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 ↓


Simple and Solvable Groups


There are many interesting concepts and structures in mathematics, especially in the field called group theory. Two important types of groups are simple groups and solvable groups. These concepts help in understanding the complexity and behavior of mathematical groups.

Introduction to the group

A group is a set combined with an operation that satisfies four main conditions known as group axioms: closure, associativity, identity, and invertibility. Formally, a group (G, *) consists of a set G and a binary operation * such that:

  • Closure: for every a, b in G, a * b is also in G
  • Associativity: for every a, b, c in G, (a * b) * c = a * (b * c)
  • Identity: There exists an element e in G such that for every element a in G, e * a = a * e = a.
  • Invertibility: for every element a in G, there exists an element b in G such that a * b = b * a = e.

Simple group

A simple group is a non-trivial group whose only normal subgroups are the trivial group and the group itself. This means that a simple group cannot be broken down into smaller groups via the operation of taking normal subgroups, making it an essential building block in group theory.

Understanding normal subgroups

A subgroup N of a group G is known as a normal subgroup if it remains unchanged under conjugation by members of G. Formally, for every n in N and every g in G, the element g * n * g -1 is still in N

Example of a simple group: alternating group A5

A classic example of a simple group is the alternating group A 5, which is the group of even permutations of five elements. Let's describe what a permutation is and explore this concept:

Permutation

A permutation of a set is a rearrangement of its elements. For example, if we have a set {1, 2, 3}, then some of its permutations are:

{1, 2, 3} → {2, 1, 3}
{1, 2, 3} → {3, 2, 1}

Even permutation

A permutation is even if it can be expressed as an even number of transpositions (a transposition is a simple swap of two elements). For example, the permutation (1 2 3 4 5) → (2 1 3 4 5) is a transposition and is, therefore, even.

Optional group A5

The group A 5 is composed of all even permutations of five elements. This group is unique in being the smallest non-abelian simple group.

A5

Solvable groups

A solvable group is a group that can be broken down into simpler groups using a series of subgroups, known as the subnormal series, each of which is a normal subgroup of the one before it, and such that the quotient groups are all abelian (commutative groups).

Understanding subnormal series

A subnormal chain is a sequence of subgroups {e} = G 0 ⊲ G 1 ⊲ ⋯ ⊲ G n = G where each G i is a normal subgroup of G i+1. A group G is solvable if all quotient groups G i+1 / G i are abelian.

Example of a solvable group: the symmetric group S3

The symmetric group S 3, which consists of all permutations of three elements, is a simple example of a solvable group. It can be represented using a subgroup series where each quotient is abelian:

  • Start with the trivial subgroup {e}.
  • Next, include the subgroup formed by the identity and a single transposition, which is itself normal.
  • Finally, group S 3 is also added.

The key property here is that the quotient groups are all abelian. For S 3, the subnormal series looks like this:

{E} ⊲ Z 3 ⊲ S 3

Interrelationships between simple and solvable groups

One can think of the relationship between simple and solvable groups as follows. Essentially, simple groups represent an "atomic" level in the structure of groups. In contrast, solvable groups can be "solved" or broken down into simpler components, all of which are abelian.

Theoretically, every finite group is built from simple groups, making simple groups the building blocks of more complex group structures. Meanwhile, solvable groups indicate the degree of "complexity" of decomposing a group through abelian layers.

Feit–Thompson theorem

An important result in this context is the Feit–Thompson theorem, which states that every finite group of odd order is solvable. This theorem helps to clearly delineate when a group can be broken down into abelian factors versus when its structure remains irreducibly complex.

Why study simple and solvable groups?

The study of these groups provides deep insight into the internal structure of groups. Simple groups, such as the alternating group A 5, cannot be broken down into any smaller groups, making them the fundamental "atoms" of group theory.

On the other hand, solvable groups involve a better understanding of how complex operations can be simplified into abelian properties. In practical applications, these groups can model various real-world systems, such as symmetric arrangements or hierarchical systems within networks.

Application

Both simple and solvable groups find numerous applications in various fields:

  • Coding theory: Solvable groups play a role in error correction and cryptography.
  • Physics: Simple groups help explain particle symmetry.
  • Chemistry: Molecular symmetries are often described using group theoretic principles.
  • Computer science: Algorithms, especially those related to graph theory, often use insights gained from group theory.

Conclusion

Simple and solvable groups are the cornerstones of the study of algebraic structures called groups. While simple groups are complex in their indecomposability, solvable groups provide insight into decomposing such structures. Understanding both of these concepts expands the framework for handling diverse mathematical and real-world problems, making these concepts indispensable in the field of mathematics.

The discovery of simple and solvable groups underlines the beauty of mathematical abstraction and theory, which contributes to progress not only in the field of mathematics but also in various disciplines of science and engineering.


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Simple and Solvable Groups

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