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Introduction to Group Theory: Groupoid to Abelian Group

Dr. Gajendra Purohit explains the foundational definitions of abstract algebra including groupoids, semigroups, monoids, groups, and abelian groups. The lecture illustrates these algebraic structures with sets such as natural numbers, integers, and matrices under various operations. It also covers concepts of finite and infinite groups along with the order of a group.

Maths
English · 7 topics
Checked against the video on 24 Oct 2026

Made by AI from the YouTube video Group Theory | Semi-Group , Monoid | Abelian Group | Discrete Mathematics (opens YouTube in a new tab) by Dr.Gajendra Purohit, then checked against it. All credit for the teaching goes to the creator. Tap a topic’s time to confirm a fact in the video.

Introduction to Group Theory: Groupoid to Abelian Group

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In short

Dr. Gajendra Purohit explains the foundational definitions of abstract algebra including groupoids, semigroups, monoids, groups, and abelian groups. The lecture illustrates these algebraic structures with sets such as natural numbers, integers, and matrices under various operations. It also covers concepts of finite and infinite groups along with the order of a group.

Introduction and Overview of Group Theory

Group theory is introduced as an abstract algebra topic relevant for B.Sc., engineering, and M.Sc. mathematics students.
The session focuses on building step-by-step structural definitions from basic closures up to commutative groups.

Groupoid or Binary Algebra

A non-empty set G equipped with one binary operation × is called a groupoid if G is closed under *.
The closure property means that for every pair of elements a, b belonging to G, their operation a × b also belongs to G.
A groupoid is also known as a quasi group.
In the set of natural numbers N, addition and multiplication are closed, whereas subtraction and division are not closed because operations like 3 - 4 or 3 / 4 yield values outside N.
In the set of integers Z, addition, multiplication, and subtraction are closed, but division is not closed since -2 / 3 does not belong to Z.
Must knowA non-empty set G is a groupoid if it satisfies the closure property under a given binary operation.

Semi Group

An algebraic structure (G, *) is called a semi group if the binary operation × satisfies the associative property.
Associativity means (a × b) × c = a × (b × c) for all elements a, b, c in G.
In natural numbers N, addition and multiplication are associative, but subtraction is not associative because 2 - (4 - 7) does not equal (2 - 4) - 7.
Algebraic structures like (N, +), (Z, +), (Z, *), and (Q, *) are semi groups, whereas (Z, -) is not a semi group due to the failure of associativity under subtraction.
The power set structure (P(S), U) under union and (P(S), ∩) under intersection are semi groups because both operations are associative.
Must knowA semi group must satisfy both closure and associativity.

Monoid and Identity Element

A semi group is called a monoid if there exists an identity element e in G such that e × a = a × e = a for all a in G.
Under addition, the identity element is 0; since 0 is not in the set of natural numbers N, (N, +) is not a monoid.
Under multiplication, the identity element is 1; since 1 belongs to N, (N, *) is a monoid.
In the set of integers Z, 0 is the additive identity and 1 is the multiplicative identity, so Z forms a monoid under both addition and multiplication.
For power sets P(S), the empty set φ serves as the identity under union, and S serves as the identity under intersection, making both structures monoids.
In matrix algebra, the zero matrix serves as the additive identity, and the identity matrix serves as the multiplicative identity.
Must knowA monoid is a semi group that contains an identity element.

Group and Existence of Inverse

An algebraic structure (G, *) is called a group if it satisfies four postulates: closure, associativity, existence of identity, and existence of inverse.
For every element a in G, an inverse element a⁻¹ must exist in G such that a × a⁻¹ = a⁻¹ × a = e, where e is the identity element.
The additive inverse of an element a is -a, which satisfies a + (-a) = 0.
The multiplicative inverse of a non-zero element a is 1/a, which satisfies a × (1/a) = 1.
(N, *) is not a group because the multiplicative inverse of an element like 2 is 1/2, which does not belong to N.
(Z, +) is a group because every integer a has an additive inverse -a in Z, and the additive identity 0 is in Z.
(Z, *) is not a group because elements like 3 do not have their multiplicative inverse 1/3 in Z.
Must knowA group requires four fundamental properties: closure, associativity, existence of identity, and existence of inverse.

Abelian Group or Commutative Group

A group (G, *) is called an abelian group or commutative group if it satisfies the commutativity condition: a × b = b × a for all a, b in G.
(Z, +) is an abelian group because addition of integers is commutative (a + b = b + a).
The rational numbers (Q, *) under multiplication form a commutative structure where commutativity holds (a × b = b × a).
Abstract structures progress hierarchically: closure yields a groupoid; closure with associativity yields a semi group; adding identity yields a monoid; adding inverse yields a group; adding commutativity yields an abelian group.
Must knowAn abelian group is a group whose binary operation satisfies commutativity.

Structure	Closure	Associativity	Identity
Groupoid	Yes	No	No
Semi Group	Yes	Yes	No
Monoid	Yes	Yes	Yes
Group	Yes	Yes	Yes
Abelian Group	Yes	Yes	Yes

Finite and Infinite Groups and Order of a Group

A group (G, *) is called a finite group if its underlying set G is finite; otherwise, it is called an infinite group.
The number of elements in a finite group is called the order of the group, denoted by O(G).
If (G, *) is an infinite group, it is said to be of infinite order.
Must knowThe order of a finite group is the total number of elements present in the group.

Key terms

Groupoid: A non-empty set equipped with a binary operation that is closed.
Quasi Group: Another name used for a groupoid.
Semi Group: An algebraic structure that satisfies closure and associative properties.
Monoid: A semi group that contains an identity element for the binary operation.
Group: A monoid in which every element has an inverse.
Abelian Group: A group whose binary operation satisfies commutativity.
Order of a Group: The number of elements contained in a finite group, denoted by O(G).

Quick revision

Read these the night before the exam.

A groupoid satisfies only the closure property under its binary operation.
(N, -) is not closed because the subtraction of two natural numbers can produce a negative integer.
Subtraction is not associative, meaning (Z, -) cannot be a semi group.
(N, +) is not a monoid because the additive identity 0 does not belong to the set of natural numbers.
(N, *) is a monoid because 1 belongs to N and acts as the multiplicative identity.
(Z, +) is a group because identity 0 and additive inverses -a exist for all integers.
(Z, *) is not a group because multiplicative inverses of integers (except 1 and -1) do not lie in Z.
Commutativity (a × b = b × a) added to group axioms forms an abelian or commutative group.
The empty set φ is the identity element in the power set monoid (P(S), U).
The order of a group, written as O(G), is the total number of elements in a finite group.

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