Algebraic System or Algebraic Structure and Group

Algebraic System or Algebraic Strucure

"A non-empty set 'G' together with a binary operation ' O' is called an algebraic system or algebraic structure and is denoted by (G,O), (N,+). "


Group
"A non-empty set 'G' together with a binary operation 'O' is said to be a group, if the following laws holds for all the operations 'O'."


Group holds the following given four laws :
  
(a) Clouser Law : 'G' is cloused under the binary operation 'O' that is ( a O b ) = ( b O a ) ∀ a, b ᗴ G.

(b) Associative Law : The associative law for the operation holds that is  a O ( b O c ) = ( a O b ) O c ∀ a, b, c ᗴ G.

 (c) Existence of Identity Element : The identity element exists in 'G' that is there exists an element e ᗴ G such that a ᗴ  e = e ᗴ a = a ∀ a ᗴ G.

 (d) Existence of Inverse Element : The inverse element of all the elements of 'G' exists that is for every a ᗴ G there exists an element b ᗴ G such that a O b = b O a = e , where 'e' bring the identity element.

Example : 
                   The set of cube roots of unity forms a group with respect to ordinary multiplication.
G = { 1, w, w² }  (finite set)

Testing the existence of clouser law 
Multiplication Table

X 1 w w2
1 1 w w2
w w w2 1
w2 w2 1 w



                      From the above table we see that clouser law holds good in 'G'.

Testing the existence of associative law

           Since, 'G' is the sub set of complex number and we know that associative law hold good in set of complex number so it also holds good in 'G'.

Testing the existence of identity element

          There exists an element 1 ᗴ G, known as identity element  such that a * 1 = 1 * a = a ∀ a ᗴ G.

Testing the existence of inverse element

          Inverse of 1 → 1 * 1  is 1
          Inverse of w is w² because w * w² = w² * w = 1 
          Inverse of w² is w because w² * w = w * w² = 1

Hence, 'G' is a group with respect to the ordinary multiplication.

Comments

Post a Comment

Popular posts from this blog

Euclid's Algorithm to find greatest common divisior

Image
Greatest common divisor One integer often needed in cryptography is the greatest common divisor of two positive integers. Two positive integer may have many common divisor but only one greatest common divisor. For example, the common divisor of 12 and 140 are 1, 2 and 4. However, 4 is the greatest common divisor.       The greatest common divisor of two integers is largest integer which can divide both given integers. The following block diagram illustrate the basic of the greatest common divisor of 12 and 140. " Euclidean Algorithm " Finding the greatest common divisor ( GCD ) of two positive integers by listing all common divisor is not practical, when the two number integers are large. Fortunately, more than 2000 years ago a mathematician named Euclid develop an algorithm that can find the greatest common divisor of two positive integers. The Euclidean algorithm is based on the following two facts: 1. gcd ( a, 0 ) = a 2. gcd ( a, b ...
Read more

Symmetric key cryptography

Image
Symmetric key cryptography started thousand years ago, when people needed to exchange secrets (for example in a war ). We still mainly uses symmetric key cryptography in our network security. However, today ciphers are much more complex. Let us first discuss traditional algorithms, which were character oriented. Then we discuss modern one which are nit oriented. Traditional ciphers : We briefly introduce some traditional ciphers, which are character oriented. Although these are now obsolete, the goal is to show how modern cipher evolved from them. We can divide traditional symmetric key ciphers in to two broad categories namely -- substitution ciphers and traditional ciphers, as shown in the following figure. Substitution ciphers : A substitution cipher substitute one symbol with another. If the symbol in the plain text are alphabetic character, we replace one character by another. For example, we can replace character 'a' with character 'd' ,  and characte...
Read more