Group Theory

PPT ON BASIC GROUP THEORY AND ITS PROPERTIES.

1. GROUP BY Durgesh Chahar ( M.Phil Scholar) I.B.S Khandari agra 1

2. History The term group was coined by Galois around 1830 to described sets functions on finite sets that could be grouped together to form a closed set. The modern definition of the group given by both Heinrich Weber and Walter Von Dyck in 1882, it did not gain universal acceptance until the twentieth century. 2

3. Binary Operation :- Let be a set. A binary operation on is a function that assigns each order pair of elements of an element of . Remark : o is a binary operation on iff a ο b Є .   3

4. Algebraic Structure :- A non empty set together with one or more than one binary operation is called algebraic structure. Examples :- (R, ) is an algebraic structure. (N, ) , (Z, ), (Q, ) are algebraic structures.   4

5. Group :- A non empty set together with an operation o is called a group if the following conditions are satisfied : Closure axiom, Associative axiom, Existence of identity, an element called identity Existence of inverse, This is called inverse of   5

6. Abelian Group :- A group is called abelian group or commutative group if Examples :- all are commutative group. are commutative group. 1 is an identity , is the inverse of in each case. The set of all matrics (real and complex) with matrix addition as a binary operation is commutative group. The zero matric is the identity element and the inverse of matric of A is –A.   6

7. Quaternion Group :- define a binary operation of multiplication as The red arrows represent multiplication on the right by and the green arrows represent multiplication on the right by This is non abelian group for this operation. This is called Quaternion group .   7

8. Klein’s four group Let with operation defined by the following table :   e a b c e e a b c a a e c b b b c e a c c b a e e a b c e e a b c a a e c b b b c e a c c b a e 8

9. Theorem :- Uniqueness of identity The identity in a group always unique. Proof- If possible, suppose that and are two identity elements in a group is an identity element is an identity element these statements prove that from which, we get .   9

10. Theorem :- The cancellation laws Suppose, are arbitrary elements of a group Then 1. ( left cancellation ) 2. (right cancellation ) Proof :- Let be the identity element in a group Let be arbitrary [by associative law]   10

11. Again Example :- 1. The positive integer form a cancellative semigroup under addition. 2. The non-negative integers form a cancellative monoid under addition. 3. The cross product of two vectors does not obey the cancellation law. if then it does not follow that even if   11

12. 4. Matrix multiplication also does not necessary obey the cancellation law. AB = BC and A Consider the set of all matrices with integer coefficients. The matrix multiplication is defined by It is associative, and is identity but the cancellation law does not follow and This implies but   12

13. Theorem :- Uniqueness of inverse The inverse of each element of a group is unique. Proof :- If possible, let and be two elements of a group so that ...(1) ...(2) be an identity in or [by right cancellation law.]   13

14. Theorem :- The left identity is also the right identity. Proof:- Let be the left identity of a group and let be arbitrary. Then …………… (1) To prove that is also that right identity. It suffices to show that suppose is the left inverse of a, then is the left inverse of a, then …………… (2) by associative law in . [ by (2)] [ again by (2)] [ by left cancellation law]   14

15. Theorem :- Reverse rule Let and be the elements of a group . Then then Proof:- consider arbitrary elements and of a group Since and are inverse of and respectively. and ………….(1) Hence, by associativity law, ……………(2) This . Generalizing this result, we obtain   15

16. Theorem:- If let be a group and then Proof:- let be the inverse of an element of a group then ……………(1) to prove that the inverse of is , premultiplying (1) by [ by associative law Remark -   16

17. THANK YOU 18

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