 So, just to put in this notation which just like SI units or your periodic table with elements having some symbols, there is some notation called Schoen-Feyes notation. C n is for an unfold z axis rotation by 2 pi by n, C n v in this case C 2 I have written C 2 v for an unfold rotation plus reflection in the y z plane, C n h is an unfold axis rotation plus reflection in the x y plane. I am going to take the axis to be along the z axis, this is what I was explaining in the last lecture before we. So, I am going to take the plane of the molecule to be the x y plane, this board plane of the board to be the x y plane, I am going to take the axis to be coming out of the board that I am taking it to be the z axis, is that clear? So, if I want to say sigma v plane, sigma v plane should have the z axis, axis of rotation is taken to be along the z axis, the principal axis of rotation and sigma v plane will be always in the involving either x z or y z, it should have the z axis. Sigma h plane will be only in the x y, because it has to be perpendicular to the z axis, the plane has to be perpendicular, here it should contain the z axis that is the definition, is that clear? So, those are the short end notations and each one is a point group ok. C n is a point group, C n v has C n as a subgroup and C n h has C n as subgroup again. The question for you is can we write C n v as C n cross sigma v always? In the C 2 v we wrote it, that is question with something which you have to look at the ammonia molecule, what is the ammonia molecule? You have the 3 hydrogens or you can take the nitrogen through which you can put a triangular 3 fold axis and rotate it right, rotate it by 2 pi by 3. So, this has a C 3 symmetry, you can also put a plane, you can put a plane containing this axis, let us call this to be y axis and then what happens? If you put a plane through this, then you have a sigma v, a question is a C 3 sigma v question mark, is it same as sigma v C 3? So, these are questions which will tell you whether it is a direct product group or semi direct product group or these are only generators by which you can write the group. To see a semi direct product, you have to see whether there is an invariant subgroup, is that clear? So, is this happening is the question. So, if I do a mirror symmetry, let us keep the mirror here, let me number the atoms. So, when I put the mirror and let us concentrate on the atom 3, atom 3 goes to atom 2. So, let us do that sigma v on atom 3 it goes to atom 2 and now I want to do a C 3 on it, C 3 on this, C 3 on that will give you atom 1. If you do C 3 on 3, this goes to the 2 pi by 3 rotation about this axis should take the 3 to 2 and then I want to do sigma v, sigma v will go to what does this tell us, it is not coming. So, you can figure it out this way that this group is a non-Abelian group unlike this which is Abelian and then you have to see once it is non-Abelian, you know it cannot be written as a direct product of C 3 and sigma v and then you can see whether there is an invariant subgroup what will be the order of this group. So, the order of this group this is generator right, you can just multiply because we have to multiply every element by every other element. So, there will be 6 elements. So, you can write the elements of C 3 v which is another point group and this is the point group symmetry of ammonia molecule, symmetry of ammonia molecule a C 3 v and the elements are E, sigma v, sigma v C 3, sigma v C 3 squared and then C 3 and C 3 square. Anything I have left? I have just multiplied the sigma v with C 3 and C 3 squared and without sigma v, now can you see any subgroups of your symmetric group to which this will be isomorphic which one. In terms of Ab you can see the same symmetry and you can also rewrite this as can we write it also as a class structure. Identity element union C 3 and C 3 squared is the belongs to the class and then this is sigma v, sigma v C 3, sigma v C 3 squared. There are 3 classes to see that it is a class you can also do sigma v C 3, sigma v here and check it out. C 3 will go into C 3 squared, you can check it out even here by operating whatever I have done, you can show that sigma v C 3, sigma v will be C 3 squared and you can show that C 3, sigma v C 3 be some answer and then you will also find C 3 squared sigma v C 3 sorry inverse. This is inverse and C 3 minus 2 which is nothing but which is C 3, what is that ok. So, you can try out this and check that it can be broken up into conjugacy classes which will also tell you that in the planes of symmetry you could try to say that you can have another plane going through this axis, but containing you know it can be like this. You can call this plane as sigma v, you can call this plane as sigma v prime and similarly this plane as sigma v double prime. So, there are 3 planes of symmetries equivalently these 3 elements which I wrote can be independently written as if this molecule has 3 planes of symmetry each of these planes has the principal axis, it contains a principal axis. So, they are all vertical planes. So, there are 3 planes of symmetries and one principal axis. So, you can show that if you have a principal axis and 3 planes, the 3 planes will be related by a similarity transformation. So, they will all be conjugate to each other. So, that is why this set of elements which are the 3 mirror planes will be conjugate to each other and C 3 and C 3 squared belongs to the conjugacy class. And once I write it this way as she has already pointed it out that you can write this to be like b and b square and then this is a a b and a b square or you can write it in the permutation group of 3 objects that it is going to be this one will be like a 1, 2, 3 and a 1, 3, 2 this is identity which is 1, 2 and 3 these 2 are in the same class and then this one will be a 1, 2 plane, 1, 3 plane and a 2, 3 permutation, transposition. So, these 2 are isomorphic to so C 3 v is isomorphic to electric group of degree ok. Whatever we did in the earlier lecture is formalism now you can see to be applied to looking at the molecular symmetries ok. So, I also briefly said we call it dihedral groups in the picture of these molecules in the picture of these molecules besides the principal axis of rotation you could also have a two-fold axis of rotation let us say it is the y axis it is not a plane it is still a proper rotation you can have a two-fold axis of rotation what is two-fold axis 180 degree rotation. So, if you have that additional two-fold axis of rotation over on top of this principal axis which is an n fold you can call that two-fold axis of rotation as u 2 you will have a C n group for the principal axis of rotation you will have an u 2 group which is similar to what we saw this one was one identity r r squared r to the power of n minus 1. I did briefly mention about the dihedral groups this one is identity n s. So, r to the power of n is identity here s squared is identity and then we try to combine all possible terms and I said that is the dihedral group which is denoted as d n which will be made of how many elements what will be the order of the group this has order and this has order 2 when you take these two together and write the whole set it will be order 2 n you will have 2 n order elements ok. So, you will have identity s then s with r s with r squared and so on and then r r squared n minus 1 conjugacy class we did this there where it was very simple, but here you need to see what are the conjugacy class ok. So, these are things which require some more information that if you have a principal axis and if you have a two-fold axis perpendicular to the principal axis ok, then you can show that r a should be same as r n minus a should be conjugate 2. So, I should say that there exist an element s s inverse which will give you r n minus a in such situations you will have this. Here also I should say that even though we did c 3 v even if you had an n-fold axis symmetry here and if you had a plane which contains by plane I mean a sigma v plane do not confuse the sigma v plane with this axis the two-fold axis we showed by the symbol the sigma v plane here if you want you can put like as if it is a mirror. If you have a sigma v plane with containing the principal axis here also you can show that sigma v c 3 k sigma v a c 3 sorry not c 3 c n in general. So, what does this mean c n k is conjugate to c n n minus k that is why c 3 and c 3 squared are conjugate elements. So, I am not proving it for you, but you can show that whenever you have a mirror plane containing the principal axis then you can show that c n k any power k power c n n s identity. So, they will be conjugate to each. So, you can use those properties here and try and do how to break this into conjugacy class and will it depend on when n is odd, when n is even. So, some of these questions if you can think over it we can discuss the structures. So, d n group u 2 is the two-fold axis symmetry two-fold axis symmetry rotation by y axis by 180 degrees. The principal axis will be a rotation about z axis by 2 pi by n clear yeah ok. So, that is what I am mentioning on the screen d n represents n-fold axis rotation plus a two-fold axis in the x y plane which is perpendicular to the z axis that is what I mean ok. So, this is this can be rotation not only in the y axis it could be in the x y plane in general axis could be, but it should be perpendicular to the principal axis. So, that is the dihedral group. In the earlier lecture I just gave it as generators and I asked you to generate the set, but now you have a physical interpretations on molecules which have these kinds of symmetries that is why these terms start coming up when you ask people in chemistry working on symmetries of molecules ok. And then it goes on you can have a d n and then you all can also have a d n is a proper group ok only pure rotations, but d n h is on top of d n you also add a mirror plane in the x y plane. So, you will have more elements ok. So, d n plus a reflection in the x y plane. So, this is the way the notation continues ok. So, this I have already explained to you I have shown the planes of symmetry and axis of symmetry and then I said find out what is the symmetry of this. This is very now simple to you to see what are the allowed symmetry you can start seeing does it have only pure rotations can it have some more reflection planes all these things you have to make it maximal it may have a subgroup like here water molecule as a C 2 symmetry which is a subgroup, but you have to make it maximal that beyond that you cannot add any more symmetries. So, C 2 v is the symmetry of the water molecule which includes rotations and the reflections. Similarly, for the methane molecule you can show that it is C 4 v because all the 4 hydrogen atoms can be rotated amongst each other you can start putting reflection planes ok and so on ok.