Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Abstract Algebra ↓


Groups


In the field of mathematics, especially in the field of abstract algebra, the concept of a group is one of the fundamental building blocks. This concept is the basis of many structures in algebra and is important in a variety of mathematical disciplines and applications. In its most basic form, a group is a set combined with an operation that satisfies certain conditions. Let us understand this concept in depth and also give ample examples.

Defining the group

A group is a set (G) paired with a binary operation, commonly denoted as (*), that satisfies the following four properties:

  1. Closure: For any two elements (a) and (b) in (G), the result of the operation (a * b) must also be in (G).
  2. Associativity: For any three elements (a, b,) and (c) in (G), the equation ((a * b) * c = a * (b * c)) must be true.
  3. Identity element: There must exist an element (e) in (G) so that the equation (e * a = a * e = a) holds for every element (a) in (G). This (e) is called the identity element.
  4. Inverse Element: For every element (a) in (G) there must exist an element (b) in (G) such that (a * b = b * a = e), where (e) is the identity element. The element (b) is the inverse of (a).

Examples of groups

Integers under addition

Consider the set of integers (mathbb{Z}) with the operation of addition ((+)).

  • Closure: If you take any two integers, the sum is also an integer.
  • Associativity: For any integers (a, b, c), the sum ((a + b) + c = a + (b + c)).
  • Identity element: The integer 0 acts as the identity element because for any integer (a), (a + 0 = 0 + a = a).
  • Inverse element: For any integer (a), the integer (-a) is the inverse because (a + (-a) = (-a) + a = 0).

Thus, ((mathbb{Z}, +)) is a group.

The set of nonzero real numbers under multiplication

Consider the group of non-zero real numbers (mathbb{R}^*) with multiplication ((cdot)).

  • Closure: The product of any two non-zero real numbers is a non-zero real number.
  • Associativity: Multiplication is associative, that is, ((a cdot b) cdot c = a cdot (b cdot c)) for any (a, b, c in mathbb{R}^*).
  • Identity element: The number 1 is the identity because for any (a in mathbb{R}^*), (a cdot 1 = 1 cdot a = a).
  • Inverse element: For any non-zero real number (a), its inverse is (frac{1}{a}) since (a cdot frac{1}{a} = frac{1}{a} cdot a = 1).

Therefore, ((mathbb{R}^*, cdot)) is a group.

Visualization of group operations

To better understand groups, let's imagine a group operation on a simple set:

G = {1, -1}
1 * 1 = 1
1 * -1 = -1
-1 * 1 = -1
-1 * -1 = 1

This set (G = {1, -1}) forms a group with multiplication. The group table (Cayley table) is illustrated above in a simple structure that shows endowment and associativity clearly.

Special types of groups

Abelian group

A group is called abelian if it satisfies the additional property of commutativity:

For all (a, b) in (G), (a * b = b * a).

The group ((mathbb{Z}, +)) is abelian because (a + b = b + a) for any integers (a) and (b).

Cyclic groups

A group is called cyclic if there exists an element (g) in (G) such that every element in (G) can be written as (g^n) for some integer (n). The element (g) is called the generator of the group.

For example, the group (mathbb{Z}) is a cyclic group, generated by 1, since every integer (n) can be expressed as (n times 1).

Groups of permutations

Permutation groups are essential in fields such as combinatorics. They help describe how elements can be rearranged or permuted. Consider a set (S = {1, 2, 3}). The permutation group of (S), denoted by (S_3), consists of all bijections from (S) onto itself.

The collection of all such permutations forms a group under the function composition. Let us list some of these permutations:

  • ((1, 2, 3) to (1, 2, 3)) (identity permutation)
  • ((1, 2, 3) to (2, 1, 3)) (1 and 2 interchanged)
  • ((1, 2, 3) to (1, 3, 2)) (2 and 3 interchanged)

These permutations are produced via combination to form a group operation in (S_3).

Visual example of cyclic groups

Let us consider a cyclic group (C_4) which can be visually represented in a circular way.

* 0 *
 /    
3  0
     /
* 2 *

This illustration shows the elements {0, 1, 2, 3} moving around a circle, where the operation can be thought of as a rotation.

Importance and applications of groups

Groups are not just an abstract concept confined to the confines of theoretical mathematics; they have wide-ranging applications:

  • Symmetry: Groups help us understand symmetry, which is a core concept in physics, chemistry, and art.
  • Cryptography: In cryptography, groups play an important role in developing secure cryptographic methods.
  • Theoretical computer science: automata theory and formal languages using group theory.

Conclusion

Groups are a powerful concept in mathematics that connects a wide range of structures and transformations. Studying them helps to understand deeper algebraic structures and reveals the relationships between different mathematical paths. By exploring groups, you gain insight into the fabric of algebra, strengthening your grasp on symmetry, transformations, and algebraic systems.


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