Unit 6: Algebraic Structures

Introduction to Algebraic Structures

Algebraic structures are essential concepts in modern mathematics, providing a framework to study various operations and their properties. They help in understanding how elements within a set interact with each other under a defined operation. In this unit, we will explore key algebraic structures such as Semigroups, Monoids, Groups, Rings, Fields, and their applications. These structures have profound implications in areas like cryptography, coding theory, and computer science.

Semigroup

Definition of a Semigroup

A Semigroup is an algebraic structure consisting of a set $S$ together with a binary operation $*$ that combines two elements to form another element of the set. The operation must satisfy the associativity property, which means that for all elements $a$, $b$, and $c$ in $S$:

However, a semigroup does not require an identity element or inverses for its operation. An example of a semigroup is the set of natural numbers $N$ under addition, where associativity holds, but there is no identity element in the semigroup.

Example

Consider the set $S = \{1, 2, 3\}$ with the operation of multiplication. This set is a semigroup because multiplication is associative:

Thus, the set forms a semigroup under multiplication.

Monoid

Definition of a Monoid

A Monoid is a semigroup with an additional requirement: it must contain an identity element. This identity element, denoted as $e$, satisfies the property that for every element $a$ in the set $M$, the following holds:

In other words, combining any element with the identity element under the operation does not change the element. The set of natural numbers $N$ with addition, and 0 as the identity element, is an example of a monoid.

Example

The set of non-negative integers $M = \{0, 1, 2, \dots\}$ under the operation of addition is a monoid. The number 0 acts as the identity element because:

Group

Definition of a Group

A Group is a monoid with the added property that every element must have an inverse. A group is a set $G$ equipped with a binary operation $*$ such that:

1. Associativity: For all $a, b, c \in G$, $(a * b) * c = a * (b * c)$.
2. Identity Element: There exists an element $e \in G$ such that $a * e = e * a = a$ for all $a \in G$.
3. Inverses: For each element $a \in G$, there exists an element $b \in G$ such that $a * b = b * a = e$, where $e$ is the identity element.

Groups are central in various mathematical fields, including geometry and algebra.

Example

Consider the set of integers $Z$ under addition. The identity element is 0, and each integer $n$ has an inverse $-n$, since:

Thus, the set of integers forms a group under addition.

Abelian Group

Definition of an Abelian Group

An Abelian Group (or commutative group) is a group in which the binary operation is commutative. This means that for all elements $a$ and $b$ in the group, the operation satisfies:

In addition to the usual group properties (associativity, identity, and inverses), Abelian groups are characterized by this commutative property, which makes them easier to work with in many applications.

Example

The set of real numbers $R$ under addition forms an Abelian group. For any two real numbers $a$ and $b$:

Thus, real numbers form an Abelian group under addition.

Permutation Groups

Definition of a Permutation Group

A Permutation Group is a set of permutations of a finite set $S$ that is closed under composition and inverses. A permutation is a rearrangement of the elements of a set. If $S$ is a set with $n$ elements, the Symmetric Group $S_n$ is the group of all permutations of $S$.

Permutation groups are particularly useful in the study of symmetry and are used in many areas of mathematics, including algebra and combinatorics.

Example

Consider the set $S = \{1, 2, 3\}$. The symmetric group $S_3$ consists of all possible permutations of $S$. These include:

These permutations form a group under composition.

Cosets and Normal Subgroups

Cosets

A coset is a subset of a group formed by multiplying all elements of the group by a fixed element. If $H$ is a subgroup of a group $G$ and $g$ is an element of $G$, the left coset of $H$ with respect to $g$ is:

Similarly, the right coset is defined as:

Cosets are used in the study of group structure and are important in understanding quotient groups.

Normal Subgroups

A Normal Subgroup is a subgroup $N$ of a group $G$ such that the left coset of $N$ is equal to the right coset for every element $g$ in $G$. In other words, $N$ is normal if:

Normal subgroups are crucial in the construction of quotient groups and play a central role in the study of group homomorphisms.

Ring

Definition of a Ring

A Ring is an algebraic structure consisting of a set $R$ equipped with two binary operations: addition and multiplication. The set must satisfy the following properties:

1. Additive Group: The set $R$ under addition is an Abelian group.
2. Multiplication is Associative: For all $a, b, c \in R$, $(a * b) * c = a * (b * c)$.
3. Distributive Laws: Multiplication distributes over addition: $a _ (b + c) = a _ b + a _ c$ $(a + b) _ c = a _ c + b _ c$

Rings can either be commutative (if multiplication is commutative) or non-commutative.

Example

The set of integers $Z$ under the usual addition and multiplication forms a commutative ring.

Integral Domain

Definition of an Integral Domain

An Integral Domain is a commutative ring with no zero divisors. A zero divisor is a non-zero element $a$ in a ring such that there exists a non-zero element $b$ with $a * b = 0$. In an integral domain, the product of two non-zero elements is always non-zero.

Example

The set of integers $Z$ forms an integral domain, as there are no zero divisors in $Z$.

Field

Definition of a Field

A Field is an algebraic structure in which both addition and multiplication are defined and behave similarly to the arithmetic of real numbers. A field is a set $F$ with two operations (addition and multiplication) that satisfy the following properties:

1. Additive Group: The set $F$ under addition is an Abelian group.
2. Multiplicative Group: The non-zero elements of $F$ under multiplication form an Abelian group.
3. Distributive Law: Multiplication distributes over addition.

Fields are important in many areas of mathematics, including algebra, number theory, and geometry.

Example

The set of rational numbers $Q$ forms a field under the usual addition and multiplication.

Codes and Group Codes

Introduction to Codes

Codes are methods used to transform data into a form that can be efficiently transmitted or stored. Group codes refer to a type of error-correcting code where the codewords form a group under a particular operation. These codes are used in data transmission and storage to detect and correct errors.

Applications of Algebraic Structures

Algebraic structures have several practical applications in fields such as cryptography, coding theory, and computer science. Groups are used in symmetry analysis in physics and chemistry, while rings and fields form the foundation of coding theory and

cryptography.