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 ↓


Normal Subgroups


In the fascinating world of abstract algebra, groups form the backbone of many conceptual frameworks and are central to understanding algebraic structures. In these structures, subgroups play a crucial role, and normal subgroups, in particular, serve as a foundation for further exploration and applications, such as the construction of quotient groups. In this narrative, we aim to comprehensively understand normal subgroups, their properties, significance, and applications.

Basic definitions

A group (G, *) is a set G equipped with a binary operation (*) that satisfies four fundamental properties: closure, associativity, identity, and invertibility. A subgroup H of a group G is a subgroup of G that is itself a group under the operation of G. For a subgroup H to be a subgroup, it must satisfy three conditions: it must contain the identity element of G, it must be closed under the group operation, and it must be closed under taking the inverse.

A subgroup H of G is a normal subgroup if it is invariant under conjugation by elements of G. More formally, H is a normal subgroup of G if, for every element g in G and every element h in H, the element g * h * g -1 is also in H. This property allows the set of left cosets of H in G to coincide with the set of right cosets, which is an intrinsic property of normal subgroups.

Notation:

The notation H ⟲ G is used to show that H is a normal subgroup of G.

Understand with examples

To better understand the concept of normal subgroups, let's look at some examples:

Example 1: Integer groups

Consider the group Z of integers under addition. A subgroup of Z is the group of all multiples of a fixed integer n, denoted by nZ = { ..., -2n, -n, 0, n, 2n, ...} Every subgroup of Z is of this form.

Let us check whether nZ is a normal subgroup. Choose any integer a and any element from nZ, say kn where k is an integer. According to the normal subgroup condition, we must check whether a + kn - a is in nZ:

a + kn - a = kn

Since kn is obviously in nZ, every subgroup nZ of Z is a normal subgroup. Essentially, in deterministic terms, every subgroup of an abelian group such as Z is normal under addition. An abelian group is one where the group operation is commutative.

Example 2: The symmetric group S3

The symmetric group S3, the group of all permutations of three elements, provides a more explicit example. S3 has six elements: the identity permutation e, two 3-cycles (123), (132), and three transpositions (12), (13) and (23)

Consider the subgroup A3 = { e, (123), (132) }, which is the group of even permutations. Let us determine whether A3 is a normal subgroup of S3. For A3 to be normal, for any σ in S3 and τ in A3, the conjugate στσ -1 must also be in A3.

Here is a visual representation of S3 and the subgroup A3:

E (12) (23) (13) (123) (132) A3

This shows that A3 is in fact a normal subgroup since every conjugation of an element in A3 results in another element in A3.

Properties of normal subgroups

Normal subgroups have special and important properties. Some key properties include:

  • If H is a normal subgroup of G, then the left and right cosets of H in G are equal. Thus, left cosets and right cosets are simply called cosets.
  • The factor groups or quotient groups G/H can be formed, where the elements are these cosets.
  • Every normal subgroup is the core of an isomorphism.
  • The intersection of any collection of normal subgroups of G is a normal subgroup of G.
  • Every subgroup of any abelian group is normal, since a * b = b * a for all a, b in the group.

Quotient group

The concept of normal subgroups allows us to define quotient groups. If H is a normal subgroup of G, then the set of cosets of H in G forms a group under the following operation:

(gh)(kh) = (g * k)h

The groups G/H, the quotient groups, have elements that are cosets, and provide a structured way to "split" groups.

Example of a quotient group

Consider the cyclic group of integers modulo 6, denoted as Z6 This group contains the elements {0, 1, 2, 3, 4, 5}. The subgroup H = {0, 3} is a normal subgroup. The quotient group Z6/H contains the following cosets:

{0, 3}, {1, 4}, {2, 5}

Thus, Z6/H is similar to Z3.

Importance of normal subgroups

Normal subgroups play a central role in group theory and have far-reaching implications in mathematics and related fields, including:

  • using them to detect group isomorphisms, since every normal subgroup can be viewed as a core of isomorphisms.
  • Laying a basis for the construction of quotient groups, supporting structures that simplify complex manifolds to groups.
  • Facilitating proofs and derivations in various branches including Galois theory and number theory.

Conclusion

Normal subgroups are fundamental constructs within group theory, whose important applications range from defining quotient groups to representing abstract isomorphic mappings. They demonstrate the beauty of mathematical structure and rigor within abstract algebra, serving as a cornerstone for a variety of advanced applications and concepts. Through examples and properties, we have seen the beauty and necessity of normal subgroups in understanding deeper and broader aspects of algebraic structures.


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