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 ↓


Sylow Theorems


The Sylow theorems are a set of powerful results in group theory, a branch of algebra that studies algebraic structures known as groups. These theorems apply to finite groups by relating the number of subgroups of a particular size to the order of the group. These theorems provide important insights into the structure of groups. Named after mathematician Ludwig Sylow, these theorems play a key role in understanding how groups behave, helping mathematicians to study and classify them more effectively.

Introduction to the group

Before diving into the Sylow theorems, it is necessary to understand some basic concepts about groups. A group is a set of elements, combined with an operation that satisfies four fundamental properties: closure, associativity, identity and invertibility. Let us discuss these properties briefly:

  • Closure: For any two elements a and b in the group, the result of the operation, a * b, is also in the group.
  • Associativity: For any three elements a, b, and c in a group, (a * b) * c = a * (b * c)
  • Identity element: There exists an element e in the group such that for any element a in the group, a * e = e * a = a.
  • Invertibility: for every element a in the group, there exists an element b such that a * b = b * a = e, where e is the identity element.

Order of groups and subgroups

The order of a group is the number of elements in the group. A subgroup is a subgroup of a group that is itself a group under the same operation. The order of a subgroup must divide the order of the whole group. This essential fact is a consequence of Lagrange's theorem, which is a fundamental result in group theory.

Lagrange's theorem

The Lagrange theorem states that for any finite group G, the order of every subgroup H of G divides the order of G. More formally, if |G| is the order of the group and |H| is the order of the subgroup, then |H| is a divisor of |G|.

|g| = n * |h|

where n is an integer. This theorem is fundamental in group theory because it provides information about the possible sizes of subgroups within a given group.

Sylow's theorem

The Sylow theorems refine our understanding of subgroups of a group by focusing on subgroups whose orders are powers of a prime number. Specifically, if G is a finite group and p is a prime number, then the Sylow p-subgroups of G are They are subgroups whose order is a power of p. These theorems help us determine the number and structure of such subgroups.

Sylow's first theorem

Sylow's first theorem guarantees the existence of p-subgroups. It states that if G is a group of order |G| = p^n * m, where p is a prime number, p^n divides |G|, and m is not divisible by p, then G contains at least one subgroup of order p^n.

To see this, consider a group G of order 12 such that |G| = 12 we have 12 = 2^2 * 3. By Sylow's first theorem, this group has order 4 (2^2). There must be a subgroup of 3 and another subgroup of order 3 (3^1).

Sylow's second theorem

Sylow's second theorem discusses the conjugacy of Sylow p-subgroups, which asserts that any two Sylow p-subgroups of a group are conjugate to each other. In simple terms, this means that if you take any two subgroups of a group, then If one chooses any two Sylow p-subgroups, there is an element of the group that can transform one into the other via conjugation.

This theorem indicates that all Sylow p-subgroups are structurally identical, which makes them distinct up to internal transformations by the elements of the group. Let's look at an example in the symmetric group S_3, which consists of all permutations of three elements. The subgroup of S_3 The order is 6, consisting of the elements `{e, (12), (13), (23), (123), (132)}`.

|S3| = 6 = 2^1 * 3^1

The group contains a Sylow 2-subgroup of order 2 ({e, (12)}) and a Sylow 3-subgroup of order 3 ({e, (123), (132)}). According to Sylow's second theorem, all subgroups generated by a cycle of length 3 are conjugate.

Sylow's third theorem

The third theorem addresses the number of Sylow p-subgroups. It states that if n_p is the number of Sylow p-subgroups of G, then n_p ≡ 1 (mod p) and n_p divides m (the expression |G| = p^n * m).

In simple terms, the number of Sylow p-subgroups divided by p must leave a remainder of 1, and it must also divide the cofactor m. For example, consider a group of order 12. We find that it has a finite number of order 4. There must be at least one subgroup of order 3 and another subgroup of order 3. By Sylow's third theorem, the number of such subgroups must be a divisor of 3 (because 12 / 4 = 3) and 1 modulo 2 for p = 2 and p = 1 modulo 2 for p = 3 must be congruent to 1 modulo 3.

Examples and applications

Let us examine some practical applications and examples of the Sylow theorems:

Example 1: Order 12 group

Consider a group G of order 12. We wish to identify the Sylow subgroups for the prime numbers involved in the factorization of 12: 2 and 3.

  • Sylow 2-subgroups: Since 12 = 2^2 * 3, m = 3 we have one or three Sylow 2-subgroups, since 1 and 3 divide 3, and 1 modulo 2 is 1. In practice there are 3 Sylow subgroups. Verifying 2-subgroups aligns with isomorphisms.
  • Sylow 3-subgroups: Again, 12 = 3^1 * 4. We have one or four Sylow 3-subgroups (1, 3 divided by 12/3 = 4, and 1 modulo 3 is 1). In general However, we find by isomorphism a Sylow 3-subgroup.

Example 2: The symmetric group S_4

Investigate the symmetric group S_4, which has order 24. Its factorization is 2^3 * 3. Finding the Sylow subgroups involves:

  • Sylow 2-subgroups: n_2 divides 3 and is equivalent to 1 modulo 2, which suggests three Sylow 2-subgroups.
  • Sylow 3-subgroups: n_3 divides 8 and is equivalent to 1 modulo 3, suggesting four Sylow 3-subgroups.

Through exploration, four Sylow 3-subgroups and three Sylow 2-subgroups were found.

Conclusion

The Sylow theorems provide profound insights into group theory. By understanding and applying these theorems, mathematicians can determine the existence, number, and structure of Sylow p-subgroups within any finite group. This powerful toolkit can be used to solve complex algebraic problems. It is important to elaborate and deepen our knowledge of group structures.

Summary

Summarize the advantages of the Sylow theorems:

  • Sylow's First Theorem guarantees the existence of a subgroup of order p^n.
  • Sylow's second theorem ensures that these subgroups are conjugate.
  • Sylow's third theorem gives the number of Sylow p-subgroups whose remainder is 1 when m divided by p.

Interactive example using code

Let us consider examples of visualization and calculations.

We can programmatically evaluate the group order and validate subgroups via computational methods:

# Consider group order and primes:
group_order = 12
prime = 2

# Calculate the possible number of Sylow p-subgroups
possible_NP = [d for d in range(1, group_order + 1) if group_order % (prime**d) == 0]

for NP in possible_NP:
    if (np * prime**d) % 2 == 1:
        print(f"Possible: {np} Sylow {prime}-subgroups")

Such calculations validate the results obtained from the Sylow theorems. This introspective journey allows an accurate visualization of group characteristics through computing powers and analogies.


PHD → 1.1.7


U
username
0%
completed in PHD

← Prev (1.1.6)
Group Homomorphisms
Next (1.1.8) →
Free Groups

Comments 



Sylow Theorems

https://www.buddymath.com/phd/1.1.7?bmal=en