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 ↓


Normal Subgroups


Group theory is a branch of mathematics that studies algebraic structures known as groups. A subgroup is a group contained within a group that itself forms a group under the same operation. A normal subgroup is a particularly interesting type of subgroup. Let's take a deeper look at understanding normal subgroups, how they work, and their importance in group theory.

Definition of normal subgroups

In group theory, a subgroup H of a group G is called a normal subgroup if it is invariant under conjugation by elements of G. This means that for every element h in H and every g in G, the element ghg-1 is also in H.

H is a normal subgroup of G if ∀ g ∈ G and ∀ h ∈ H, ghg -1 ∈ H.

This property can be briefly expressed as follows:

If N is a normal subgroup, we write N ⊲ G

Visual representation

Yes H

In this illustration, the large circle represents the group G, and the small circle inside it represents the subgroup H. If H is a normal subgroup, then its position within G remains unchanged despite any internal changes made by G.

Mathematical properties

Normal subgroups have several interesting mathematical properties that make them fundamental in group theory:

1. Quotient group

If N is a normal subgroup of G, we can form the quotient group G/N. The elements of this group are the cosets of N in G. Cosets are a form of partition that uses subsets to divide the entire group into sets, each of which contains elements that are naturally related under the group operation.

2. Kernel of homeomorphism

The subgroup N is normal and is the kernel of the group homomorphism. The kernel is the set of elements that map to the identity under the homomorphism. Since every kernel forms a normal subgroup, the analysis of homomorphisms is another way to understand normal subgroups.

For homomorphism φ: G → G', kernel is Ker(φ) = { g ∈ G | φ(g) = e' }, where e' is the identity in G'.

3. Commutator subgroup

The commutator subgroup, or derived subgroup, of a group G is the subgroup generated by all commutators. It is always a normal subgroup of G. This is useful in understanding commutativity in group structure. The derived group often helps in constructing a series of experiments aimed at studying the solvability of groups.

Commutator: [a, b] = aba -1 b -1

Example: Integers under addition

Consider the group of integers Z under addition. Every subgroup nZ (multiples of n) is a normal subgroup. To see this, note that for any integer k and subgroup nZ, we have:

k + nZ + (-k) = nZ

This is equivalent to nZ, which shows invariance under conjugation.

Example: Symmetric group

Let's look at the symmetric group S3 which consists of all permutations of three objects. Consider A3, which is an alternating group consisting of all even permutations in S3. A3 is a normal subgroup of S3. If g is an even permutation and we conjugate by any h in S3, the result remains an even permutation and is thus within A3.

Importance of normal subgroups

Normal subgroups play an important role in understanding the structure of groups. Here are some key aspects:

  • Factorization: Normal subgroups allow the construction of factor groups, which are helpful in simplifying the structure of larger groups.
  • Group extensions: They are important in the formation of new groups through extensions, which produce more complex groups from known groups.
  • Classification: The classification of groups often involves the analysis of common subgroups and their interactions within a given group.

Understanding normal subgroups is essential to in-depth study of group theory and has implications for a range of mathematical topics, including algebraic topology, geometry, and number theory.

Conclusion

Normal subgroups are a fundamental concept in abstract algebra, providing deep insight into the behavior and structure of groups. Through the lens of normality, mathematicians can better understand group operations, create quotient groups, and ultimately untangle the complex web that makes up modern group theory.

By developing a strong understanding of these fundamental building blocks, one can explore rich and complex mathematical landscapes with a strong foundational knowledge that connects disparate areas of mathematics into a unified whole.


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Normal Subgroups

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