Group theory presentation by vasundhara kumari

Group theory presentation by vasundhara

1. Group Theory and it’s Applications Seminar presentation By:- Vasundhara B.Sc.(CS) Enroll no.:- 8822204007

2. Table of contents:- 1. About Modern Algebra 2. Definition of Groups 3. Order of a group and order of an element 4. Modular Arithmetic 5. Subgroup 6. Lagrange's theorem 7. Applications

3. Modern Algebra:- Modern algebra, also called abstract algebra, branch of mathematics concerned with the general algebraic structure of various sets (such as real numbers, complex numbers, matrices, and vector spaces), rather than rules and procedures for manipulating their individual elements.
Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras.

4. Groups:- Let G be a non-empty set and * be a binary operation defined on it, then the structure (G,*) is said to be a group, if the following axioms are satisfied, (i) Closure property : a * b ∈ G, a,b ∈ G (ii) Associativity : The operation is associative on G . i.e. a*(b*c) = (a*b) *c, a, b, c ∈ G (iii) Existence of identity : There exists an unique element e ∈ G, such that a * e = a = e * a, a∈G e is called identity of * in G.

5. (iv) Existence of inverse: for each element a∈ G , there exist an unique element be G such that a*b=e=b*a The element b is called inverse of element a with respect to * and we write b = a^-1 Abelian Group :- A group (G,*) is said to be abelian or commutative, if a*b=b*a, a,b∈G

6. Some Examples of Group The set of all 3x3 matrices with real entries of the form is a group. This group sometimes called the Heisenberg group after the Nobel prize-winning physicist Werner Heisenberg, is intimately related to the Heisenberg uncertainity principle of quantum physics.

7. Order of a Group and Order of an element of a group:- Order of a Group The number of element in a finite group is called the order of a group. It is denoted by o(G). An infinite group is a group of infinite order.
e.g. 1. Let G = {1,-1}, then G is an abelian group of order 2 with respect to multiplication. 2. The set Z of integers is an infinite group with respect to the operation of addition but Z is not a group with respect to multiplication.

8. Order of an Element of a Group Let G be a group under multiplication. Let e be the identity element in G. Suppose, a is any element of G , then the least positive integer n , if exist, such that a^n=e is said to be order of the element a , which is represented by o(a) = n In case, such a positive integer n does not exist, we say that the element a is of infinite or zero order. e.g., The additive group Z={...,-3,-2,-1,0,1,2,3,...} 1.0 = 0 order of zero is one(finite).
But n=!0 for any non zero integers a. O(a) is infinite

9. Modular Arithmetic:-