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 ↓


Permutation Groups


Permutation groups are an essential concept in abstract algebra. They form a fundamental part of group theory, a branch of mathematics that explores algebraic structures known as groups. To understand permutation groups, we need to start by understanding what permutations are and then move on to see how they fit into the broader theory of groups.

What is permutation?

A permutation of a set is a rearrangement of its elements. For example, if we have a set of three elements, {1, 2, 3}, then one permutation could be {3, 2, 1}, another could be {2, 1, 3}, and so on. In a permutation the order of the elements matters, while in a set the order does not matter.

Formally, for a finite set with n elements, a permutation is a binary function from the set to itself. The binary function is both injective (one-to-one) and surjective (onto), meaning that it rearranges all the elements of the set without leaving behind or repeating any element.

Notation for permutations

There are many ways to represent permutations. A common way is to use cycle notation. In cycle notation, we write permutations as the product of cycles. A cycle is a permutation of a subset of elements that rotates the elements in a certain orbit.

For example, if we have the permutation (1 3 2), it shows that 1 goes to the position previously occupied by 3, 3 goes to the position previously occupied by 2, and 2 goes to the position previously occupied by 1. This forms a cycle. The element not mentioned in the cycle remains in its original position.

Permutation as a function: σ = 1 -> 3 2 -> 1 3 -> 2 Permutation in cycle notation: (1 3 2)

Here, the permutation replaces 1 with 3, 3 with 2 and 2 with 1.

What is a group?

Before diving specifically into permutation groups, we must understand the mathematical concept of a group. A group is a set combined with an operation that satisfies four fundamental properties:

  1. Closure: for every a, b in the group, the result of the operation (a * b) is also in the group.
  2. Associativity: For every a, b, and c in the group, (a * b) * c = a * (b * c).
  3. Identity element: There is an element e in a group such that for any element a the equation e * a = a * e = a is valid.
  4. Inverse element: For every element a in a group, there exists an element b such that a * b = b * a = e, where e is the identity element.

With these properties, groups form a structure that helps mathematicians understand symmetries, transformations, and many other concepts from a unified perspective.

Permutation group as a group

Now that we understand what permutations and groups are, we can define a permutation group. A permutation group is a group where the elements are permutations of a set, and the group operation is the composition of these permutations.

The combination of two permutations is another permutation. This operation is associative, one is the identity permutation (which leaves all elements in their original positions), and every permutation has an inverse (which is the permutation that reverses the action of the original permutation).

Symmetric group

One of the most important examples of permutation groups is the symmetric group. The symmetric group on n elements, denoted S n, consists of all possible permutations of a set of n elements.

For example, the symmetric group on three elements, S 3, consists of the following permutations of the set {1, 2, 3}:

Identity permutation: ( ) Single transpositions: (12), (13), (23) Cycles of length 3: (123), (132)

This group has a total of 6 elements, which is the factorial of 3, 3!. In general, the number of elements in S n is n!.

Visual example

1 2 3 2 1 3

Cayley's theorem

Cayley's theorem states that every group G is isomorphic to a subgroup of the symmetric group. Essentially, this means that any group can be represented as a permutation group. This is a profound result because it tells us that permutation groups are not just abstract constructions, but instead underlie all of group theory.

Examples and exercises

Let's work through some examples and exercises to understand permutation groups better.

Example 1: Permutation composition

Consider the permutations on the set {1, 2, 3}: σ = (1 2) and τ = (2 3). To find the structure σ ∘ τ, we first apply σ to τ and then to each element of the set.

Element Apply τ Apply σ
1 1 2
2 3 3
3 2 1 
σ ∘ τ = (1 3 2)

The resulting permutation is equivalent to moving from 1 to 3, 3 to 2, and 2 to 1, forming the cycle (1 3 2).

Example 2: Inverse of a permutation

Let's find the inverse of the permutation (1 3 2 4). Remember that the inverse of a permutation undoes the permutation.

Permutation: 1 -> 3 3 -> 2 2 -> 4 4 -> 1 Inverse permutation: 1 -> 4 4 -> 2 2 -> 3 3 -> 1

The inverse can be written in cycle notation as (4 1 3 2).

Example 3: The symmetric group S 4

Consider the symmetric group S 4, which is the group of all permutations of the set {1, 2, 3, 4}. This group has 24 (4!) permutations. Here are some examples:

Identity: () Two-cycles: (1 2), (1 3), (1 4), (2 3), (2 4), (3 4) Three-cycles: (1 2 3), (1 3 4), (2 3 4), ... Four-cycle: (1 2 3 4)

Every permutation in S 4 can be expressed as a product of disjoint cycles, or as a product of transpositions (two-cycles).

Properties of permutation groups

Even and odd permutations

A permutation can be classified as even or odd depending on the number of transpositions required to express it. A permutation is even if it can be expressed as the product of an even number of transpositions, and odd if it requires an odd number.

Alternative groups

The alternating group, denoted A n, is the group of all even permutations of a set with n elements. It is a subgroup of the symmetric group S n and has n! /2 elements.

For example, in S 3, the alternating group A 3 contains the following permutations:

Identity: () Three-cycles: (1 2 3), (1 3 2)

Conclusion

Permutation groups are a fascinating and rich area of study within abstract algebra. By studying how permutations can be structured into groups, we gain insight into many areas of mathematics and practical applications that require understanding symmetric and combinatorial structures. Whether it is a symmetric or alternating group, the conceptual power of each permutation group helps us understand complex patterns and operations of elements in a well-defined mathematical setting.

With permutation groups, the ways of further exploration are limitless, touching deep topics such as algebraic topology, field theory, and even cryptographic algorithms. Mastering these concepts provides a solid foundation for understanding advanced mathematical theories.


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Permutation Groups

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