 So, welcome you all to this today video class, I am Haragish Nadekar, assistant professor, department of mathematics, is now under the hands of a state of an university. So, today we will discuss unitary that is group theory from the course abstract algebra and discrete mathematics of BA mathematics second semester. In last two video we have already discussed on the topic relation and function and also on binary operation. So, learner are expected to have clear concept on binary operation before we start today class. So, it is very important to have a clear understanding on binary operation before we start group theory. So, learner are requested to see last video on the topic binary operation. So, let us start today class on the topic group theory. So, group theory is an important topic of the course abstract algebra and discrete mathematics. So, now we start today class, before we start today class we have some objective of today and this video lecture. So, the learner will be able to define group, what is group and then you know some properties of group. So, after completing this video you will be able to know about group and various properties of group. So, now we discuss what is algebraic structure, algebraic structure. So, first we define what is algebraic structure. So, a non-empty set G together with one or more operations say star we consider that operation as a star that is operation is a binary operation is called an algebraic structure a set with one or more operation is called a algebraic structure. An structure that we denoted by G bracket G comma star and that operation here plus addition, subtraction, multiplication, star, union, intersection etcetera are some binary operations. So, G plus G comma multiplication, G comma union these are some structure of these are called some algebraic structure. So, now we define group, what is a group. So, a non-empty set G equip with a binary operation star is called a group if the following postulates are satisfied there are some condition which must be satisfied by a set G which is equip with a binary operation and that postulate are number one is associative. So, first postulate is associative and associative already we define in last video on the topic binary operation what is associative, what is existence of identity, existence of inverse these properties are already discussed in the last video. So, therefore learners are requested to have a look on that video. So, first associative, so associative we define by this property, this is called associative property bracket A star B star C is equal to A star bracket star C. If this condition is satisfied then it is called associative property and for all elements of A D C belongs to the set G then existence of identity. So, what is the existence of identity there exist an element E belongs to G such that A star E is equal to A is equal to E star A. So, for any element of G here E is called identity this property is called existence of identity property then existence of inverse. So, for every element A there should be an element B such that if we operate this element then we get an identity element A star B is equal to E is equal to B star A. Here B is called the inverse of A and it is called A inverse of B this means B is equal to A inverse this is the property of existence of identity and one more properties are there that is called commutative further more if E star is commutative then G is called Abellion group. So, if E star is commutative then it is called Abellion group. So, what we get here? So, for any set G equip with an operation E star is called a group if these three conditions are there associative existence of identity and existence of inverse and there is another property is there if E star is commutative then it is called Abellion group. So, we have to for a set equip with a operation E star is called group if these three conditions are satisfied that is called group. Here you see in order to be a group there must be an unempty set A set to be a group. So, there must be an unempty set equip with a binary operation satisfying the professor mentioned above. An unempty set G equip with a binary operation is called semi group if binary operation is only associative. So, there is another type of algebraic structure that is called semi group if binary operation is associative then we define what is finite and infinite group. So, A group is finite if the set G is finite otherwise it is infinite group if the number of element of set G is finite then it is called finite group and otherwise it is infinite. Then order of group the number of elements of a finite group is called order of the group and symbolically it is denoted by order of this O order of G or this symbol absolute value absolute this absolute value of G that is the number of elements this order of group order of group G number of elements of number of elements of G number of elements of G. Next we discuss properties of a group there are some properties of group that is here we define some properties we discuss some here properties number one identity element is unique identity element of a group is unique number two inverse of each element in G is unique number three here A inverse of inverse is equal to A. So, for all elements of G where A did not minus one is denoted the inverse of E the number four A B inverse is equal to B inverse into A inverse number five A B is equal to A C implies B is equal to C this is called left cancellation law B A is equal to C A is equal to B equal to C this is called right cancellation this is actually right cancellation for all elements of A B C belongs to belongs to G. So, here proof for a proof of these properties are not given. So, learner requested to go through the study material in study material the proof of these properties are given in details. So, kindly go through the SLM. So, this is a reference. So, if you know more about the group then you go through this some reference books are here given. So, kindly go through this reference. Now, we discuss some questions. So, we now discuss some examples. So, example one show that R with Israel addition as a binary operation is an abelian group. So, Israel addition means here addition this. So, here we have to show that R plus is a abelian group is a abelian group. So, now solution. So, we consider three elements A B C belongs to R. So, first we check closure property means addition is a binary operation on the set R. So, A plus B for any A B A B real. So, A plus B always belongs to R. So, here closure property satisfied then associative. So, A plus bracket B plus C equal to A plus B bracket plus C this will be always satisfied then identity element. So, 0 always belongs to R. So, we see. So, A plus 0. So, A plus 0 is equal to 0 plus A is equal to A. So, 0 will be identity element on the set R then additive inverse for A belongs to R we have always negative of A belongs to R such that A plus negative of A is equal to 0 is equal to negative of A plus A is equal to also 0. So, inverse of the element A is minus A. So, there will be a additive inverse in the set R with binary operation is addition then commutative property. So, A plus B always B plus A for any elements of A B belongs to R. So, all properties of a group is satisfied on the set R with binary operation is addition. So, R plus is an A B L M group. So, we have proved that the set R with binary operation is addition is an A B L M group. Let us we go to the example 2 here another things is there since R contains infinite number of elements. So, R plus is an infinite A B L M group. So, here R plus infinite A B L M group. Next example 2 here we have to say that R with usual multiplication as a binary operation is an A B L M group. So, here learner requested to try this example here you have to show that R with usual multiplication is an A B L M group. So, you are requested to solve this example. Next another example 3 the set of all integers Z set of integers with ordinary addition as a binary operation is an infinite A B L M group. This means Z with binary operation plus is an infinite A B L M group. So, this example also you try yourself then example 4 set of integers in Z with binary operation as a multiplication is not a group. So, this is not a group. So, you see why this is not a group Z with multiplication as a binary operation is not a group. See there is no multiplicative inverse in Z. So, you know that the set of Z is set of integers. So, this is the set of integers. So, here 0 is the 0 is the additive identity and additive identity 1 is minus 1. But in case of multiplication as a binary operation here you see there is no multiplicative inverse. So, we cannot find this type of element. So, we cannot find any multiplicative inverse in this set. So, therefore Z with multiplication as a binary operation is not a group.