An algebraic structure is a set equipped with one or more binary operations that satisfy specific properties. These structures provide a framework for studying different mathematical objects like groups, rings, and fields, and are fundamental in abstract algebra.

Definition of an Algebraic Structure:

An algebraic structure consists of:

• A set $S$.
• One or more binary operations defined on $S$.

The properties of the binary operations (such as associativity, commutativity, etc.) define the type of algebraic structure, like groups, rings, and fields.

Common Types of Algebraic Structures:

1. Semigroup:
   • A set $S$ with a binary operation $\ast$ that is associative.
   • Associativity: $(a\ast b)\ast c=a\ast (b\ast c)$ for all $a,b,c\in S$.
   • Example: The set of integers under addition is a semigroup.
2. Monoid:
   • A semigroup with an identity element.
   • Identity element: An element $e\in S$ such that $a\ast e=e\ast a=a$ for all $a\in S$.
   • Example: The set of natural numbers $\mathbb{N}$ under addition, with 0 as the identity element, forms a monoid.
3. Group:
   • A set $G$ with a binary operation $\ast$ that satisfies:
     1. Associativity: $(a\ast b)\ast c=a\ast (b\ast c)$ for all $a,b,c\in G$.
     2. Identity element: There exists an identity element $e\in G$ such that $a\ast e=e\ast a=a$ for all $a\in G$.
     3. Inverse: For each element $a\in G$, there exists an inverse element $b\in G$ such that $a\ast b=b\ast a=e$, where $e$ is the identity element.
   • Example: The set of integers $\mathbb{Z}$ under addition forms a group, with 0 as the identity element and $-a$ as the inverse of $a$.
4. Abelian Group (Commutative Group):
   • A group where the binary operation is commutative.
   • Commutativity: $a\ast b=b\ast a$ for all $a,b\in G$.
   • Example: The set of real numbers $\mathbb{R}$ under addition is an Abelian group.
5. Ring:
   • A set $R$ equipped with two binary operations, typically addition and multiplication, satisfying the following:
     1. $R$ is an Abelian group under addition.
     2. Multiplication is associative.
     3. Multiplication distributes over addition: $a\times (b+c)=(a\times b)+(a\times c)$.
   • Example: The set of integers $\mathbb{Z}$ under addition and multiplication forms a ring.
6. Field:
   • A set $F$ with two binary operations (addition and multiplication) that satisfies:
     1. $F$ is an Abelian group under addition.
     2. F∗F^*F∗ (the set $F$ excluding the additive identity 0) is an Abelian group under multiplication.
     3. Multiplication distributes over addition.
   • Example: The set of rational numbers $\mathbb{Q}$, real numbers $\mathbb{R}$, and complex numbers $\mathbb{C}$ are all fields.

Key Properties of Algebraic Structures:

1. Associativity:
   • A binary operation $\ast$ is associative if $(a\ast b)\ast c=a\ast (b\ast c)$ for all $a,b,c\in S$.
   • Example: Integer addition is associative, as $(1+2)+3=1+(2+3)$.
2. Commutativity:
   • A binary operation $\ast$ is commutative if $a\ast b=b\ast a$ for all $a,b\in S$.
   • Example: Real number multiplication is commutative, as $2\times 3=3\times 2$.
3. Identity Element:
   • An element $e\in S$ is an identity for a binary operation $\ast$ if $a\ast e=e\ast a=a$ for all $a\in S$.
   • Example: 0 is the identity element for addition in the set of integers, and 1 is the identity element for multiplication.
4. Inverse Element:
   • For each element $a\in S$, there exists an inverse $b\in S$ such that $a\ast b=b\ast a=e$, where $e$ is the identity element.
   • Example: For addition, the inverse of $a$ is $-a$, as $a+(-a)=0$.

Examples of Algebraic Structures:

1. Group Example: The set of integers $\mathbb{Z}$ under addition forms a group:
   • Identity element: 0.
   • Inverse of $a$: $-a$.
   • Associative and commutative properties hold.
2. Ring Example: The set of integers $\mathbb{Z}$ under addition and multiplication forms a ring:
   • Addition: $(a+b)\in \mathbb{Z}$.
   • Multiplication: $(a\times b)\in \mathbb{Z}$.
   • Distributive property holds: $a\times (b+c)=(a\times b)+(a\times c)$.
3. Field Example: The set of real numbers $\mathbb{R}$ under addition and multiplication forms a field:
   • Additive identity: 0.
   • Multiplicative identity: 1.
   • Every non-zero element has a multiplicative inverse.

6.1.3 Group and its properties

A group is a fundamental concept in abstract algebra that consists of a set of elements combined with a binary operation that satisfies certain axioms. Groups are widely used in various areas of mathematics, including geometry, number theory, and algebra, and have applications in physics and computer science.

Definition of a Group:

A set $G$ with a binary operation $\ast$ is called a group if the following four properties (called group axioms) are satisfied:

1. Closure:
   • For all elements $a,b\in G$, the result of the operation $a\ast b$ is also in $G$.
   • In symbols: $a\ast b\in G$.
2. Associativity:
   • For all elements $a,b,c\in G$, the binary operation is associative, meaning: $$(a\ast b)\ast c=a\ast (b\ast c)$$
3. Identity Element:
   • There exists an element $e\in G$ (called the identity element) such that for every element $a\in G$: $$a\ast e=e\ast a=a$$
   • The identity element does not change any element when applied in the binary operation.
4. Inverse Element:
   • For each element $a\in G$, there exists an element $b\in G$ (called the inverse of $a$) such that: $$a\ast b=b\ast a=e$$
   • Here, $e$ is the identity element, and the inverse of $a$ effectively “undoes” the operation with $a$.

Examples of Groups:

1. Integers under Addition:
   • Set: $\mathbb{Z}$ (integers)
   • Operation: Addition $+$
   • Properties:
     • Closure: $a+b\in \mathbb{Z}$ for all $a,b\in \mathbb{Z}$.
     • Associativity: $(a+b)+c=a+(b+c)$.
     • Identity: The identity element is 0, because $a+0=0+a=a$.
     • Inverse: For any $a\in \mathbb{Z}$, the inverse is $-a$, since $a+(-a)=0$.
2. Non-zero Real Numbers under Multiplication:
   • Set: R∗=R∖{0}\mathbb{R}^* = \mathbb{R} \setminus \{0\}R∗=R∖{0} (non-zero real numbers)
   • Operation: Multiplication $\times$
   • Properties:
     • Closure: a×b∈R∗a \times b \in \mathbb{R}^*a×b∈R∗ for all a,b∈R∗a, b \in \mathbb{R}^*a,b∈R∗.
     • Associativity: $(a\times b)\times c=a\times (b\times c)$.
     • Identity: The identity element is 1, because $a\times 1=1\times a=a$.
     • Inverse: For any a∈R∗a \in \mathbb{R}^*a∈R∗, the inverse is 1a\frac{1}{a}a1​, since a×1a=1a \times \frac{1}{a} = 1a×a1​=1.
3. Symmetric Group:
   • The set of all permutations of a finite set $\{1,2,\dots ,n\}$ forms a group under the operation of composition of permutations.
   • The identity element is the identity permutation (which maps every element to itself).
   • The inverse of a permutation is the permutation that undoes it.

Abelian (Commutative) Groups:

A group is called an Abelian group (or commutative group) if, in addition to the four group axioms, the binary operation is commutative. That is, for all elements $a,b\in G$:

$$a\ast b=b\ast a$$

• Example: The group of integers $\mathbb{Z}$ under addition is an Abelian group, as $a+b=b+a$ for all $a,b\in \mathbb{Z}$.

Non-Abelian Groups:

A group is called non-Abelian if the binary operation is not commutative.

• Example: The group of permutations of three elements (the symmetric group S3S_3S3​) is a non-Abelian group because composition of permutations is not always commutative.

Group Properties:

1. Closure:
   • The operation must always result in an element within the set.
   • Example: In $\mathbb{Z}$, adding two integers always produces an integer.
2. Associativity:
   • The way elements are grouped in the operation does not matter.
   • Example: $(1+2)+3=1+(2+3)$.
3. Identity Element:
   • There is an element that does not affect other elements under the operation.
   • Example: In $\mathbb{Z}$ under addition, 0 is the identity because $a+0=a$.
4. Inverse Element:
   • Every element has a counterpart that combines with it to produce the identity element.
   • Example: For any integer $a$, $-a$ is its inverse because $a+(-a)=0$.

Examples of Groups:

• Group of Real Numbers under Addition:
  • Set: $\mathbb{R}$
  • Operation: Addition
  • Identity: 0
  • Inverse: For $a\in \mathbb{R}$, the inverse is $-a$.
• Group of Non-zero Real Numbers under Multiplication:
  • Set: R∗=R∖{0}\mathbb{R}^* = \mathbb{R} \setminus \{0\}R∗=R∖{0}
  • Operation: Multiplication
  • Identity: 1
  • Inverse: For a∈R∗a \in \mathbb{R}^*a∈R∗, the inverse is 1a\frac{1}{a}a1​.

Group Examples in Real Life:

• Modular Arithmetic: The integers under addition modulo $n$ form a group, which is used in cryptography.
• Symmetry Groups: The set of symmetries of geometric objects (rotations, reflections) forms a group, important in physics and chemistry.

6.1.4 Sub-groups, Cyclic groups and Permutation groups

In group theory, several specialized types of groups are essential to understanding more complex algebraic structures. These include subgroups, cyclic groups, and permutation groups. Let’s explore these concepts.

1. Sub-groups

A subgroup is a subset of a group that is itself a group under the same binary operation. If $G$ is a group, then a subset $H\subseteq G$ is a subgroup of $G$ if $H$ satisfies the group axioms under the operation defined on $G$.

Conditions for a Subgroup (The Subgroup Test):

A non-empty subset $H$ of a group $G$ is a subgroup of $G$ if and only if:

1. Closure: For all $a,b\in H$, the product $a\ast b\in H$.
2. Identity: The identity element of $G$ is in $H$.
3. Inverse: For all $a\in H$, the inverse a−1∈Ha^{-1} \in Ha−1∈H.

Notation:

If $H$ is a subgroup of $G$, we write $H\le G$.

Example:

• The set of integers $\mathbb{Z}$ under addition forms a group. The set of even integers $2\mathbb{Z}$ is a subgroup of $\mathbb{Z}$ because it satisfies closure, contains the identity element (0), and each element has an inverse.

2. Cyclic Groups

A cyclic group is a group that can be generated by a single element. That means all elements of the group can be expressed as powers (or multiples in additive notation) of a particular element called the generator.

Definition:

A group $G$ is called cyclic if there exists an element $g\in G$ such that every element $a\in G$ can be written as a=gna = g^na=gn for some integer $n$.

• The element $g$ is called the generator of the group, and we write $G=⟨g⟩$.

Examples:

1. Integers under Addition:
   • The group $\mathbb{Z}$ under addition is a cyclic group with generator $1$, because every integer can be written as a multiple of $1$. So $\mathbb{Z}=⟨1⟩$.
2. Modulo Groups:
   • The group of integers modulo $n$, denoted Zn\mathbb{Z}_nZn​, is cyclic with generator $1$, because every element of Zn\mathbb{Z}_nZn​ can be expressed as a multiple of $1$.

Finite and Infinite Cyclic Groups:

• Finite Cyclic Group: If $G$ is finite, say $|G|=n$, then $G$ consists of g0,g1,g2,…,gn−1g^0, g^1, g^2, \ldots, g^{n-1}g0,g1,g2,…,gn−1.
  • Example: Zn={0,1,2,…,n−1}\mathbb{Z}_n = \{0, 1, 2, \ldots, n-1\}Zn​={0,1,2,…,n−1}.
• Infinite Cyclic Group: An infinite cyclic group is isomorphic to $\mathbb{Z}$, the set of integers under addition.
  • Example: $\mathbb{Z}$ is an infinite cyclic group with generator $1$.

3. Permutation Groups

A permutation group is a group whose elements are permutations of a set, and the group operation is the composition of permutations.

Definition:

A permutation of a set $S$ is a bijective function from $S$ to itself. The set of all permutations of $S$ forms a group, denoted as the symmetric group SnS_nSn​, where $n$ is the number of elements in $S$.

Symmetric Group SnS_nSn​:

• The group of all permutations on a set of $n$ elements is called the symmetric group on $n$ elements, denoted SnS_nSn​.
• The order (number of elements) of SnS_nSn​ is $n!$ (factorial of $n$) because there are $n!$ possible ways to permute $n$ objects.

Examples:

1. Symmetric Group S3S_3S3​:
   • Let $S=\{1,2,3\}$. The symmetric group S3S_3S3​ consists of all 6 possible permutations of the set $\{1,2,3\}$:
     • $e$ (the identity permutation): $(1\to 1,2\to 2,3\to 3)$
     • $(12)$: $(1\to 2,2\to 1,3\to 3)$
     • $(13)$: $(1\to 3,2\to 2,3\to 1)$
     • $(23)$: $(1\to 1,2\to 3,3\to 2)$
     • $(123)$: $(1\to 2,2\to 3,3\to 1)$
     • $(132)$: $(1\to 3,2\to 1,3\to 2)$
2. Alternating Group:
   • The alternating group AnA_nAn​ is the subgroup of SnS_nSn​ consisting of all even permutations (those that can be written as a product of an even number of transpositions).
   • Example: A3={e,(123),(132)}A_3 = \{e, (123), (132)\}A3​={e,(123),(132)}.

Properties of Permutation Groups:

• Closure: The composition of two permutations is again a permutation.
• Associativity: Permutation composition is associative.
• Identity: The identity permutation, which leaves all elements unchanged, acts as the identity element.
• Inverses: Every permutation has an inverse, which is also a permutation.