the essence of symmetries that remain invariant under group operations, and bridge the gap between abstract algebra and geometric and physical symmetries.

  • Homeomorphisms: Understanding normal subgroups further helps in understanding homeomorphisms (structure-preserving mappings between groups). They directly correspond to the kernels of homeomorphisms, which helps in the first isomorphism theorem.

In conclusion, normal subgroups provide a framework for exploring groups and their complexities more deeply. Understanding this concept expands our ability to work with fundamental algebraic structures and establishes connections to a variety of applications in mathematics and science.


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Normal Subgroups

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