WEBVTT

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hello everyone hope you are doing great so
in the last class we learnt about the symmetry

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elements and symmetry operations we leant
how to find out the symmetry elements and

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symmetry operations also and then towards
the end we said that after learning this whole

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set of symmetry elements and knowing how to
find them out we need to utilize some

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some sort of you know mathematical frame work
to so that we are able to employee this

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symmetry elements belonging to a particular
molecule you know in order to get some

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molecular properties out of it so in
other word we need to learn about the

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you know applicability of this symmetry aspect
of molecule and in order to have this applications

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we need a you know quantity fame work
mathematical fame work which is known as group

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theory

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so in this particular theory which is a
purely a mathematical fame work by his origin

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so long before you know chemistry stated
using it mathematician started this

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particular beautiful topic called group
theory so on this symmetry elements

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can be utilized to form a particular
group will come to those things in a while

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now before we can you know form a
so called group something and based on the

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symmetry elements of a given molecule and
utilize them to find out various things like

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you know how to form a you know linear
combination of or you know how to find

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out the probability of transition between
two molecular electronic or vibrational

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states before we do that we need to learn
a few basic things about this group theory

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so a few a criteria of something
that can be called a group

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so lets start with the definition of this
particular thing called group so what is a

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group so a group is a collection of object
or its a set of you know some distinct

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object so if i can say this is the set of
some elements ok so a set of element can form

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a group now i hope all of you understand what
is a set set is nothing but a collection of

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some distinct objects ok suppose i take few
numbers like one two and three together it

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will form a set so one two three this three
numbers they are distinct objects right when

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i put them in a collection they form a set
so here we are considering such a set and

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this set can have any type of elements ok
it can be some numbers it can be some you

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know different objects anything and the number
of elements of any set can be anything starting

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from one to infinite so if i take any particular
set having certain elements in it and then

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if there is a particular rule of combination
ok so what i need number one is a set of

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elements number two a well defined you
know rule of combination ok so rule of combination

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that can be anything suppose i you know told
you about a set like one two three so so if

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i tell that ok as there is a big addition
is a is a rule of combination which can

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actually combine any two elements of the set

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so i have a set of elements and i have a particular
rule of combination which is well defined

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in this then i can have a group this set will
be called a group provided certain rules are

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satisfied ok so what i have now i have a set
of elements and a particular definition of

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you know rule or a particular rule of combination
is defined in it it will be called as

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group once it follow certain rules so what
are those rules so those rules are as follows

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so there are four rules or four laws ok so
rule one says that all the all the elements

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in a particular particular set if i
consider them and if i take any two from

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any two elements from that tool of
elements and then combined them with that

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defined combination of combination tool
then the product or if i say that you know

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the effect of the combination ok this product
may not necessarily mean the in the mathematical

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so called product

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but if i say that you know product and combination
is synonymous so we can take it in that way

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so a product of any two elements so suppose
i have a set which is formed by some element

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like a b c and so on ok up to say x so there
are x number of elements in that particular

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set and there is a particular a particular
method of combination ok so this is the

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method of combination set is defined in it
then what this rules says that product of

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any two elements say a and b both belongs
to this set a right so a and b they belongs

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to set s then the result of this combination
of a and b suppose that produces something

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called say m ok so then this laws says
this m must be an element of set s ok

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similarly i can say that i don't have to multiply
only two different elements i can have something

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like this i can have the product product of
a with a ok so in other word the square

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of any different element say if that is n
then n also then n should also belong to s

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and here m should belongs to s then i will
call this set as closed set ok so when this

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is satisfied so if this rule is satisfied
we will call this set as closed ok and this

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particular property is also known as closure
property so the first rule for a set that

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can be called a group so we have to call
call a set or group this first rule must be

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obeyed if it fails if any say fails to
satisfy this condition or this property then

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it will not form a group ok

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so the first one is the closure property so
lets write down all of them one by one over

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here so in order to form a group it must satisfy
closure property number one ok now i said

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there four such condition if they are with
then they will make the set a group and

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we learned about first property that is the
closure property now which one is the number

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two so number two says that there must be
an element within this set lets consider this

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set again so there it says the law you know
two says that must be an element in this particular

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set such that it commutes with any of the
elements so mathematically if i try to say

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this one it will be that there must be an
element e belonging to this group s such that

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b commutes with any elements of this group
of this set this not yet grouped so e if i

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operate on any element here so if i operate
so we have this this as my defined operation

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which is also binary operation because it
combines two elements ok so i have e star

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say a now a is an elements of these group
i can take any such element then it should

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be same as a star e which is nothing but equal
to a right

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so it says that e commutes with a commute
means it doesnt matter you know the order

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of combination whether its e star a or a star
e ok here only only difference is the order

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of combination right so the order combination
doesnt matter and ends up giving back the

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element itself this element e is known as
identity element in that particular set so

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we we can correlate between this identity
element and the identity element that we talked

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about when we discussed the symmetry elements
ok so you can see they are exactly the same

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so according to rule two we have that there
must be an identity element defining that

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group a so whatever you said so far we
can say you know you know single sentence

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that we need an identity element to be present
in the set then only i can call this as a

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group ok

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so this is called that means i need an existence
of unique identity

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element ok so you cannot have two such elements
in a set and still it is a group ok so there

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will be only one element of its kind ok so
that's why it is assuming and this elements

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is known as identity elements so we are found
two conditions for satisfying that

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means to be satisfied in order to a form a
group so lets move on to law three which says

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that the law of associativity must hold
for that particular set in order to be called

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a group so this is law of associativity
must hold

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what does that mean we know what associativity
means a associativity means the order of

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combination ok suppose i again consider this
particular set having a b c to x then if i

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take out three elements from that ok so i
consider a b and c this three elements i take

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and then combined them in an order like a
in this way now if i change this order

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of combination in such a way that
lets not put this equality now now this and

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this what is the difference the difference
is here this bracket is to be making on the

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difference it says that first i will combine
b and c and their resultant we will be combined

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two a on the other side here i combined a
and b first and with this results i combined

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with c

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now if the associativity holds then this two
should be equal this is law of associativity

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and in other to form a group the set must
satisfy this law of associativity for any

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element belonging to this particular set ok
so we got our third condition that the

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set must have associative property and then
we must have another rule that has to

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be satisfied ok so we come to the last rule
that is the law of the reciprocal ok

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law of reciprocal or we can say law of reciprocation
what does it say it says that for this particular

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set if i choose any element any one element
if i pick up so arbitrarily i will pick up

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say c then there must be an unique reciprocal
or inverse for this particular element so

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i choose say c c belongs to s then there must
be some some element which will be reciprocal

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of c which is written as c inverse right such
that this combination will give e identity

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so according to that law says that there must
be some element here say i call it like you

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know q so this c inverse and q will be equal
so that means for any given elements c belonging

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to s i will find that an elements q which
will combine with c to give identity ok and

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that is true for any elements belonging to
this set s so if i take a there must be some

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some elements their r that yields a star r
equals to e ok so the forth law that we have

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here is the existence of
unique inverse ok now again here we have this

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term unique because in a set you can have
only one inverse of that particular element

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that we are choosing ok if i choose a then
there must be only one element say r such

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that r is equal to a inverse ok and you will
not find any other elements which will be

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a reciprocal of a again ok so this reciprocal
or inverse of a must be one and that must

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be a part of this set ok so then i will say
that this set has you know you know all the

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element of the set has their own unique inverse

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so once i check all these properties there
will one closure property if it has an you

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know identity element in the set if the elements
of the set follows assosiative law and

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if there is an unique inverse for each and
every element of the set then i can call this

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is set s or group ok so this s will be a called
group and most of the time we give a symbol

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g not necessary but you will find out that
mostly we use g because groups starts with

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g quickly use g to define a group so g and
they you write like a set

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now lets revise what do we have so for
any given set ok that is a collection of elements

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will be called a group when a particular rule
of combination is defined in it and all the

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elements follow certain rules one closure
second existence of identity third law

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of associativity holds and fourth is the existence
of unique inverse then it will be called a

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group you know if if the elements of a
set cannot satisfy even one of this conditions

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it will not form a group ok so that has to
be verified all the time when we are trying

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to find out if a set can form a group if i
define an define a rule of combination

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ok

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now if we had defined group now at this
point i will mention one an additional

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point about group a group will definitely
follow all this four rules ok they will

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obey all these four rules now if the elements
of that set which follows all this rules and

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they have well defined binary operation
in that additionally if it follows a rule

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as i am going to show you which is known as
commutativity so additionally if the elements

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which follows all this rules if they follow
rule of commutativity or rule of commutation

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then if this hold ok if holds good then obviously
that will be group because they have already

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satisfied all the condition for being called
a group now additionally they are you know

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following this particular condition of
commutativity which means for any two element

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a and b belonging to a particular group if
a star b equals to b star a that means

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a and b commutes right so once this holds
good for any element a and b belonging to

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g then that particular group is called an
abelian group ok so this is another very

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important a types of group that we will
come across they have some unique properties

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and all

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so we have learnt so far how to define
a group ok and also we learnt that ok there

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is a special kind of group which is known
as abelian group where apart from all this

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four properties an additional property of
commutation also hold good they are called

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abelian group now lets take some actual example
ok and then try to find out if we really understand

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this topic ok so a lets define a set lets
define a set s which contains all the rational

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numbers ok so which is a set by rational numbers
ok that is possible right and we defined

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we defined say an algebraic multiplication
ok as my rule of combination

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now is this set going to form a group under
this binary operation which is multiplication

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that's the question lets try out so in the
set of rational number i have everything right

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i have infinite here i can have one of an
infinite ok so all the numbers are there

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so now i i just check if there is a defined
operation that i already said x so ok multiplication

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is by operation now first thing is that closure
property now i take any two you know rational

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numbers because i have all the rational numbers
in my set so i take two and three and then

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see if all these properties hold so two and
three if i combine it will give me six six

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is also rational number so it belongs to this
one so i say closure property it is satisfied

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ok now is there any identity element its
very easy to find out right identity element

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will do will not change that you know
element with which it is been combined right

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and it will commute with that so if i take
any any elements say i again take three can

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i have something say can i have something
i am putting it under a code i am not writing

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what it can what is it such that i will have
three into this equals to three answer is

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very obviously right it has to be unity

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so now check with any given you know number
you take it five five thousand anything multiplication

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with one it will return you the same number
right so i have one as the identity element

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so i found my identity element now does it
obey associative law lets see again so

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i have two into three here right so lets take
a two into three into four now if i change

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the order two into three into four obvious
answer is these two are equal so associativity

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holds good ok so
associativity holds now we are left with the

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existence of unique inverse ok so what
does an unique inverse due to any elements

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belonging to a set that that inverse will
combine with the elements to produce identity

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so lets take three so i have to combine with
something which will produce my identity which

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i already have got as unity ok and similarly
if i you know this will be commuted you

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know so the answer is obvious i should have
something like this ok so i have for any given

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element such as three always i have one upon
three and this combination that is the multiplication

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of three and one upon three is going to give
me unity which is the identity element for

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this so this particular set of rational number
is a group under binary operation multiplication

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so this i can call it as a group under multiplication
ok so i will quickly give you another example

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that is the set of say all integers ok
so the previous one was a group under multiplication

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ok

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the next set i will take say all integers
both positive as well as negative integers

31:26.559 --> 31:44.169
including zero ok so including zero and i
define an operation as addition now quickly

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check if the set of all positive and negative
integers including zero will form a group

31:51.119 --> 31:57.919
under the binary operation addition first
to our closure property take any two elements

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again two and three two plus three is equals
to five three plus two is equal to five five

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is a positive integer which is already a member
of this you take five thousand ten thousand

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combine them you get fifteen thousand what
is that is an integer belongs to this particular

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this set so closure property holds for this
particular set under binary operation addition

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now is there any unique identity there is
one you can see i have the set which has all

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the integers which includes zero specifically
so if you add zero to any number it is going

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to return you the same number so it will also
commute with any number right under addition

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so zero plus five equals to five plus zero
equals to five that means zero is an identity

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elements in this particular set when i use
addition at my binary operation let look at

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the third one associativity property it holds
good because i have say i have something like

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minus two plus three plus five and then i
can write minus two plus three plus five both

33:26.649 --> 33:37.429
of them are going to give me six so associativity
law holds so third point is also clear for

33:37.429 --> 33:44.429
this set under addition existence of unique
inverse so inverse demands that for any given

33:44.429 --> 33:52.850
element i will have another element with which
if i combine my previous element it will result

33:52.850 --> 34:06.909
in having the identity element so if i take
say for example any element say plus two do

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i have something with which if i add this
plus two i will get my identity what is my

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identity i got my identity as zero so plus
two will be adding to something which will

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give zero and that something must be minus
two so if takes say another example like minus

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fifteen so i have to add with plus fifteen
in order to get my identity element

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now all these number are integers so i have
unique inverse for each and every element

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with you know in this particular
set which has unique inverse so i have set

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you know i found out that all the conditions
closure property unique and identity associativity

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as well as the existence of unique inverse
are hold good i mean they hold good under

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the binary operation plus so we can call that
the set of all positive negative integers

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including zero forms a group under binary
operation addition so we should try many

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more such thing for example if you remove
this zero from this set try it out find out

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if it forms a group under addition does it
form a group under multiplication or the first

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problem that you solve today why we define
multiplication as the binary operation will

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it form a group under addition so try out
many such operations and that will you

35:55.590 --> 36:01.710
know help you getting the basic essence
of a this mathematical group theory

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so with this will stop here and will
come back with a little bit more about

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the group theory particularly how to combine
different elements of a group using something

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called group multiplication table that will
be necessary to you know understand the

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you know combinations of different symmetry
elements which is our actual goal so we

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will learn about good multiplication table
and then will move to the combination of symmetry

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operations so with that i come to the end
of this class

36:40.660 --> 36:43.369
thank you very much for your attention and
see we next week