## 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:

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

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:

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:

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.

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

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:

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:

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]

Discrete Structure

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ACA: ADVANCED COMPUTER ARCHITECTURE EasyExamNotes.com covered following topics in these notes.
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Discrete Structure EasyExamNotes.com covered following topics in these notes.
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Cloud Computing:

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1. Buyya, Selvi ,” Mastering Cloud Computing “,TMH Pub
2. Krutz , Vines, “Cloud Security “ , Wiley Pub
3. Velte, “Cloud Computing- A Practical Approach” ,TMH Pub
4. Sosinsky, “ Cloud Computing” , Wiley Pub
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Principles of Programming Languages
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#### Language Evaluation Criteria Influences on Language Design  Language Categories Programming Paradigms Compilation Virtual Machines  Programming Environments Issues in Language Translation Parse Tree  Pointer and Reference type Concept of Binding Type Checking Strong typing  Sequence control with Expression Subprograms Fundamentals of sub-programs Scope and lifetime of variable Static and dynamic scope Design issues of subprogram and operations Local referencing environments Parameter passing methods Overloaded sub-programs Generic sub-programs  Design issues for functions  Co routines  Abstract Data types Abstraction and encapsulation Static and Stack-Based Storage management Garbage Collection OOP in C++ OOP in Java  OOP in C#  OOP in PHP Concurrency Semaphores  Monitors Message passing Java threads  C# threads  Exception handling  Exceptions Exception Propagation  Exception handler in C++  Exception handler in Java Introduction and overview of Logic programming Basic elements of Prolog Application of Logic programming Functional programming languages Introduction to 4GL  Practicals: Memory Implementation of 2D Array.  Memory Implementation of 3D Array.  Implementation of pointers in C++.  Write a program in Java to implement exception handling. Write a program in C++ to implement call by value parameter passing Method. Write a program in C++ to implement call by reference parameter passing Method.  Write a program in Java to implement concurrent execution of a job using threads. Implement Inheritance in C#. Implement Encapsulation in C#. Implement static/compiletime Polymorphism in C#. Implement dynamic/runtime Polymorphism in C#. Previous years solved papers: PPL|RGPV|May 2018  PPL|RGPV|June 2017  A list of Video lectures Click here References: Sebesta,”Concept of programming Language”, Pearson Edu  Louden, “Programming Languages: Principles & Practices” , Cengage Learning  Tucker, “Programming Languages: Principles and paradigms “, Tata McGraw –Hill.  E Horowitz, "Programming Languages", 2nd Edition, Addison Wesley

Computer Organization and Architecture ↓ 👆

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

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3. KavehPahlavan, Prashant Krishnamurthy, “Networking Fundamentals”, Wiley Publication.
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