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.
In part I, we acquired the following tools:
Set, binary operation and Algebraic Structure.
A Group is denoted by its Set S and the binary operation * defined on that Set as (S,*). (Same as an Algebraic Structure)
Now we call (S,*) a Group, if it satisfies the following 4 axioms:
Closure Law:
If a and b are two elements of the Set S, then the resultant of the binary operation * on a and b should also be an element of S.
i.e. a, b ? S => a * b ? S
For ex: Multiplication of 2 Integers satisfies the Closure Law as the product of any 2 integers is also an Integer. (-3)x(4)=-12
Associative Law:
If a,b,c are the elements of a Set S then,
a * (b * c) = (a * b) * c
In other words, it does not matter on which 2 elements you perform the binary operation * first as long as the order is the same i.e. a,b,c.
For ex: 3 x (5 x 4)=(3 x 5) x 4
Identity Law:
There exists an unique element e (called the identity of the Group (S,*)) belonging to the Set S such that, a*e= a , where a is any element of the Set S.
For ex: Under the binary operation of multiplication of Integers we have 1 as the Identity element i.e. e=1 bcos the product of any Integer with 1 is the Integer itself.
In other words whenever the binary machine swallows any element of the Set S along with the identity element e, the other element comes out of the binary machine unharmed.
The identity element e of a Group is unique i.e. you cannot have 2 different identities for the same Group (S,*).
Inverse Law:
For every element a of the Set S, there exists an element a-1 called the inverse of a belonging to the same Set S, such that
a * a-1 = e.
i.e. if the binary operation machine swallows an element and its inverse as the input then it will definitely spit out the identity element e of the Group.
For ex: for the Set S of rational numbers p/q where p¹0 and q¹0, we have for each element p/q its inverse element as q/p under multiplication where the identity e=1 bcos (p/q)x(q/p)=1.
Thus any Set S which has a binary operation * defined on it such that the above mentioned 4 axioms are satisfied will form a Group (S,*).
example: The Set S of the 4th roots of 1 form a Group under multiplication. i.e. S ={1,-1,i,-i} is a Group under multiplication. In this group again e=1. Here,
Inverse of 1 is 1 bcos 1×1=1
Inverse of -1 is -1 bcos (-1)x(-1)=1
Inverse of i is -i bcos ix(-i)=1
Inverse of -i is i bcos (-i)xi=1
What is an Abelian Group?
A Group (S,*) in which Commutative Law is valid is called an Abelian Group.
i.e. if a,b ? S, then a * b = b * a.
In other words, it does not matter in which order you perform the binary operation * on the elements of the Group.
example: The above mentioned Set S of the 4th roots of 1 is also an Abelian Group.
NOTE: Group Theory is an abstract mathematical theory which deals with a variety of mathematical systems. This is because it does not impose any restrictions on the type of the Set S, or on the nature of the binary operation *, as long as they satisfy the above mentioned axioms.
very clear explanation but less about group theory
Very good explanation.
Thanks Rani!