Discrete Maths: Algebraic Structures

Contents

**Define and classify the Binary Operation** 1

**Recall the following terms: Algebraic Structure, Semi-groups, Monoid, Group etc.** 2

**Verify various properties of the Group** 4

**Define Order of an element in a Group** 5

**Define Subgroup and explain its properties** 6

**Explain the following terms: Normal Subgroup, Cosets, Cyclic group, Homomorphism, and Isomorphism** 8

**Verify various properties of Subgroups** 11

**Recall Fundamental Theorem of Arithmetic** 13

**Find the Prime Factors of a given positive integer number** 14

**Define the term Permutation** 15

**Recall the properties of Permutation** 17

**Recall Group of Permutation** 18

**Describe the Cyclic Permutations or Cycle and its type** 18

**Define and differentiate Even Permutation and Odd Permutation** 18

**Recall the Ring and its properties** 18

**Verify the given Algebraic Structure is a Ring** 18

**Recall the following terms: Zero Divisor, Cancellation Laws, Integral Domain, etc** 18

**Recall the Field and its properties** 18

**Verify the given Algebraic Structure is a Field** 18

**Define and classify the Binary Operation**

In discrete mathematics, a binary operation is an operation that combines two elements (operands) to produce a single result. The term “binary” refers to the fact that it operates on a pair of elements. Binary operations are fundamental in various mathematical structures, including sets, groups, rings, and fields.

A binary operation is typically denoted by a symbol, such as “+”, “−”, “×”, “÷”, or “*”, which represents the operation itself. The two operands are usually written as a pair of elements on which the operation is applied.

Binary operations can be classified into several types based on their properties and the mathematical structure in which they are used. Here are some common classifications of binary operations:

- Commutative: A binary operation is commutative if the order of the operands does not affect the result. In other words, for any elements a and b, a binary operation satisfies a * b = b * a. Examples include addition (+) and multiplication (×) of real numbers.
- Associative: A binary operation is associative if the grouping of the operands does not affect the result. In other words, for any elements a, b, and c, a binary operation satisfies (a * b) * c = a * (b * c). Examples include addition (+) and multiplication (×) of real numbers.
- Distributive: A binary operation is distributive if it interacts with another operation in a specific way. For example, multiplication (×) is distributive over addition (+) since a * (b + c) = (a * b) + (a * c) for any elements a, b, and c.
- Identity Element: A binary operation has an identity element if there exists an element, denoted by “e” or “1”, such that a * e = a and e * a = a for any element a. Examples include addition (+) with the identity element 0 and multiplication (×) with the identity element 1.
- Inverse Element: A binary operation has an inverse element if for every element a, there exists an element b such that a * b = e and b * a = e, where e is the identity element. Examples include subtraction (−) with the inverse element of additive inverse and division (÷) with the inverse element of reciprocal.

These classifications help us understand the properties and behaviors of binary operations in different mathematical structures and aid in the study of algebraic systems.

**Recall the following terms: Algebraic Structure, Semi-groups, Monoid, Group etc.**

Here are the definitions of the terms you mentioned:

- Algebraic Structure: An algebraic structure, also known as an algebraic system, refers to a set equipped with one or more operations that satisfy certain properties. It provides a framework for studying mathematical objects and their properties. Examples of algebraic structures include groups, rings, fields, and vector spaces.
- Semi-group: A semi-group is an algebraic structure consisting of a set together with an associative binary operation. In a semi-group, the operation combines two elements of the set to produce another element of the set. However, a semi-group does not necessarily have an identity element or inverse elements. Example: The set of non-negative integers under addition forms a semi-group.
- Monoid: A monoid is an algebraic structure that extends a semi-group by adding an identity element. It is a set together with an associative binary operation and an element that serves as the identity for the operation. Monoids do not necessarily have inverse elements. Example: The set of non-negative integers under addition forms a monoid with the identity element 0.
- Group: A group is an algebraic structure that extends a monoid by adding inverse elements. It is a set together with an associative binary operation, an identity element, and each element having an inverse element. The inverse element of an element, when combined with the original element using the binary operation, yields the identity element. Example: The set of integers under addition forms a group.
- Abelian Group: An abelian group, also known as a commutative group, is a group in which the binary operation is commutative. In other words, the order of elements does not affect the result of the operation. Example: The set of integers under addition forms an abelian group.

These terms are fundamental in algebra and provide a framework for studying various mathematical structures and their properties.

Here are examples of each term:

- Semi-group:
- The set of positive integers under addition. The operation of addition is associative, but there is no identity element or inverse elements.
- The set of all non-empty strings under concatenation. The operation of concatenation is associative, but there is no identity element or inverse elements.

- Monoid:
- The set of natural numbers (including zero) under addition. The operation of addition is associative, and the identity element is 0. However, there are no inverse elements.
- The set of all strings over a given alphabet under concatenation. The operation of concatenation is associative, and the identity element is the empty string. Again, there are no inverse elements.

- Group:
- The set of integers under addition. The operation of addition is associative, the identity element is 0, and every integer has an inverse element.
- The set of invertible 2×2 matrices under matrix multiplication. The operation of matrix multiplication is associative, the identity element is the identity matrix, and every invertible matrix has an inverse matrix.

- Abelian Group:
- The set of integers under addition. Addition is both associative and commutative, so it forms an abelian group.
- The set of real numbers (excluding zero) under multiplication. Multiplication is associative and commutative, and the identity element is 1. The non-zero real numbers also have multiplicative inverses, making it an abelian group.

These examples illustrate the concepts of semi-groups, monoids, groups, and abelian groups and how they differ based on the properties of their operations and elements.

**Verify various properties of the Group**

Here are five examples of groups along with the verification of their properties:

- The set of integers under addition:
- Closure property: For any two integers a and b, the sum a + b is also an integer.
- Associative property: For any three integers a, b, and c, (a + b) + c = a + (b + c).
- Identity element: The integer 0 acts as the identity element since for any integer a, a + 0 = 0 + a = a.
- Inverse element: For every integer a, the additive inverse -a exists such that a + (-a) = (-a) + a = 0.
- Commutative property: Addition of integers is commutative, i.e., for any two integers a and b, a + b = b + a.

- The set of real numbers (excluding zero) under multiplication:
- Closure property: For any two non-zero real numbers a and b, the product a * b is also a non-zero real number.
- Associative property: For any three non-zero real numbers a, b, and c, (a * b) * c = a * (b * c).
- Identity element: The real number 1 acts as the identity element since for any non-zero real number a, a * 1 = 1 * a = a.
- Inverse element: For every non-zero real number a, the multiplicative inverse 1/a exists such that a * (1/a) = (1/a) * a = 1.
- Commutative property: Multiplication of real numbers is commutative, i.e., for any two non-zero real numbers a and b, a * b = b * a.

- The set of 2×2 invertible matrices under matrix multiplication:
- Closure property: The product of any two invertible matrices is also an invertible matrix.
- Associative property: For any three invertible matrices A, B, and C, (A * B) * C = A * (B * C).
- Identity element: The identity matrix acts as the identity element, i.e., for any invertible matrix A, A * I = I * A = A.
- Inverse element: For every invertible matrix A, the inverse matrix A^(-1) exists such that A * A^(-1) = A^(-1) * A = I.
- Non-commutative property: Matrix multiplication is generally non-commutative, i.e., for two invertible matrices A and B, A * B ≠ B * A.

- The set of rational numbers under addition:
- Closure property: For any two rational numbers a and b, the sum a + b is also a rational number.
- Associative property: For any three rational numbers a, b, and c, (a + b) + c = a + (b + c).
- Identity element: The rational number 0 acts as the identity element since for any rational number a, a + 0 = 0 + a = a.
- Inverse element: For every rational number a, the additive inverse -a exists such that a + (-a) = (-a) + a = 0.
- Commutative property: Addition of rational numbers is commutative, i.e., for any two rational numbers a and b, a + b = b + a.

- The set of positive integers under multiplication:
- Closure property: For any two positive integers a and b, the product a * b is also a positive integer.
- Associative property: For any three positive integers a, b, and c, (a * b) * c = a * (b * c).
- Identity element: The positive integer 1 acts as the identity element since for any positive integer a, a * 1 = 1 * a = a.
- Inverse element: In this case, there is no inverse element for every positive integer since division may not yield a positive integer.
- Commutative property: Multiplication of positive integers is commutative, i.e., for any two positive integers a and b, a * b = b * a.

These examples demonstrate different groups and their properties, including closure, associativity, identity element, inverse element, and commutativity, depending on the specific set and operation defined for each group.

**Define Order of an element in a Group**

In the context of group theory, the order of an element in a group refers to the smallest positive integer n such that raising the element to the power of n gives the identity element of the group. The order of an element is denoted as ord(a), where a is the element of interest.

Formally, let (G, *) be a group and let a be an element of G. The order of a, denoted as ord(a), is defined as the smallest positive integer n such that a^n = e, where e is the identity element of the group G.

The order of an element determines the number of distinct powers of that element before it cycles back to the identity element. For example, in the group of integers under addition, the order of any non-zero integer is infinity because there is no positive power of any integer that yields the identity element (0) in this group.

The concept of order is essential in group theory as it helps classify elements and analyze the structure of groups. It provides insights into the behavior and properties of elements within a group.

Let’s consider the group (Z₇, +), where Z₇ represents the integers modulo 7 (0, 1, 2, 3, 4, 5, 6) and + denotes addition modulo 7.

In this group, let’s take the element a = 3. We can determine the order of a by repeatedly applying the group operation until we reach the identity element (0).

Starting with a = 3:

3 + 3 = 6

3 + 6 = 2

3 + 2 = 5

3 + 5 = 1

3 + 1 = 4

3 + 4 = 0

After 6 iterations, we reach the identity element (0). Therefore, the order of the element 3 in the group (Z₇, +) is 6, denoted as ord(3) = 6.

This means that when we repeatedly add 3 to itself modulo 7, we cycle through all the elements of the group (except for 0) before reaching the identity element (0). The order of an element represents the length of this cycle.

It is important to note that for any element in a group, the order divides the order of the group itself. In this case, the order of the group (Z₇, +) is 7, and the order of the element 3 (ord(3)) is 6, which is a divisor of 7.

**Define Subgroup and explain its properties**

In abstract algebra, a subgroup is a subset of a group that forms a group itself under the same operation. In other words, a subgroup is a smaller group contained within a larger group.

Formally, let (G, *) be a group, and let H be a non-empty subset of G. H is called a subgroup of G if it satisfies the following properties:

- Closure: For any elements a, b in H, the operation a * b is also in H. In other words, if a and b are in H, their product under the group operation is also in H.
- Identity: The identity element of the group G is also in H.
- Inverses: For every element a in H, its inverse (denoted as a⁻¹) is also in H. In other words, for every element a in H, there exists an element b in H such that a * b = b * a = identity.
- Associativity: The operation * is associative on H. For any elements a, b, c in H, the expression (a * b) * c is equal to a * (b * c).

The properties of a subgroup are derived from the properties of the larger group. Some key properties of subgroups include:

- Subgroup is a closed under the group operation: The operation on the larger group restricts to the subgroup, ensuring closure.
- Subgroup has an identity element: The identity element of the larger group is also an element of the subgroup.
- Subgroup has inverses: Every element in the subgroup has an inverse within the subgroup.
- Subgroup inherits associativity: The associative property holds for the subgroup since it is a subset of the larger group.
- Subgroup forms a group: The subset H, with the same operation as the larger group, forms a group on its own with the properties of closure, identity, inverses, and associativity.

Subgroups allow us to study the structure of groups by examining smaller, self-contained groups within them. They provide insights into the symmetry and patterns within groups and are an important concept in group theory.

Let’s consider the group (Z, +) of integers under addition. We can define a subgroup of this group by considering a subset that consists of all the even integers.

H = {2n | n ∈ Z}

To show that H is a subgroup of (Z, +), we need to verify the properties of closure, identity, inverses, and associativity.

- Closure: Take any two even integers a and b from H. Their sum a + b will also be an even integer since the sum of two even integers is always even. Therefore, H is closed under addition.
- Identity: The identity element of (Z, +) is 0. Since 0 is an even integer, it is in H.
- Inverses: For every even integer a in H, its inverse -a is also in H. This is because the negation of an even integer is also even.
- Associativity: Addition is associative in (Z, +), so it remains associative within the subset H.

Since H satisfies all the properties of closure, identity, inverses, and associativity, we can conclude that H = {2n | n ∈ Z} is a subgroup of (Z, +). The subgroup H represents all the even integers, and it forms a self-contained group under addition.

**Explain the following terms: Normal Subgroup, Cosets, Cyclic group, Homomorphism, and Isomorphism**

- Normal Subgroup:

A normal subgroup is a subgroup that is invariant under conjugation by elements of the larger group. In other words, if N is a subgroup of a group G, N is normal if for every element g in G and every element n in N, the conjugate gng⁻¹ is also in N. Notationally, it is denoted as N ◁ G. Normal subgroups play a significant role in quotient groups and factor groups.

Example:

Consider the group G = S₃, the symmetric group of degree 3. The subgroup N = {(), (1 2)} consisting of the identity element and the permutation (1 2) is a normal subgroup of G. This can be verified by checking that for every element g in G and every element n in N, the conjugate gng⁻¹ is also in N.

- Cosets:

Given a subgroup H of a group G, the left coset of H with respect to an element g in G is the set of all products of elements of H with g. Similarly, the right coset of H with respect to g is the set of all products of elements of H with g from the right side. Cosets provide a partitioning of the larger group into distinct sets.

Example:

Consider the group G = Z, the integers under addition, and the subgroup H = 2Z, the even integers. The left coset of H with respect to the integer 3 is {3 + 2n | n ∈ Z}, which consists of all integers that are congruent to 3 modulo 2. The right coset of H with respect to 3 is also the same set.

- Cyclic Group:

A cyclic group is a group that can be generated by a single element, called a generator, under the group operation. The generator is raised to different powers to generate all the elements of the group. Cyclic groups exhibit a cyclic pattern and have a finite or infinite number of elements.

Example:

The group Z₅ = {0, 1, 2, 3, 4} under addition modulo 5 is a cyclic group. It is generated by the element 1. Raising 1 to different powers modulo 5 gives all the elements of the group:

1^0 ≡ 1 (mod 5)

1^1 ≡ 1 (mod 5)

1^2 ≡ 1 (mod 5)

1^3 ≡ 1 (mod 5)

1^4 ≡ 1 (mod 5)

- Homomorphism:

A homomorphism is a mapping between two groups that preserves the group operation. It is a structure-preserving map that takes elements from one group to another while maintaining the group structure. In other words, if (G, *) and (H, ⋅) are two groups, a homomorphism φ: G → H satisfies φ(a * b) = φ(a) ⋅ φ(b) for all elements a, b in G.

Example:

Consider the groups G = (Z, +) and H = (Z₆, +) both under addition. The mapping φ: G → H defined by φ(x) = x mod 6 is a homomorphism. It preserves the group operation, as φ(a + b) = (a + b) mod 6 = (a mod 6) + (b mod 6) = φ(a) + φ(b) for all integers a and b.

- Isomorphism:

An isomorphism is a bijective homomorphism between two groups. It is a special type of homomorphism that preserves not only the group operation but also the structure and properties of the groups. If there exists an isomorphism between two groups, they are considered isomorphic, and the groups are essentially the same under a different labeling or representation.

Example:

The groups G = (Z, +) and H = (2Z, +) are isomorphic. The mapping φ: G → H defined by φ(x) = 2x is an isomorphism. It is a bijective homomorphism that preserves the group operation and structure.

**Verify various properties of Subgroups**

Here are five examples of subgroups with their corresponding properties:

Example 1: The group G = (Z, +) of integers under addition

Subgroup: H = {0}

Properties:

- Closure: The sum of any two integers in H is always 0, which is also an integer.
- Identity: The identity element of G, 0, is in H.
- Inverses: The inverse of 0 is also 0, which is in H.
- Associativity: Addition is associative in G, so it remains associative within H.

Example 2: The group G = (Z, +) of integers under addition

Subgroup: H = 5Z = {…, -10, -5, 0, 5, 10, …}

Properties:

- Closure: The sum of any two multiples of 5 in H is always another multiple of 5.
- Identity: The identity element of G, 0, is in H.
- Inverses: For every element x in H, its inverse -x is also in H.
- Associativity: Addition is associative in G, so it remains associative within H.

Example 3: The group G = (R, +) of real numbers under addition

Subgroup: H = (0, ∞) = {x | x > 0}

Properties:

- Closure: The sum of any two positive real numbers is always another positive real number.
- Identity: The identity element of G, 0, is not in H.
- Inverses: H does not contain inverses for all elements since it does not contain negative numbers.
- Associativity: Addition is associative in G, so it remains associative within H.

Example 4: The group G = (Z, +) of integers under addition

Subgroup: H = 2Z = {…, -4, -2, 0, 2, 4, …}

Properties:

- Closure: The sum of any two even integers in H is always another even integer.
- Identity: The identity element of G, 0, is in H.
- Inverses: For every element x in H, its inverse -x is also in H.
- Associativity: Addition is associative in G, so it remains associative within H.

Example 5: The group G = (R, *) of non-zero real numbers under multiplication

Subgroup: H = {1, -1}

Properties:

- Closure: The product of 1 and -1 is -1, which is in H.
- Identity: The identity element of G, 1, is in H.
- Inverses: Every element in H is its own inverse.
- Associativity: Multiplication is associative in G, so it remains associative within H.

These examples demonstrate different subgroups with their respective properties, showcasing the diverse nature of subgroups within various groups.

**Recall Fundamental Theorem of Arithmetic**

The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be uniquely expressed as a product of prime numbers, up to the order of the factors. In other words, any positive integer can be factorized into a unique set of prime numbers.

The theorem can be stated as follows:

For any positive integer n > 1, there exist prime numbers p1, p2, …, pk (where k can be any positive integer) such that:

n = p1 * p2 * … * pk

Furthermore, this factorization is unique up to the order of the prime factors. In other words, if we have two different factorizations of the same number, the prime factors will be the same, although their order might be different.

For example:

- The number 12 can be factorized as 2 * 2 * 3, where 2 and 3 are prime numbers.
- The number 60 can be factorized as 2 * 2 * 3 * 5.
- The number 100 can be factorized as 2 * 2 * 5 * 5.

The Fundamental Theorem of Arithmetic is a key result in number theory and forms the foundation for various concepts and algorithms, such as prime factorization, greatest common divisor calculation, and primality testing.

**Find the Prime Factors of a given positive integer number**

To find the prime factors of a given positive integer number, you can use a process called prime factorization, which involves repeatedly dividing the number by its smallest prime factor until the quotient is no longer divisible by that factor.

In general, to find the prime factors of a positive integer n, you can follow these steps:

- Divide n by the smallest prime number, 2. If n is not divisible by 2, move on to the next prime number (which is 3).
- If n is divisible by the current prime number, divide it by that number and continue dividing by that same prime until it is no longer divisible by that prime.
- Repeat step 2 for the next larger prime number until n is reduced to 1.
- Write down the prime factors that you have used to divide n, along with their powers as needed, to get the prime factorization of n.

Here are five examples of finding the prime factors of given positive integer numbers:

Example 1:

Number: 24

Prime Factors: 2, 3

Explanation: The prime factorization of 24 is 2 * 2 * 2 * 3, where 2 and 3 are prime numbers.

Example 2:

Number: 45

Prime Factors: 3, 5

Explanation: The prime factorization of 45 is 3 * 3 * 5, where 3 and 5 are prime numbers.

Example 3:

Number: 70

Prime Factors: 2, 5, 7

Explanation: The prime factorization of 70 is 2 * 5 * 7, where 2, 5, and 7 are prime numbers.

Example 4:

Number: 96

Prime Factors: 2, 3

Explanation: The prime factorization of 96 is 2 * 2 * 2 * 2 * 2 * 3, where 2 and 3 are prime numbers.

Example 5:

Number: 121

Prime Factors: 11

Explanation: The prime factorization of 121 is 11 * 11, where 11 is a prime number.

These examples illustrate the process of finding the prime factors of given positive integer numbers. By decomposing the number into its prime factors, we can express it as the product of these prime numbers.

**Define the term Permutation**

In mathematics, a permutation refers to an arrangement of objects in a specific order. It is an ordered arrangement of elements, where the order matters. Permutations are often used in combinatorial mathematics and are important in various areas such as probability theory, group theory, and computer science.

Formally, a permutation of a set of distinct elements is a reordering of those elements. For example, if we have a set {1, 2, 3}, the possible permutations of this set are:

{1, 2, 3}

{1, 3, 2}

{2, 1, 3}

{2, 3, 1}

{3, 1, 2}

{3, 2, 1}

Each of these arrangements is considered a different permutation because the order of the elements is different. Permutations can be thought of as rearrangements of objects in a specific order.

The number of permutations of a set with n distinct elements is given by n! (n factorial), which is the product of all positive integers less than or equal to n. For example, for a set of 3 elements, the number of permutations is 3! = 3 * 2 * 1 = 6.

Permutations can be used to solve problems involving arrangements, selections, and orderings, and they have various applications in mathematics, statistics, computer science, and other fields.

Here are some formulas and examples related to permutations:

- Formula for the number of permutations of n objects taken r at a time:

P(n, r) = n! / (n – r)!

This formula calculates the number of permutations when selecting r objects from a set of n objects without repetition.

Example:

Let’s say we have a set of 5 letters {A, B, C, D, E}. We want to find the number of permutations when selecting 3 letters.

P(5, 3) = 5! / (5 – 3)! = 5! / 2! = 5 * 4 * 3 = 60

So, there are 60 different permutations when selecting 3 letters from the given set.

- Formula for the number of permutations of n objects:

P(n) = n!

This formula calculates the total number of permutations of a set of n objects without any restrictions.

Example:

Consider a set of 4 numbers {1, 2, 3, 4}. We want to find the total number of permutations.

P(4) = 4! = 4 * 3 * 2 * 1 = 24

So, there are 24 different permutations of the given set.

- Formula for the number of permutations with repetition:

If there are n objects with repetition, where some objects are identical, the number of permutations is given by:

P(n, n1, n2, …, nk) = n! / (n1! * n2! * … * nk!)

Here, n1, n2, …, nk represent the frequencies of the repeated objects.

Example:

Let’s say we have the word “MISSISSIPPI”. We want to find the number of permutations of the letters in this word.

P(11, 1, 4, 4, 2) = 11! / (1! * 4! * 4! * 2!) = 34,650

So, there are 34,650 different permutations of the letters in the word “MISSISSIPPI”.

These formulas and examples demonstrate how to calculate the number of permutations in different scenarios, considering factors such as selection, repetition, and total objects.

**Recall the properties of Permutation**

Here are some properties of permutations:

- Order Matters: In a permutation, the order of elements is important. Changing the order of elements results in a different permutation.
- No Repetition: In a permutation, each element can only appear once. There are no repetitions of elements within a permutation.
- All Elements Included: A permutation includes all the elements from the original set. None of the elements are left out or added.
- Unique: Each permutation is unique. No two permutations have the same arrangement of elements.
- Total Number of Permutations: The total number of permutations of a set with n distinct elements is given by n!.
- Permutations with Repetition: In some cases, permutations allow for repeated elements. In such cases, the total number of permutations is calculated using the formula n! / (n1! * n2! * … * nk!), where n is the total number of elements and n1, n2, …, nk represent the frequencies of repeated elements.
- Composition of Permutations: Permutations can be composed or combined using the composition operator, which represents performing one permutation followed by another. The composition of permutations is not commutative.
- Inverse of Permutation: Every permutation has an inverse, which, when composed with the original permutation, results in the identity permutation. The inverse of a permutation undoes its effect.

These properties describe the fundamental characteristics and rules associated with permutations. They are essential in understanding and working with permutations in various mathematical and computational contexts.

**Recall Group of Permutation**

In mathematics, a group of permutations, also known as a permutation group, refers to a set of permutations that form a group under composition. A group is a mathematical structure that consists of a set of elements and an operation that satisfies certain properties.

Specifically, a group of permutations is a group where the elements are permutations and the operation is the composition of permutations. The group contains all the possible permutations of a set, and the composition of any two permutations in the group results in another permutation that is also in the group.

The group of permutations has the following properties:

- Closure: The composition of any two permutations in the group results in another permutation that is also in the group.
- Identity Element: The group contains an identity element, which is the permutation that leaves all elements unchanged. Composing any permutation with the identity element does not change the permutation.
- Inverse Element: Every permutation in the group has an inverse, which, when composed with the original permutation, results in the identity permutation.
- Associativity: The composition of permutations is associative, which means that when three permutations are composed together, the order of composition does not matter.

The group of permutations is denoted by the symbol Sn, where n represents the number of elements being permuted. For example, S3 represents the group of permutations of a set with three elements.

The group of permutations has applications in various areas of mathematics, such as group theory, combinatorics, and cryptography. It provides a framework for studying and analyzing the properties and relationships between permutations.

Let’s consider the group of permutations, S3, which represents the permutations of a set with three elements.

The elements of S3 are the six possible permutations of the set {1, 2, 3}:

- Identity permutation: (1, 2, 3)
- Permutation 1: (1, 3, 2)
- Permutation 2: (2, 1, 3)
- Permutation 3: (2, 3, 1)
- Permutation 4: (3, 1, 2)
- Permutation 5: (3, 2, 1)

Now, let’s verify the properties of the group of permutations, S3:

- Closure: Take any two permutations from S3, and their composition will result in another permutation in S3. For example, if we compose Permutation 1 and Permutation 3, we get (1, 3, 2) o (2, 1, 3) = (2, 3, 1), which is also in S3.
- Identity Element: The identity permutation, (1, 2, 3), is an element of S3. Composing any permutation with the identity permutation leaves the permutation unchanged. For example, (2, 3, 1) o (1, 2, 3) = (2, 3, 1).
- Inverse Element: Each permutation in S3 has an inverse. For example, the inverse of Permutation 2, (2, 1, 3), is (2, 1, 3) itself, as composing them gives the identity permutation: (2, 1, 3) o (2, 1, 3) = (1, 2, 3).
- Associativity: The composition of permutations is associative. For example, if we have three permutations: (1, 3, 2), (2, 1, 3), and (3, 1, 2), then ((1, 3, 2) o (2, 1, 3)) o (3, 1, 2) = (1, 2, 3) o (3, 1, 2) = (1, 3, 2).

Thus, the set of permutations S3 forms a group under composition, satisfying all the properties of a group.

**Describe the Cyclic Permutations or Cycle and its type**

In the context of permutations, a cyclic permutation, also known as a cycle, is a permutation that cyclically permutes a subset of elements while leaving the remaining elements unchanged. It represents a circular shifting or rotation of elements within a set.

A cycle is represented using parentheses, where the elements inside the parentheses indicate the cyclic permutation. For example, (1 2 3) represents a cycle that permutes the elements 1, 2, and 3 in a circular fashion.

There are different types of cycles based on their length:

- 2-Cycle or Transposition: A 2-cycle is a cycle that swaps two elements. For example, (1 2) represents a transposition that swaps the positions of elements 1 and 2.
- 3-Cycle: A 3-cycle is a cycle that cyclically permutes three elements. For example, (1 2 3) represents a cycle that shifts the positions of elements 1, 2, and 3 in a circular fashion.
- k-Cycle: A k-cycle is a cycle that cyclically permutes k elements. It represents a circular shifting or rotation of k elements within a set.

Cycles can also be combined to form longer permutations. For example, (1 2 3)(4 5) represents a permutation that cyclically permutes elements 1, 2, 3 while leaving elements 4, 5 unchanged.

Cyclic permutations have interesting properties. Any permutation can be decomposed into a product of disjoint cycles, and the order of a permutation is the least common multiple of the lengths of its cycles.

Cyclic permutations are widely used in group theory and combinatorics to study the properties and structure of permutations and other algebraic structures.

Here are some examples of cyclic permutations:

- (1 2 3): This represents a cyclic permutation of the elements 1, 2, and 3. It cycles through these elements in a circular fashion: 1 becomes 2, 2 becomes 3, and 3 becomes 1.
- (4 7 5): This represents a cyclic permutation of the elements 4, 7, and 5. It cycles through these elements: 4 becomes 7, 7 becomes 5, and 5 becomes 4.
- (2 6): This represents a transposition, or a 2-cycle, that swaps the positions of elements 2 and 6.
- (1 3)(5 2): This represents a permutation that cyclically permutes elements 1 and 3, and elements 5 and 2, while leaving any other elements unchanged. It first performs the cyclic permutation (1 3), which cycles elements 1 and 3, and then performs the cyclic permutation (5 2), which cycles elements 5 and 2.

Cyclic permutations can have different lengths and can involve any set of elements. The length of a cycle corresponds to the number of elements that are cyclically permuted.

**Define and differentiate Even Permutation and Odd Permutation**

Even Permutation:

- Definition: An even permutation is a permutation that can be obtained by an even number of swaps from the original arrangement.
- Parity: Even permutations have an even parity.
- Sign: Even permutations have a positive sign.
- Example: (1 2 3) is an example of an even permutation.

Odd Permutation:

- Definition: An odd permutation is a permutation that can be obtained by an odd number of swaps from the original arrangement.
- Parity: Odd permutations have an odd parity.
- Sign: Odd permutations have a negative sign.
- Example: (2 1 3) is an example of an odd permutation.

In summary, even permutations require an even number of swaps and have an even parity and positive sign, while odd permutations require an odd number of swaps and have an odd parity and negative sign.

Here are some examples to help illustrate the concept:

- The permutation (1 2 3) can be expressed as (1 2) * (2 3), which is a product of two transpositions, so it is an even permutation.
- The permutation (1 3 2) can be expressed as (1 2) * (2 3) * (1 2), which is a product of three transpositions, so it is an odd permutation.
- The identity permutation (1) can be expressed as an empty product of transpositions, so it is an even permutation.

It’s worth noting that the parity of a permutation (i.e., whether it is even or odd) is a fundamental property that is invariant under conjugation by any other permutation. In other words, if you apply any other permutation to a given permutation, its parity does not change.

Here’s a tabular form comparing the differences between even permutations and odd permutations:

Even Permutation |
Odd Permutation |

Requires an even number of swaps | Requires an odd number of swaps |

Parity: Even | Parity: Odd |

Sign: Positive | Sign: Negative |

Example: (1 2 3) | Example: (2 1 3) |

In summary, even permutations require an even number of swaps, have an even parity, and a positive sign. On the other hand, odd permutations require an odd number of swaps, have an odd parity, and a negative sign.

**Recall the Ring and its properties**

A ring is an algebraic structure consisting of a set equipped with two binary operations, usually denoted as addition (+) and multiplication (·), that satisfy certain properties.

Here are the properties of a ring:

- Closure under Addition: For any elements a and b in the ring, the sum a + b is also in the ring.
- Associativity of Addition: Addition is associative, meaning that for any elements a, b, and c in the ring, (a + b) + c = a + (b + c).
- Commutativity of Addition: Addition is commutative, meaning that for any elements a and b in the ring, a + b = b + a.
- Existence of Additive Identity: There exists an element 0 in the ring such that for any element a in the ring, a + 0 = a.
- Existence of Additive Inverse: For every element a in the ring, there exists an element -a in the ring such that a + (-a) = 0.
- Closure under Multiplication: For any elements a and b in the ring, the product a · b is also in the ring.
- Associativity of Multiplication: Multiplication is associative, meaning that for any elements a, b, and c in the ring, (a · b) · c = a · (b · c).
- Distributive Properties: The ring satisfies the distributive laws, which state that for any elements a, b, and c in the ring, a · (b + c) = (a · b) + (a · c) and (a + b) · c = (a · c) + (b · c).

Note: Rings can be further classified as commutative rings if the multiplication operation is commutative, or as non-commutative rings if the multiplication operation is not commutative.

Additionally, a ring may have additional properties such as the existence of multiplicative identity and the cancellation property, which further characterize its structure.

**Verify the given Algebraic Structure is a Ring**

To verify if a given algebraic structure is a ring, we need to check if it satisfies all the properties of a ring. Here are five examples to verify if the given structures are rings:

Example 1:

Algebraic Structure: Integers under addition and multiplication

- Closure under Addition: The sum of any two integers is always an integer.
- Associativity of Addition: Addition of integers is associative.
- Commutativity of Addition: Addition of integers is commutative.
- Existence of Additive Identity: The integer 0 serves as the additive identity.
- Existence of Additive Inverse: For every integer a, the integer -a serves as its additive inverse.
- Closure under Multiplication: The product of any two integers is always an integer.
- Associativity of Multiplication: Multiplication of integers is associative.
- Distributive Properties: The distributive laws hold for integers.

Since all the properties of a ring are satisfied, the algebraic structure of integers under addition and multiplication is a ring.

Example 2:

Algebraic Structure: Rational numbers under addition and multiplication

Similar to the previous example, we can verify that the rational numbers under addition and multiplication satisfy all the properties of a ring.

Example 3:

Algebraic Structure: Real numbers under addition and multiplication

Similarly, the real numbers under addition and multiplication form a ring.

Example 4:

Algebraic Structure: Matrices of a fixed size with real entries under matrix addition and matrix multiplication

We can verify that matrices of a fixed size with real entries satisfy all the properties of a ring under matrix addition and matrix multiplication.

Example 5:

Algebraic Structure: Polynomials with real coefficients under polynomial addition and polynomial multiplication

Similarly, polynomials with real coefficients form a ring under polynomial addition and polynomial multiplication.

In all these examples, the given algebraic structures satisfy all the properties of a ring, thus confirming that they are indeed rings.

To verify if a given algebraic structure is a ring, we need to check if it satisfies all the properties of a ring. Here are five numerical examples to verify if the given structures are rings:

Example 1:

Algebraic Structure: (Z, +, *)

Z represents the set of integers, + represents integer addition, and * represents integer multiplication.

- Closure under Addition: For any two integers a and b, a + b is always an integer.
- Associativity of Addition: Addition of integers is associative.
- Commutativity of Addition: Addition of integers is commutative.
- Existence of Additive Identity: The integer 0 serves as the additive identity.
- Existence of Additive Inverse: For every integer a, the integer -a serves as its additive inverse.
- Closure under Multiplication: For any two integers a and b, a * b is always an integer.
- Associativity of Multiplication: Multiplication of integers is associative.
- Distributive Properties: The distributive laws hold for integers.

Since all the properties of a ring are satisfied, the algebraic structure (Z, +, *) is a ring.

Example 2:

Algebraic Structure: (R, +, *)

R represents the set of real numbers, + represents real number addition, and * represents real number multiplication.

Similar to the previous example, we can verify that the real numbers under addition and multiplication satisfy all the properties of a ring.

Example 3:

Algebraic Structure: (Q, +, *)

Q represents the set of rational numbers, + represents rational number addition, and * represents rational number multiplication.

Similarly, the rational numbers under addition and multiplication form a ring.

Example 4:

Algebraic Structure: (M2(R), +, *)

M2(R) represents the set of 2×2 matrices with real entries, + represents matrix addition, and * represents matrix multiplication.

We can verify that matrices of size 2×2 with real entries satisfy all the properties of a ring under matrix addition and matrix multiplication.

Example 5:

Algebraic Structure: (P(R), +, *)

P(R) represents the set of polynomials with real coefficients, + represents polynomial addition, and * represents polynomial multiplication.

Similarly, polynomials with real coefficients form a ring under polynomial addition and polynomial multiplication.

In all these numerical examples, the given algebraic structures satisfy all the properties of a ring, thus confirming that they are indeed rings.

**Recall the following terms: Zero Divisor, Cancellation Laws, Integral Domain, etc**

- Zero Divisor: In a ring, a zero divisor is a nonzero element that, when multiplied by another nonzero element, results in zero. In other words, if a and b are nonzero elements of a ring and a * b = 0, then a and b are zero divisors.

Example: In the ring of integers modulo 6 (Z6), both 2 and 3 are zero divisors because 2 * 3 ≡ 0 (mod 6).

- Cancellation Laws: In a ring, the cancellation laws state that if a, b, and c are elements of the ring such that a * b = a * c or b * a = c * a, where a is nonzero, then b = c.

Example: In the ring of real numbers (R), if a * b = a * c, where a is nonzero, then b = c. For example, if 3x = 3y, where x and y are real numbers and 3 is nonzero, then x = y.

- Integral Domain: An integral domain is a commutative ring with unity (multiplicative identity) in which there are no zero divisors.

Example: The ring of integers (Z) is an integral domain because it is commutative, has unity (1), and there are no zero divisors. In other words, in Z, if a * b = 0, where a and b are integers, then either a or b must be zero.

- Field: A field is a commutative ring with unity in which every nonzero element has a multiplicative inverse.

Example: The field of rational numbers (Q) is a field because it is commutative, has unity (1), and every nonzero rational number has a multiplicative inverse. For any nonzero rational number a/b, its multiplicative inverse is b/a.

- Integral: An integral is an element of a ring that satisfies the property that if a * b = 0, then either a = 0 or b = 0.

Example: In the ring of polynomials with real coefficients (R[x]), the polynomial x^2 + 1 is an integral because if it is multiplied by any other polynomial and the result is 0, then the other polynomial must be 0.

These terms are fundamental concepts in ring theory and help to define and understand the properties of various algebraic structures.

**Recall the Field and its properties**

A field is an algebraic structure that consists of a set of elements along with two operations, addition and multiplication. It satisfies the following properties:

- Closure under Addition and Multiplication: For any two elements a and b in the field, the sum a + b and the product a * b are also in the field.
- Associativity: Addition and multiplication in a field are both associative operations. That is, for any three elements a, b, and c in the field, (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c).
- Commutativity: Addition and multiplication in a field are commutative operations. That is, for any two elements a and b in the field, a + b = b + a and a * b = b * a.
- Additive Identity: There exists an element 0 in the field such that for any element a in the field, a + 0 = a.
- Additive Inverse: For every element a in the field, there exists an element -a in the field such that a + (-a) = 0.
- Multiplicative Identity: There exists an element 1 in the field such that for any element a in the field, a * 1 = a.
- Multiplicative Inverse: For every nonzero element a in the field, there exists an element a^-1 in the field such that a * a^-1 = 1.
- Distributive Property: For any three elements a, b, and c in the field, the distributive property holds: a * (b + c) = a * b + a * c.

The properties of a field ensure that it is a well-defined and structured mathematical object. Fields are more general than rings and have a higher level of symmetry and structure. Examples of fields include the rational numbers (Q), real numbers (R), and complex numbers (C).

**Verify the given Algebraic Structure is a Field**

To verify whether an algebraic structure is a field, we need to check if it satisfies all the properties of a field. Here are five numerical examples to verify if the given algebraic structures are fields:

Example 1:

Algebraic Structure: Integers modulo 5 under addition and multiplication.

Properties to Verify:

- Closure under Addition and Multiplication: For any two elements a and b in the algebraic structure, the sum a + b and the product a * b should also be in the structure.
- Verify if the integers modulo 5 under addition and multiplication are closed.

- Associativity: Addition and multiplication in the algebraic structure should be associative operations.
- Verify if addition and multiplication of integers modulo 5 are associative.

- Commutativity: Addition and multiplication in the algebraic structure should be commutative operations.
- Verify if addition and multiplication of integers modulo 5 are commutative.

- Additive Identity: There should exist an element 0 in the algebraic structure such that for any element a, a + 0 = a.
- Verify if there is an additive identity element in integers modulo 5.

- Additive Inverse: For every element a in the algebraic structure, there should exist an element -a such that a + (-a) = 0.
- Verify if every element in integers modulo 5 has an additive inverse.

- Multiplicative Identity: There should exist an element 1 in the algebraic structure such that for any element a, a * 1 = a.
- Verify if there is a multiplicative identity element in integers modulo 5.

- Multiplicative Inverse: For every nonzero element a in the algebraic structure, there should exist an element a^-1 such that a * a^-1 = 1.
- Verify if every nonzero element in integers modulo 5 has a multiplicative inverse.

- Distributive Property: For any three elements a, b, and c in the algebraic structure, the distributive property should hold: a * (b + c) = a * b + a * c.
- Verify if the distributive property holds for integers modulo 5.

By verifying these properties for the given algebraic structures, we can determine if they are fields.

Example 2:

Algebraic Structure: Rational numbers (Q) under addition and multiplication.

Properties to Verify:

- Closure under Addition and Multiplication: Rational numbers are closed under addition and multiplication.
- Associativity: Addition and multiplication of rational numbers are associative.
- Commutativity: Addition and multiplication of rational numbers are commutative.
- Additive Identity: The additive identity in rational numbers is 0.
- Additive Inverse: Every rational number has an additive inverse.
- Multiplicative Identity: The multiplicative identity in rational numbers is 1.
- Multiplicative Inverse: Every nonzero rational number has a multiplicative inverse.
- Distributive Property: The distributive property holds for rational numbers.

By verifying these properties, we can conclude that rational numbers form a field under addition and multiplication.

Example 3:

Algebraic Structure: Real numbers (R) under addition and multiplication.

Properties to Verify:

- Closure under Addition and Multiplication: Real numbers are closed under addition and multiplication.
- Associativity: Addition and multiplication of real numbers are associative.
- Commutativity: Addition and multiplication of real numbers are commutative.
- Additive Identity: The additive identity in real numbers is 0.
- Additive Inverse: Every real number has an additive inverse.
- Multiplicative Identity: The multiplicative identity in real numbers is 1.
- Multiplicative Inverse: Every nonzero real number has a multiplicative inverse.
- Distributive Property: The distributive property holds for real numbers.

By verifying these properties, we can conclude that real numbers form a field under addition and multiplication.

Example 4:

Algebraic Structure: Complex numbers (C) under addition and multiplication.

Properties to Verify:

- Closure under Addition and Multiplication: Complex numbers are closed under addition and multiplication.
- Associativity: Addition and multiplication of complex numbers are associative.
- Commutativity: Addition and multiplication of complex numbers are commutative.
- Additive Identity: The additive identity in complex numbers is 0.
- Additive Inverse: Every complex number has an additive inverse.
- Multiplicative Identity: The multiplicative identity in complex numbers is 1.
- Multiplicative Inverse: Every nonzero complex number has a multiplicative inverse.
- Distributive Property: The distributive property holds for complex numbers.

By verifying these properties, we can conclude that complex numbers form a field under addition and multiplication.

Example 5:

Algebraic Structure: Integers (Z) under addition and multiplication.

Properties to Verify:

- Closure under Addition and Multiplication: Integers are closed under addition and multiplication.
- Associativity: Addition and multiplication of integers are associative.
- Commutativity: Addition and multiplication of integers are commutative.
- Additive Identity: The additive identity in integers is 0.
- Additive Inverse: Every integer has an additive inverse.
- Multiplicative Identity: The multiplicative identity in integers is 1.
- Multiplicative Inverse: Not every nonzero integer has a multiplicative inverse. Hence, integers do not form a field.
- Distributive Property: The distributive property holds for integers.

By verifying these properties, we can conclude that integers form a ring under addition and multiplication but not a field since not every nonzero

integer has a multiplicative inverse.