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 ↓


Cyclic Groups


In the field of abstract algebra, one of the fundamental concepts studied is the concept of a group. There are many types of group classifications contained under this concept, one of which is the cyclic group. Cyclic groups are not only a foundational element but also an important step towards understanding more complex structures in algebra. This lesson is going to explore cyclic groups in depth, starting with their basic definitions and properties, moving through explanations and examples, and thus building a strong understanding of these groups.

What is a cyclic group?

A cyclic group is defined as a group that can be generated by a single element. In formal terms, a group ( G ) is cyclic if there exists an element ( g ) in ( G ) such that every element in ( G ) can be expressed as a power of ( g ) (using the group operation). In other words, every element in the group is of the form ( g^n ) for some integer ( n ).

Mathematically, we say that a group ( G ) is cyclic if:

G = { g^n  |  n in mathbb{Z} }

Here, ( mathbb{Z} ) denotes the set of all integers, both positive and negative, including zero.

The element ( g ) is called the generator of the group. If such a generator exists, then the cyclic group is represented by ( langle g rangle ).

Properties of cyclic groups

Cyclic groups have some interesting and useful properties that distinguish them from other types of groups.

1. Every subgroup of a cyclic group is cyclic:

If ( G ) is a cyclic group generated by ( g ), and ( H ) is a subgroup of ( G ), then ( H ) is also cyclic. Moreover, if ( H ) is a non-trivial subgroup (containing only elements other than the identity), then ( H ) will have an element ( h ) of minimal positive order, which can then generate all of ( H ).

2. Cyclic groups are either infinite or have finite order:

If ( G ) is cyclic and ( g ) is a parent of ( G ), then there are two possibilities:

  • ( G ) is infinite, and the elements are of the form ( g^n ) for all integers ( n ).
  • ( G ) is finite, which means that there is a smallest positive integer ( n ) such that ( g^n ) is the identity element. This is the order of the group ( n ).

3. Every infinite cyclic group ( mathbb{Z} ) is isomorphic to:

Infinite cyclic groups are structurally similar to ( mathbb{Z} ), meaning that they can be mapped binaryly in a way that preserves the group structure. Thus, if ( G ) is an infinite cyclic group generated by ( g ), then the mapping defined by ( g^n longrightarrow n ) is an isomorphism.

4. Finite cyclic groups ( mathbb{Z}_n ) are isomorphic to:

A finite cyclic group of order ( n ) is isomorphic to ( mathbb{Z}_n ), the group of integers in modulus of ( n ). The operation is addition in modulus of ( n ), and the elements are equivalence classes ( {0, 1, 2, ..., n-1} ).

Examples and visualizations

Example 1: Infinite cyclic group

Consider the integers under addition, ( mathbb{Z} = { ..., -3, -2, -1, 0, 1, 2, 3, ... } ). This forms a cyclic group under the operation of addition, with 1 generator.

G = langle 1 rangle = { ..., -3cdot1, -2cdot1, -1cdot1, 0cdot1, 1cdot1, 2cdot1, 3cdot1, ... }

We can also represent this group visually using a simple path diagram:

-3 -2 -1 0 1 ,

Example 2: Finite cyclic group

Consider the group ( mathbb{Z}_4 = {0, 1, 2, 3} ) under addition modulo 4. In this group, 1 is a generator because:

1 + 1 = 2  (text{mod}  4) 1 + 1 + 1 = 3  (text{mod}  4) 1 + 1 + 1 + 1 = 0  (text{mod}  4)

Visually, this group can be represented as a circle:

0 1 2 3

Example 3: Small finite group from geometry

Consider the group formed by the rotations of a square (by increments of 90 degrees). This is a cyclic group of order 4, since any rotation can be represented by 0 degrees, 90 degrees, 180 degrees, or 270 degrees.

{e, r, r^2, r^3},  where  r^4 = e

Here, ( e ) is the identity rotation (0 degrees), and ( r ) represents a 90-degree rotation.

R R² R³ E

Many-to-one correspondence and generator specification

An important aspect of cyclic groups is the concept of generators. While a cyclic group may have multiple generators, all generators are related by a cycle or order relation with the size of the group.

In a finite cyclic group of order ( n ), the generators are precisely the elements of order ( n ). In particular, an element ( g^a ) is a generator if and only if ( gcd(a, n) = 1 ), where ( gcd ) denotes the greatest common divisor.

For example, in ( mathbb{Z}_6 = {0, 1, 2, 3, 4, 5} ), the generators are 1 and 5. This is because:

gcd(1, 6) = 1  and  gcd(5, 6) = 1

Practical applications of cyclic groups

Cyclic groups play an important role in various fields of mathematics as well as applied areas. Here are some notable applications:

1. Cryptography:

Cyclic groups are the backbone of many cryptographic protocols, including public-key systems. Exponential operations in cyclic groups help to securely encrypt and decrypt messages due to their predictable structure.

2. Computer science:

The management of passwords or keys often relies on cyclic properties of numbers, and security is maximized by taking advantage of these mathematical properties.

3. Number theory:

Cyclic groups help in the analysis of symmetry, modular arithmetic, and solving complex algebraic equations.

Conclusion

Understanding cyclic groups is most fundamental in the study of abstract algebra. They present a perfect blend of simplicity and structural elegance that provides a stepping stone towards mastering more complex algebraic structures. Whether clarifying the concept of symmetry in geometry, aiding in discrete algorithms, or underpinning secure communications in cryptography, cyclic groups form an integral element in both theoretical and applied mathematics.


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Cyclic Groups

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