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Group
Theory simplified - part II
back
to part I
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 Є S => a
* b
Є 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 1x1=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 Є 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.
-by Gurudev
MADE IN
INDIA
gurudevp@vsnl.net
On 13 November 2002
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