PRESENTATION JAINITHISH.pptxfdddddddddddddddddddddddd

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2. GROUPS AND SUB GROUPS

3. ADD A FOOTER 3 In discrete mathematics, the concepts of groups and subgroups are fundamental in the study of algebraic structures . They are primarily used in abstract algebra, which is a branch of mathematics that studies algebraic systems such as groups, rings, and fields . Évariste Galois coined the term "group" and established a connection, now known as Galois theory, between the nascent theory of groups and field theory. INTRODUCTION

4. ADD A FOOTER 4 A non empty set G together with a binary operations * is called a group if that satisfies the following four properties Closure Associativity Identity Inverses * A binary operation is a function on G which assigns an element of G to each ordered pair of elements in G . For example, multiplication and addition are binary operations GROUPS

5. ADD A FOOTER 5 1. Closure : For any two elements a and b in G, the result of the operation a * b is also in G . If we combine any two elements in the group under the binary operation, the result is always another element in the group . Example : The Integers under Addition, (Z, +) 1 and 2 are elements of Z 1 + 2 = 3 also an element of Z PROPERTIES

6. ADD A FOOTER 6 2. Associativity Properties : For any three elements a, b, and c in G, ( a*b)c = a(b*c) 3. Existence of Identity: There exists an element e in G such that for every element a in G, the equation e*a= a*e=a holds. This e is called the identity element . Example: Let's consider the set of real numbers 𝑅 and the operation of multiplication. The identity element for multiplication is the number 1 if a = 5,then 1*5=5 and5*1=5 PROPERTIES

7. ADD A FOOTER 7 4. Existence of Inverse : For each element a in G, there exists an element b in G such that a*b=b*a=e, where e is the identity element. The element b is called the inverse of a . An example of an inverse element can be seen with the operation of addition in the set of integers 𝑍 If a=5, then the inverse of 5 is −5 because: 5 +(−5)=0 and(−5)+ 5=0 When a set G with a binary operation satisfies all these properties, it is called a group. PROPERTIES

8. ADD A FOOTER 8 A group is said to be Abelian if it already a group and Commutative property is also satisfied ie . (a*b)(b*a) for all a, b in G. Order of a Group: The number of elements in a group is called the order of the group . Examples : < Z +> and < Q ^ * > are abelian groups ABELIAN GROUP

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11. ADD A FOOTER 11 A subgroup is a subset of a group that is itself a group under the same binary operation as the original group. More formally, Let < G ,*> be a group and let H be a non-empty subset of G. Then H is called a subgroup (<H,*> ). If H satisfies the following conditions : Closure Identity Inverse SUB GROUPS

12. ADD A FOOTER 12 EXAMPLE CONSIDER (Z6,+) ,Z6={0,1,2,3,4,5}

13. ADD A FOOTER 13 Cryptography Public Key Cryptography: Uses the properties of groups , such as in the RSA algorithm which relies on the difficulty of factoring large integers . Elliptic Curve Cryptography (ECC): Utilizes the algebraic structure of elliptic curves over finite fields. 2. Symmetry and Geometry Symmetry Groups: Groups describe the symmetries of geometric objects and structures. For instance, the symmetry group of a regular polygon is used in crystallography and molecular chemistry. APPLICATION OF GROUPS AND SUB GROUPS

14. ADD A FOOTER 14 3. Algebraic Structures in Graph Theory Graph Automorphism : Groups of permutations of vertices that preserve graph structure . Graph Coloring: Application of group theory to find coloring patterns that satisfy certain constraints . 4 . Computer Science and Algorithms Group Theory in Algorithms: Used in algorithms for data organization, pattern recognition, and search algorithms . Hash Functions: Some cryptographic hash functions are based on properties of groups and permutations. APPLICATION

15. ADD A FOOTER 15 Groups and subgroups are crucial in discrete mathematics as they provide a framework for understanding algebraic structures and symmetry. They have applications in various fields, including computer science, cryptography, and combinatorics. The study of these structures enables the analysis of more complex systems by breaking them down into simpler, well-understood components. CONCLUSION

16. ADD A FOOTER 16 TEXTBOOK https://www.geeksforgeeks.org/subgroup-and-order-of-group-mathematics / https:// www.javatpoint.com/discrete-mathematics-subgroup https:// www.tutorialspoint.com/discrete_mathematics/discrete_mathematics_group_theory.htm REFERENCE

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