Group

GROUP


Group ?


A non-empty set G of some elements (a, b, c, etc.), with one or more operations is known as a group.
A set needed to be satisfied following properties to become a group:


1)      Closure Property-
a.b ∈  G , ∀ a, b ∈  G


2)      Associative Property-
(a . b) . c = a . (b . c), ∀ a, b, c ∈  G


3)      Existence of Identity-
e →  identity element
e.a = a = a.e, ∀  a ∈ G


4)      Existence of Inverse-
a-1→  inverse of a
a.a-1 = e = a-1.a , ∀  a ∈ G


ABELIAN OR COMMUTATIVE GROUP


A set needed to be satisfied following properties to become an abelian group:


1)      Closure Property-
a.b ∈  G , ∀ a, b ∈  G


2)      Associative Property-
(a . b) . c = a . (b . c), ∀ a, b, c ∈  G


3)      Existence of Identity-
e →  identity element
e.a = a = a.e, ∀  a ∈ G


4)      Existence of Inverse-
a-1 →   inverse of a
a.a-1 = e = a-1.a , ∀  a ∈ G


5)      Commutativity-
a.b = b.a , ∀  a , b ∈ G


SUBGROUP


A subgroup is a subset H of group elements of a group G that satisfies all the four properties of a group.


“ H is a subgroup of G” can be written as H ⊆ G


A subgroup H of a group G, where H ≠ G, is known as proper subgroup of G.


Problems based on above study:


Prob. Show that the set I of all integers (…, -4, -3, -2, -1, 0, 1, 2, 3, 4,…}
Is a group with respect to the operation of addition of integers?
Solution: Click here.


Prob. Let ({a, b}, * ) be a semigroup where a*a = b show that-
                                           i.            a*b = b*a
Solution: Click here.


Prob. Let (A, *) be a semigroup. Show that for a, b, c ∈  A, if a*c = c*a and b*c = c*b, then (a*b)*c = c*(a*b).
Solution: Click here.


Prob. Suppose (A, *) be a group, show that (A, *) is an abelian group and only if a3 * b3 = (a*b)3 for all a and b in A.
Solution: Click here.


Prob. Prove that the set Z of all integers with binary operations defined by-
a*b = a+b+1 , ∀  a, b ∈  z.  Is an abelian group.
Solution: Click here.


Prob. Prove that the cube roots of unity namely ( 1, w, w2) abelian under multiplication of complex numbers.
Solution: Click here.


Prob. Prove that the set G = {0, 1, 2, 3, 4, 5} is a finite abelian group of order 6 with respect to addition modulo 6.
Solution: Click here.


Prob. Prove that the set G = {1, 2, 3, 4, 5, 6} is a finite abelian group of order 6 with respect to multiplication modulo 7.
Solution: Click here.


Prob. Which of the following property/ies a Group G must hold, in order to be an Abelian group? (a) The distributive property (b) The commutative property (c) The symmetric property. [CBSE NET December 2015]



Python Programming ↓ 👆
Java Programming ↓ 👆
JAVA EasyExamNotes.com covered following topics in these notes.
JAVA Programs
Principles of Programming Languages ↓ 👆
Principles of Programming Languages
EasyExamNotes.com covered following topics in these notes.

Practicals:
Previous years solved papers:
A list of Video lectures References:
  1. Sebesta,”Concept of programming Language”, Pearson Edu 
  2. Louden, “Programming Languages: Principles & Practices” , Cengage Learning 
  3. Tucker, “Programming Languages: Principles and paradigms “, Tata McGraw –Hill. 
  4. E Horowitz, "Programming Languages", 2nd Edition, Addison Wesley

    Computer Organization and Architecture ↓ 👆

    Computer Organization and Architecture 

    EasyExamNotes.com covered following topics in these notes.

    1. Structure of desktop computers
    2. Logic gates
    3. Register organization
    4. Bus structure
    5. Addressing modes
    6. Register transfer language
    7. Direct mapping numericals
    8. Register in Assembly Language Programming
    9. Arrays in Assembly Language Programming

    References:

    1. William stalling ,“Computer Architecture and Organization” PHI
    2. Morris Mano , “Computer System Organization ”PHI

    Computer Network ↓ 👆
    Computer Network

    EasyExamNotes.com covered following topics in these notes.
    1. Data Link Layer
    2. Framing
    3. Byte count framing method
    4. Flag bytes with byte stuffing framing method
    5. Flag bits with bit stuffing framing method
    6. Physical layer coding violations framing method
    7. Error control in data link layer
    8. Stop and Wait scheme
    9. Sliding Window Protocol
    10. One bit sliding window protocol
    11. A protocol Using Go-Back-N
    12. Selective repeat protocol
    13. Application layer
    References:
    1. Andrew S. Tanenbaum, David J. Wetherall, “Computer Networks” Pearson Education.
    2. Douglas E Comer, “Internetworking with TCP/IP Principles, Protocols, And Architecture",Pearson Education
    3. KavehPahlavan, Prashant Krishnamurthy, “Networking Fundamentals”, Wiley Publication.
    4. Ying-Dar Lin, Ren-Hung Hwang, Fred Baker, “Computer Networks: An Open Source Approach”, McGraw Hill.