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 ↓


Basic Properties of Groups


In algebra, the concept of groups is fundamental. Groups are a mathematical structure widely used in abstract algebra. Studying groups helps you understand other structures such as rings, fields, and vector spaces. This discipline has wide applications in various fields such as physics, chemistry, and computer science, providing a consistent way to deal with symmetries and transformations.

Definition of group

A group is a set, G, combined with an operation (*) that satisfies four fundamental properties: closure, associativity, identity, and inverse. These are known as the group axioms.

Group axioms

  • Closure: for all elements a, b in the set G, the result of the operation a * b is also in G
  • Associativity: for all a, b, c in G, the equation (a * b) * c = a * (b * c) is valid.
  • Identity element: There exists an element e in G such that the equation e * a = a * e = a holds for all elements a of G
  • Inverse element: for every a in G, there exists an element b in G such that a * b = b * a = e, where e is the identity element.

Visualization of basic properties

Let's look at some of these properties using simple pictures.

Finale idea

A B operation(a*b) Now

This diagram shows the closure. If you have two elements a and b from the set, then combining them by the binary operation * yields another element ab that is also in the set.

Text example: closing

Let us consider the integers under addition. The set of integers (..., -3, -2, -1, 0, 1, 2, 3, ...) is a group. For any two integers a and b, the sum a + b is also an integer. Thus, this set is closed under addition.

Visualization of affiliations

A B C (a * b) * c = a * (b * c)

The visualization shows that no matter how you perform the pairing operations, in a group, the result is the same. It doesn't matter if you first add a and b, then multiply the result by c; or if you first add b and c, then multiply by a.

Textual example: association

Consider again the integers under addition. Here, for any integers a, b, and c, it is valid that (a + b) + c = a + (b + c) For example, if a = 1, b = 2, and c = 3, then both ways of grouping give the result 6: 

        (1 + 2) + 3 = 3 + 3 = 6
        1 + (2 + 3) = 1 + 5 = 6

Visualization of the identity element

A E (ident) A

The identity element e acts like a "do-nothing" element when applied to any element a. This means a * e = e * a = a.

Textual example: identity element

In the group of integers under addition, the identity element is 0 because adding 0 to any integer a gives a itself. For example, 5 + 0 = 5 and 0 + 5 = 5.

Visualization of inversions

A b (the inverse of a) A B E

Every element a has an inverse b, so that their operation yields the identity element e. This is represented as a * b = b * a = e.

Textual example: inverse

In integers under addition, the inverse of any integer a is -a because a + (-a) = 0, which shows that every element has an inverse. For example, the inverse of 3 is -3 because 3 + (-3) = 0.

Special types of groups

Abelian group

If the operation is commutative for all elements, the group is called an abelian (or commutative) group. This means that for all a, b in G, the equation a * b = b * a is valid.

Example: The group of integers including addition is an abelian group because for any integers a and b, a + b = b + a.

Non-abelian groups

In contrast, a non-abelian group is one where the operation is not necessarily commutative. This means that there exist elements a, b in G such that a * b ≠ b * a.

Example: Consider the group of 2x2 invertible matrices under multiplication. In general, matrix multiplication is non-commutative, which makes this group non-abelian.

Subgroups

A subgroup is a subgroup of a group that is itself a group with the same operation. If H is a subgroup of a group G, then every element of H is also in G, and H satisfies the group properties within itself.

Example of a subgroup

Consider the group of integers Z under addition. The even integers 2Z form a subgroup of Z because they satisfy all the group axioms: closure, associativity, identity (0 is an even number), and inverse (the inverse of an even number is also even).

Cyclic groups

A group is cyclic if it can be generated by a single element. This means that every element in the group can be expressed as powers (or multiples) of this single element, called the generator.

Example of a cyclic group

The integers Z are a cyclic group, with 1 as a generator, since every integer can be written as a multiple of 1: 

        0 = 0 * 1, 1 = 1 * 1, 2 = 2 * 1, -1 = -1 * 1, etc.

Conclusion

Understanding the basic properties of groups is essential to delve deeper into algebra and solve complex problems in various scientific fields. Groups provide a framework for symmetry and help us understand the nature of mathematical operations on sets. Whether dealing with symmetry in molecules or cryptographic keys in computer security, the study of groups provides valuable insights and tools for these fields.


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Basic Properties of Groups

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