This document provides an overview of abstract algebra and key concepts such as groups. It discusses how the word "algebra" is derived from an Arabian word meaning "union of broken parts." It also outlines important mathematicians who contributed to the development of algebra, such as Al Khwarizmi, the "father of algebra." The document defines what a set and group are, including the properties a group must satisfy like closure, associative, identity, and inverse elements. Examples of groups are given like integers, rational numbers, and matrices. Applications of group theory in fields like physics, chemistry, and technology are mentioned.

1. Abstract algebra
BY
J. MANJULA
Assistant Professor of Mathematics
Bon Secours College for Women

2. Warm up
The word ‘Algebra’ is derived from
Al-jabr, an Arabian word.
Here, Al means that &
Jabr means union of broken parts.

3. AL KHWARIZMI –
FATHER OF ALGEBRA

4. Al Khwarizmi
is an Arabian Mathematician
He wrote the Book –
“Kitab al-jabr wa
l-mugabala”
The word ‘Algebra’ got from the title of the
Book.
That means – Compilations and Equations.

5. Indian Mathematicians
Aryabhatta
Bhaskara 
Bramma Gupta 

6. SETS
Among many branches of
Modern Mathematics, the
theory of Sets occupies a
unique place which was
founded by Georg Cantor

7. Definition:
A set is a collection of well
defined objects.
Example:
A student is defined to be “tall” if
her height is greater than 5’6”. The
collection of all students of Bon
Secours college For Women, Thanjavur.

8. Example for a non set:
The collection of all beautiful
ladies in town.
The collection of all talent
students in your class.

9. GROUP
Definition :
A non empty set together with a binary
operation * : G x G  G is called a GROUP if the
following properties are satisfied.
(i) Closure Property
(ii) Associative Property
(iii) Identity Element
(iv) Inverse Element

10. Closure Property:
It means a set is closed
for some Mathematical
operations.
i.e., if a,b ϵ G, then a*b ϵ G.

11. Associative Property:
* is associative (i.e.)
a * (b*c) = (a*b)*c
for all a,b,c ϵ G

12. Identity Element:
There exist an element
e ϵ G, such that,
a*e = e*a = a.
For all a ϵ G.

13. Inverse Element:
For any element, a in G
there exists an element a’ϵG.
such that
a*a’ = a’*a = e
a’ is called the Inverse of a.

14. Example
1. Z – The set of all integers
2. Q – The set of all Rational numbers
3. R – The set of all real numbers
4. C – The set of all complex numbers
are Groups under usual addition.
5. The set of all 2 x 2 matrices
where a, b, c, d ϵ R is a Group under matrix
Addition.
6. Q* - The set of all non-zero rational numbers
7. Q + The set of all positive rational numbers are
Group under usual multiplication.

15. Abelian Group
Definition :
A Group is set to be abelian if
a*b= b*a for all a, b ϵG.
A Group which is not abelian
is called a Non-abelian Group.

16. Example
1. Z – The set of all integers
2. Q – The set of all Rational numbers
3. R – The set of all real numbers
4. C – The set of all complex numbers
are abelian Groups under
usual addition.

17. Applications of Groups
 Modern Particle physics is based on symmetry
principles and by the application of Group theory,
the existence of several particles were predicted
before they were experimentally observed.
 In Chemistry, the Symmetry of a Molecule
provides us with the information of what energy
levels the orbitals will be, what the orbitals
symmetries are, what transitions can occur
between energy levels, even bond order and all
of that is calculated using Group theory.
 Group theory is used in Robotics, Computer
vision / Graphics and Medical image analysis