 Hello and welcome to the session on the topic groups under the course Discrete Mathematical Structures at second year of Information Technology Engineering semester 2. At the end of this session, students will be able to demonstrate groups and its properties. These are the contents we are going to cover. Groups with definitions and examples followed by subgroups and homomorphisms. Let us start with group. Definition for a group is a group g, star is an algebraic system in which the binary operation star on g satisfies three conditions. For all x, y, z which belong to g, x star y star z is equal to x star y star z which is known as the property of associativity that is the operation star is associative on g. Number two, there exists an element e belongs to g such that for any x belongs to g, x star e is equal to e star x is equal to x which is termed as the identity of g. And number three, for every x belongs to g, there exists an element denoted by x inverse which also belong to g such that x inverse star x is equal to x star x inverse is equal to e and which is known as the inverse element of x in g. The next definition based on the earlier definition of group, a group g, star in which the operation star is commutative is called an abelian group. Now here we notice that earlier we have defined the group with the property of associativity, identity and inverse element. So now in addition to these three properties, if star is the operation which also is commutative then the group is given a special name termed as an abelian group. Here are number of examples to understand. Example number one, let i be the set of integers the algebra i, plus is an abelian group because we know that the operation of addition is always commutative such that for any two elements a, b which belongs to i, a plus b is same as b plus a and it also holds the property of associativity because a plus b plus c is same as a plus b plus c also it holds the properties of identity where the identity for addition operation will be 0 and also the inverse being also present in set of integers i. So along with these three properties which are required for a group the operation plus is also commutative that is why the algebra or the algebraic structure i, plus is termed as an abelian group. Another example the set of rational numbers excluding 0 is an abelian group under multiplication. The explanation is left to the students as we have done it for the earlier example. Now pause the video for a while and try to answer this question. In the earlier videos we have learnt about semi-groups, monoids, sub-semi-groups and sub-monoids. Now in this video we have learnt about a group, now try to find out what may be a sub-group. Here is the answer, let g, star be a group and s is any subset of g such that it satisfies the following conditions. Number one e belongs to s where e is identity of g, star which means the identity of given group g is also present in the subset which is defined over g. Number two for any a belongs to s, a inverse also belongs to s which means if you take any element from given subset s the inverse of the same must be also present in the same set s. Number three for a, b belongs to s, a star b also belongs to s. It is simply the property of we say the operation star is closed under the operation star. So that is why we get the condition for a, b belongs to s, a star b that is a result of the operation performed on two elements a and b with star is also present in the same subset s. Then s, star is called a sub-group of g, star. So on the same line as we have defined a sub-semi-group and a sub-monoid we have also come out with a definition for a sub-group. So what is the additional condition is for any group which must be a sub-group or subset of given group. The conditions that must be true are the identity element must also be present in the subset. The element as well as its inverse must also be present in the same subset as well as if you take any two elements the result of the operation performed on these two elements must also be present in the same set. Next we define similar definitions which we have already learnt in case of semi-groups and monoids which is termed as group homomorphism. Here is the definition, let g, star and h, delta be two groups. A mapping small g from g to h is called a group homomorphism from g, star to h, delta if for any a, b belongs to g, g of a star b is equal to g of a delta g of b. If you recollect similar is a definition for a semi-group homomorphism as well as a monoid homomorphism. A group homomorphism g is called a monomorphism epimorphism or isomorphism depending upon whether g which is a mapping from g to h is of the type one to one or on two or one to one on two respectively and a homomorphism from a group g, star to itself that is g, star is called an endomorphism while an isomorphism of g, star to itself is called an automorphism. We have a special definition related to a group which we have not defined earlier for semi-group or a monoid which is termed as a kernel. Now what is a kernel? There is a definition let g be a group homomorphism from g, star to h, delta the set of elements of g which are mapped into e of h or other e h which is termed as the identity of h is called the kernel of the homomorphism of g and denoted by k, e, r which stands for the short form of kernel in bracket g. So we have come to an end of this particular topic that is groups and homomorphism. These are the references. Thank you.