Group Theory is one of those rare
mathematical theories which look boring and dull when unexplored, but as
one gets deeper into it, it appears to me like a deep cave filled with valuable diamonds and precious stones.
So here we begin our journey towards and then right into, the Groups.
Initial Requirements:
Requirement #1
Sets are the basic ingredients of our 'to be explored' Groups.
What is a Set?
Well, A Set is a collection of unique but similar elements. For ex: A
Classroom is a set of the students in it.
In Groups the sets we are interested in are collections of numerical
values.
For ex:
N is the set of all Natural
numbers.
i.e., N={1,2,....}
I is the set of Integers
i.e., I={0,±1,±2,...}
Requirement #2 If Sets are the basic ingredients of Groups, then there must be some
mechanism to churn Groups out of these Sets. Such a mechanism does in fact exist and is called a binary operation.
It is represented by *.
What does this * do?
Speaking loosely, it can be thought of as a machine which swallows two
numerical values (hence called binary operation) as input and fuses them
into one value and spills out that value as the output.
For example, Addition is a binary operation which takes in two numbers,
performs the binary operation (i.e. adds them) and spills out the
resultant number i.e. the sum of the two.
Algebraic Structure: If the 2 elements
swallowed by * and the one spilled out by it all belong to the same Set S
then S and * together form what is called an Algebraic Structure, and is
denoted by (S,*).
To understand this, consider a Set of Colors consisting of different colors.
(See below). If you mix any of the two colors from the Set of Colors, the
resulting Color will also be another member of the same Set. So now we can
say that the Set of Colors and the binary operation of mixing any two
colors form an algebraic structure as (Color Set, Mixing).
If S and * form an Algebraic
Structure (S,*), then S is said to be closed under *. This simply means
that, 'No matter on which 2 elements of S you perform the binary operation
*, the resultant element will also belong to the same Set S'. As simple as
that. Math statements speak less, but convey more.
Now that we have a Set S and a binary operation * defined over it together
forming an Algebraic Structure (S,*) we are well equipped to understand
the Groups.
