04 Groupes (Introduction)

today I want to start talking about groups um this will take a few sessions um next week I should say I'm visiting a family in Israel I might be able to make it but H it's not completely certain so I I will uh I will let let you know as soon as I can probably um I'll know for sure or next Wednesday um but as of now let's let's keep the meeting um okay so um suppose you have a set a binary operation on X is a function uh let's call it mu from x * X to X and namely you can take uh some a pair of elements A and B in each and a binary operation ought to give you a new element which ER which we denote as muab officially note that officially in our notation I should I should have added another pair of brackets here because an element in x * X is H is already with brackets and and then we we need to have the brackets of mu but um we we usually simply omit a repetition of brackets to to simplify the notation and we write new Ab ER and now a group is a set [Music] G um together with a a binary operation and mu from G * g 2 G and um uh a chosen Element e in G such that oh um and let me just organize it a bit different um and function aota from G to itself uh such that the following conditions hold so first um this is called um e is neutral uh with respect to Mu uh this means that for any uh X in G mu x e is equal to e and also to Mu e x and the second axom is associativity of uh mu um this this means that for any x y and Zed in G now mu is a binary operation so I can I can plug in two things into mu and I get a new element in G but then I can take the result and um do me with a a third element um and there are two ways I can I can do this process um one is that I take muxy and then I do mu of muxy with zed and the other is that I do mu y z and I do mu of X with mu y z and these two H these two elements should be equal that's associativity and three is the um existence of inverses for any X and G if I apply if I apply a Yota into X and then I multiply either x with aota X or aota x with X I get the neutral element now this is this is a formal definition um let me elaborate a bit uh there are a few terminological uh ER and notational conventions so usually the operation mu we denote as um a b goes to a multiplication by B so this this is just to to simplify the notation uh you you can choose either multiplication or addition or or some other notation I mean you could do of course star or something else but um multiplication is the standard way um if we adopt this notation of multiplication then the unit element is usually denoted as one instead of e and either way aota of an element aota of an element is denoted as um x to the power of of minus one if again this is a if we use the um the not the multiplicative notation um so just to to re rephrase um a group is a set G with binary operation Dot and a chosen element one such that the first axum says that x * 1 is 1 * X and is X for any EX in G the second axom says that for any XY and z in G XY * Z is equal to x y z and the third axom says that for n ex and G x * x -1 is equal to 1 and is also equal 2 x -1 * X okay this um purple definition is is um is usually what you find in a introduction books um and this um this definition in white is um maybe a more obstruct version um another another common way that we we may write a group is instead of multi multiplication we call MU addition so this would be a goes to a + b and then the unit element or the neutral element is denoted as zero and the inverse is denoted as minus X okay so uh you can write the same axom with um with addition but our convention usually would be I mean if we are given a group with no particular er um description of the operation then we will result in the the purple uh notation we just write it as multiplication but of course if if the group group it comes from a a specific operation on let's say on numbers that is either addition or multiplication then we use this this operation ER I will soon give give a lot of examples but let me point out on a meta principle um of this definition a a notion you can say maybe an algebraic notion but this this is true even wider uh um um is defined via um specification ification of um data and and data here really means just a collection of sets sets and functions plus conditions that and this data needs to satisfy so really what I what I mean here we go back to the definition of um in white what is the data the data let me put it in a a red color so the data would be um the set G the binary operation a a chosen Element e and the function iot this is data and the conditions are 1 2 and three right these These are conditions that the data needs to satisfy okay and you if you you will follow this principle I think it will be easier for you to uh to understand new definitions I mean we will we will Define ER many more algebraic structures like Fields And even Rings eventually elliptic curves but all of them can be understood with this meta principle okay so um it choose an object e a and set G to be the set containing the unique Element e this is a group in a trivial trivial manner because um there is only one function from g to g * g to g this is a function that takes e and e it has to take it to uh to and this is called the trivial group um second is um you take G to be the the set of integers and I will write it like this um the binary operation is plus and the neutral element is zero right this is like e and this is me and um the the inverse I mean for any a in z minus a is is um the the inverse of a right because H A + minus a equals the neutral element but of course I don't have to h h do addition I can do I can take G to be the set of real numbers that are not zero and the binary operation now I take to be multiplication and the neutral element I take to be one and for any X in g x - one is the inverse of X any questions so far um now of course you can say um non examples um the first non example is the natural numbers natural numbers with addition and let's say natural numbers include zero for this example um this is not a group right because I don't have a a additive inverse ER for for a natural number the second um the real numbers with multiplication and one is also not a group can anybody say what you have no idea why this is not this is not a group I mean let's check the action what what can go wrong I mean the associative it let's let's put it like this neutral element one is a neutral element because for any X in R um x * 1 = X right and of course it's also one * 6 h two for any x y and Zed we have associativity x y z is equal to X why is it and what is the third ation you remember no commutativity is not is not on the cards at the moment what we need in a group what what is the third ER what is the third condition that is required for a set with a binary operation to be a group okay but what what is the what is the precise formulation of the axum you're right that that's that's the missing axum the the inverses but what is the precise prolation of this ax to scroll okay I scroll up like sorry there should be inversive the inverse of the and yeah but what is the what is the quantification of a I mean you see the the axum says let's say here for NX in G there exist an inverse yeah okay and is there exist an element so here I could have said there exist some element which we denote as xus one such that this equation holds okay and in the example of the real numbers is it is it true for any X there is an x x to the power minus one no Z right does the same also for the yeah well no okay I will I will talk about the natural numbers afterwards but um so here the problem is um for for the element zero there is no element which you can denote as 0 to the minus one such that 0 times this element is equal to one because 0 times everything is equal to zero so so but but but if X is not zero element then it's okay then then one / X is the the the inverse of X with respect to to this binary operation right because x * 1 / X is one which is also um another x * X okay so if I just take out the zero element then I get a group this was the example that I mentioned before so if I take R and remove the zero element and take multiplication and the unit element or neutral element to be one is group okay now for a a um for the integers it's it's really okay that I don't have multiplicative inverse because my operation is addition so if G is the integers with the operation of addition n zero that's that's okay right because um for X in Zed I need to find um some element I remind you if if we have a plus then we usually denote the the inverse of an element by minus X but whatever it is I need to find element Y in Zed such that x + y is equal to the neutral element but it's okay it take take yal - x and you get it so for every X the inverse of X is is minus X you see inverse inverse is a notion that depends on the context depends on the binary operation inverse of an depends on the binary operation and depends on what you call the neutral element so an inverse of an element is an is a new element or I possibly even the same element but it's an element such that if you apply the binary operation to both of these element you get the neutral element whether the neutral element is zero or one or something else that's depending on the context and is z actually also group I mean z without zero no Z zero is a group but of course z with the binary operation multiplication this is not going to be a group okay and then here you can take whatever neutral element you want but usually if you you take multiplication then the neutral element needs to be one right but it's not going to work this is not a group yeah because for for a in Z there exist no element B in Z such that um a * B is equal to 1 let's say if if a is not one Ian if a is one it's okay but but for any other guy you don't have an inverse yeah okay so again it the inverse is a notion depending on the context if I consider the natur the the integers with multiplication then the inverse needs to be invers with respect to this multiplication whether it exists or not that's that's a different question but that's how we interpret the definition that's it okay um another example what we denote is integers modu n with addition module n and the zero element I remind you that in our construction integers module n is simply the set of of numbers zero up to n minus one so I CL that this is a group as well what is the inverse of an element in what is the inverse of some element a in ZN what do you think sorry is that a trick question no it's not a trick question it's a question of basic understanding I mean you know what is a modulus I give you an I give you I mean let's say say n equals s and I take h i take a equals 5 what is the inverse of a you need to find what is what is minus a again because we we have operation plus then inverses are written as so what is minus a minus a is if it exists I mean needs to be some element an element in the in this in the set of the group I mean so in the set zero up to n minus one such that operation minus a would be equal to the neutral element zero now here this operation ER I can write simply as it means that I need an element such that a plus minus a is equal to zero modu n right this is another equivalent way of writing this do you know any element in in the set it's now it's very concrete it's zero up to seven you can try all the options a is equal to five so I need to find um an element such that five plus element that I call minus five such that 5 + - 5 modu n is equal to Z is equal module 7 right mod 7 if I do 5 + 1 how much is it modu 7 5 + 1 mod 7 Yes No 5 + 1 is six modu 7 Z no it's like modulus is like is like clock arithmetic right if you do uh 13 is equal to 1 and uh 14 is equal to 2 and so on I mean clock arithmetic is module 12 and so but so if I do a I mean if I I tell you 11 11 in clock arithmetic is still 11 right so six mod 7 is still six okay 5 + 1 mod 7 is is six 5 + 2 mod zero so we take - 5 to be 2 and then 5 + - 5 5 is equal to 7 which is equal to 0 mod 7 okay so two and two is indeed in the set in the the set of element of the groups it's like it's a it's a legitimate element from zero up to six generally if I give you if generally if n is some a natural number and a is an element in integers modul n that is just the numbers 0 up to n minus one with addition module n what is the inverse of a minus a what should I Define minus A to B just go back to this example of a um seven if I now want the inverse of a three what is going to be the inverse of three in this in this Z mod 7 what is going to be the inverse of three yeah yeah so in general this is minus a is going to be n minus a depending on the but because obviously a + nus a is equal to n which is equal to Z okay and this only natural numbers what what is I mean and to take modules you need a natural number you cannot do an integer okay now um let H C be the complex numbers that is um the set of numbers of the form x + y i such that X and Y are real numbers and Define multiplication um if I have Z equals to x + y i and Z Prime equal x Prime + y Prime I then Z times Z Prime is defined to be well I do it's like doing a a the usual algebraic operation on these on these variables so this is a x * X prime plus x * y Prime I plus X Prime y i minus y y Prime okay so this minus means really that that um I * I or I sare is = to minus1 and this I can rewrite I mean this this means that I have x * X Prime - y * y Prime plus XY prime plus X Prime y all multiplied by n you you notice um um complex numbers multiplication have you seen it before I mean you you remember no I was thinking netive yes so this is we we simply Define it like this this is a definition and you can check that this is an associative operation um dot is is a associative and the element 1 + 0 I is one neutral with respect to this operation right if you you you can picture it like this you have um two coordinates basically so this is a this is called X is called the real part and Y is called the imaginary part and I I can I can draw complex numbers on a a on like a two AIS plane this is going to be the real axis this is going to be the imaginary axis and if I have a number a complex number Z it has to be equal to x + Yi and so this is how I go on the the real axis and this is how I go on the imaginary axis okay the imaginary refers to this this I because it's the imaginary part of X is like the the coefficient of the of the i i is is referred to as like a formal symbol that satisfy so I is a formal symbol let's satisfy guys I * I which is i s right is equal to -1 of course if you think of it it's the same as saying that I is like the square root of minus one but this is this is only in in quotes quotes un quotes because we cannot do square root of ordinary numbers as I mean the operation of square root as we know it cannot be done for um for negative numbers but we so we introduce a symbol that satisfy this kind of imaginary condition we can do it it's not a problem because it's for us it's just a formal symbol and we we Define multiplication very I mean very concretely very formally you take any any so-called comp complex number if you want to multiply it by another one I give you a formula and you can check that this formula is associative and and and has a mutual element one and so on it's very easy um to clear that's just a trick we are using right so this is just a mathematical trick to kind of yes I mean of course it has it has a good reasons I mean there are good reasons for talking about complex numbers and I will mention it soon but um for now I I want to say the following you take the complex numbers and you remove zero you take the operation dot that I just defined above and you take the element one that that is equal to right to 1 + 0 I and I claim that this is a group okay so um associativity and a a neutral element aoms this is it's very easy I leave it as an exercise the nontrivial part is inverse existence of inverses so let Z be a general complex number I write it as x + y i and this is a complex number that is not zero so now I want one/ Z so one / Z should be 1 / x + y i this is what I want but this is not in the form of the the complex numbers as we defin them so I'm going to start from what I want this is kind of a reverse engineering I'm start I'm starting with what I want and I'm going to modify it until it becomes in the form of complex numbers as I Define it so what I do is I multiply and divide by xus y i right this is this should this should be a completely legitimate action because I'm just multiplying by one but if you multiply x + y i with xus y i what you get is well here in the the numerator I get x - y i and in the denominator I get x² + Y 2 H sorry let's see this is a let's just write it like this x + y i x - y i and this is x - y i divided and now I do I have x² and minus XY I plus XY I right and then plus y s so this is x - y i ided by X2 + y^ 2 okay and this I can write of course as X ided by x² + Y 2 plus sorry minus y ided by x² + Y 2 I okay so this is going to be my inverse you see that um so we we simply Define Z minus one to be this this number and if you reverse the the sequence of equalities you will see that a z * zus one is equal to 1 right why H why why cannot I have Z equal Z why why is this construction not working for Z equal Z what is what is the zero element in complex numbers complex numbers are this X plus y i right what is the zero element here what what X and Y you you need to take in order to get the zero element what X and Y do you need to take in order to get the the the neutral element the element one this weet right to get the element one what you need to take is X and Y to get the element one you need to take x = 1 and y = 0 and then you get like Z 1 + 0 I but 0 * anything is is assumed to be zero that's one and so what do I need what X and Y do I need to take in order to get the element zero Z zero yes so if I take the zero if I take 0 0 I mean xal 0 y equal 0 then this Construction does not work because X2 + y^ 2 is equal to Z and I cannot divide by so this this construction only works so Z is different than the zero element if and only if x² + Y 2 is Z right okay so but you see now the complex numbers this finishes the proof the complex numbers uh are a group with respect to multiplication that's we yeah so we had a group iation I mean that was not a grow I introducing this concept we know having the same yeah almost the same complex number but why is it still called Z no this this Z is like a just a a a lowercase alphab like a English alphabet Z okay the the integers I denote like this which I mean you see the difference it's first of all it's a capital letter and second I I do this H this extra line ER any questions before the break okay okay so as here in 10 e e okay so staying with complex numbers um what we can do is uh represent the complex number geometrically uh that is I take some complex number Zed is some X Plus y i i remind you X is like the real part and Y is the imaginary part and so this this length this length is y and this length is X and now I mean suppose this complex number Zed is is not zero then I draw a line like this I put an arrow this is like a vector it has some a angle that is generated between the arrow and the the positive direction of the real uh the real axis and as a result is I can I can do um cus Theta is um y/ X and cinus Theta H sorry um I Define this quantity r as x² + y s and then C Theta is y / R and cinus Theta is x/ R do do you uh do you follow this is a basic definition of these trigonometric functions right so so now I can write why didn't just start by us the instead us the I I would anyway need to use the I because I need to to encode a way to multiply two complex numbers you see if you take just so you you you understand the the the role of I here I could represent Z as just a pair right X comma Y and then I if I have another one I have X Prime comma y Prime but how do I multiply it's not the same I mean z * Z Prime if I if I Define it to be x x Prime comma y y Prime this is not it's not going to satisfy the axom of a group right because for example I I would have um a a z comma comma 1 Time 1 comma 0 is going to be equal to zero I get two I start with two nonzero elements I multiply them and I get zero but if this was true and I would have a multiplicative inverse I I would be able to multiply the the left hand side and the right hand side of two this equation by let's say the inverse of the element is one Z right and then I would have 0 1 Time 1 0 Time 1 Z inverse is equal to zero here zero I say zero but this zero is z zero right and this would be equal to 0 0 times whatever whatever this is one one 0 time minus one but 0 0 times anything if I Define multiplication here as here 0 0 times anything is0 0 so I would get 0 0 on the right hand side and on the left hand side this guy needs to be equal to to one to the unit to the neutral element so I would get 0 comma 1 times the neutral element whatever e but by the ax of a group 0 * one times times the neutral element is 0 * 1 0 comma 1 I would get that 0 comma 1 is equal to 0 comma 0 that's a contradiction okay so this this multiplication this maybe we call it a remark with this multiplication this is not a group okay because inverse of an element cannot be but if I Define multiplication with this twist if I Define multiplication with this twist then I do get an invers this is what we we saw before the break right so that's the importance of this I it it allows you to remember of course look I I didn't have to introduce I any in any sense I could have said we could have said complex numbers is simply pairs of real numbers x x y H the zero element is H 0 comma 1 the the element one sorry the zero element is 1 comma 0 the the element this sorry the zero element is 0 comma 0 the one element is 1 comma 0 and now multiplication if I have some pair XY and I want to multiply it with another pair X Prime y Prime then I do um just like I'm I'm used to in the formulas so it's going to be let's look at the the formula of uh you see here I'm given in fact the formula so this is a here it would be x x primeus y y Prime comma X Y Prime + y Prime X or or plus X Prim y okay so this it's the same here there is no I I just simply Define a formula for for pairs of real numbers and you can check that this formula satisfies assoc associativity and and inverse and everything but it's not so easy to work with such a formula it's much easier to work with this representation x + Yi and then I know that if I I have another guy I and I want to multiply I multiply just like I do in algebra and all I need to remember is when that when I multiply I with with itself I get minus one okay so this you can regard this I as as a just a way to remember the the formula of the product but in any case if you look at the formula of the the product geometrically we have z = x + y i this is y this is X and this is R which is x² + Y 2 and if this is Theta the the angle between the line of R and the positive h a part of the real axis then um C Theta is y / R square yeah square root square root sin Theta is y / R cinus Theta is x / R and now I can write z with x + y I so here it means that a y is R sinus Theta and X is R cinus thet so this is r c theta plus r cosinus Theta I and I can extract r so this is R cinus theta plus I plus sorry C Theta I I I confused the sin and and the cinus I mean X is is R cinus Theta and Y is our Theta so I get R cinus theta plus I Theta but now let's abbreviate I call it kiss kiss Theta kiss is Just this H cinus theta plus I Theta okay now let's suppose Z Prime is equal to X Prime + y Prime I and now I I write I write z Prime as RP Prime K Theta Prime okay and now I want to multiply Zed with Z Prime this is like doing I have R his Theta and I want to multiply it by R Prime K Theta Prime and so this is R * R Prime and now I have this cinus I open it up cus theta plus I Theta times cinus Theta prime plus I C Theta Prime right so I I get R * R Prime and here what do I get I get cinus Theta cosinus thet cus Theta cus Theta Prime um I guess I have this formula somewhere but the second is um I cinus Theta Theta prime plus a i c Theta cinus Theta um or in fact minus C Theta sin Theta Prime and if you you check so I I I put it like this r r Prime um I have cinus Theta cinus Theta primeus C Theta C Theta prime plus cinus Theta cinus Theta prime plus sin thet coin thet Prime uh sorry uh I have I cinus Theta Theta Prime and then C Theta cinus Theta Prime and all that is multiplied by I now there is a formula for um for these these things let me just uh verify it yes so this guy is simply cinus of theta plus Theta Prime and this guy is sinus of theta plus Theta Prime this is a ER trigonometric identities you can do I mean you usually see in school and so what I get is R * R Prime kiss theta plus Theta Prim in other words if you want to picture multiplication geometry ically you have let's say Z and Z Prime you take the radius of Z and the radius of Z Prime this is R and R Prime you take the multiplication of the radiuses and you add the uh the two angles so I have let's say we do it in another color so here I have Theta Prime and here I have Theta and the result would be some guy Z * Z Prime the the radius of Z time Z Prime is going to be simply the multiplication of the radiuses r * R Prime and the the angle here is going to be theta plus Theta Prime okay so it's it's a very convenient way to picture the the multiplication you multiply the radiuses and add the the you add the angles in particular if R is equal to 1 and R Prime is also equal to one that is a a z and Z Prime sit on the unit circle they um S1 the unit circle is simply the collection of all complex numbers whose radius is one or if you want in formulas it's the collection of all Z equals x + y i such that x² + y² = 1 in square root or without square root okay and if I do this then multiplication of complex numbers if I have Z and Z Prime then to multiply Z and Z Prime is to multiply the radiuses but the radiuses are one so I I I don't need to do anything and I just need to add the the two angles so Z * Z Prime is going to be I mean let's say let's say this is a um pi/ 2 and this is a or we can do it without radians let's say this is 45 degrees and this is a 120 then I multiply it I multiply the two of these together I get 165 165 is somewhere okay and in particular I get um consider an equation Z to the N equals 1 over the complex numbers a solution to this equation is called an N root of unity let's say z0 is an n root of unity now how does an N FR of unity look like say n equals um five I take the unit circle I divided into five equal parts so this this guy let's call it z0 this this is a um one right one is always a solution to this equation because one today this is 1 + z i right this is the real number the real axis this is the imaginary axis so 1 + z i i i simply write as one this is a solution right it's a solution to the equation Z to the N equals one so it's an N root of unity but this should be expected one is always a solution to this now I divide the the circle into five equal parts starting from starting from one starting from here so I have H 360 divided by 5 what is it 72 right much er sorry no 36 you're right ah 72 okay 7 72 so 72 I I do 72 here this is like here this is going to be Z1 I move another 72 so it means a 144 actually here and this is going to be Z3 sorry Z2 and I have furthermore Z3 and Z4 and these are all solutions to the to the equation Z to the 5 equal 1 how come if you do Z Z1 to the five this is simply doing 72 degrees times five right it just to do to multip we said that to multiply two complex numbers that have radius one you simply add their their Ang so if I do Z1 to the the five I simply need to add the angle five times so this 5 * 72 this is 360 which is zero it's so um or I should say ER because this is zero Z1 to the 5 is going to be equal to one right it's going to be equal to the the guy one has Theta equal Z the unit the element z0 has has Theta equal to zero so when I do Z1 to the file I need to add the the angle of Z1 five time to itself I get the angle 360 which is angle zero so I I know that Z1 to the 5 is equal to 1 right how do I add how do I add two numbers sorry how do I multiply two complex numbers on the unit circle I simply add their angle yeah so if you take 72 you you take this Vector in purple you multiply it it with itself five times you need to move to move it five times 72 degrees so you go back to the beginning here but that should be defined in exactly this Z because suddenly I mean you're in Z 10 this doesn't hold anymore no in Z 10 I would need to divide the circle into 10 equal parts okay this is not going to be so Z1 is not going to be a a a 10 root of unity it's going to be a a five root okay but but the same argument would show you that Z2 to the to the five sorry to the five is also equal to one right because you you add 144 to itself five time you get 76 720 which is zero no I think that's the buttery but um so just to finish the point um so Z1 up to sorry Z z0 up to Z4 are the are the solutions and this is in fact the only solutions uh to the equation Z Z to the 5 equals 1 now these Solutions let's call it um mu of C mu n sorry mu in this case mu5 C so note that mu5 of C is a group with respect to multiplication of complex numbers because if I multiply Z1 and Z2 I get Z3 right if I to multiply Z1 and Z2 I need to add their angles so I get Z3 If I multiply Z three with itself I get something else but it's still it's still going to be it's all going to to move in around these uh these Solutions right you multiply any of these two these two guys I mean it means that you you have an angle Theta which is 360 divided by either 1 2 3 4 5 right and you add these two together you get still something of this point you get an angle an angle of this form so these guys form a group these these five vectors they form a group the unit element is of course the the the vector z0 which is one you multiply it has an angle zero you multiply any other Vector with this with this guy you get the other vector right so so this is a group and this is called ah and the inverse what is the inverse what is going to be for example the inverse of Z1 here what is Z1 to the minus one yes exactly Z1 to Theus one is going to be Z4 okay and what is the inverse of Z2 Z3 okay so this is a this is another group um in fact the argument I said here is valid for any N I just I did five for kind of illustration but is zero this is always true in a group the inverse of the the neutral element is the neutral element so this actually this is maybe we have um we have 10 more minutes so I'll just say generally um mu n c is called the group of nth roots of unity okay um now let me come back this you reminded me that I wanted to to say something er er fundamental on on the definition of a group uh and this is H this is the following um if G with let's say multiplication and and neutral element one or let's let's call it e is a group then um the the neutral Element e is unique unique in what sense in that if e Prime is another element in G that kind of pretends to be the unit element so such that for any X in g x * e Prime is equal to X which is also equ Al to [Music] um e Prime * X then I claim that e has to be equal to e Prime you cannot have two neutral elements why if I had such a an imposer e Prime then a a e * e Prime would have to be equal to e because e is a neutral element I mean e e Prime is a neutral element it means I I multiply the the somebody who says that is a neutral element I multiply it by any other guy I get the other and but also so so this is since e Prime is neutral but also e Prime Time e is going to be e e Prime because e is neutral I idea done if we go back to the Z we apply this to Z wait wait a minute I want to I want to stay here um so so e * e Prime is equal to e h because e Prime is a neutral Element e Prime Time e h is is equal to e Prime H but e e Prime * e is is also equal to e * e Prime actually I I could have written it let's let's just do it e times e Prime is is equal to e Prime because e is a neutral element neutral element if you multiply it from the right or from the left with any other element you get this other element so I get that e has to be equal to e Prime now I claim oh okay so you had a question on z0 so we have Z where the was like new element and if I take this new element by itself it will be C whole squ which is in order Z1 no z0 square is not Z1 z0 Square what is the angle of z0 Z what how do you multiply two complex numbers with two angles Theta okay so that's why the neutral element has to be unique now there is another thing um if um let's call it x -1 Prime is such let's say if H yeah let's let's call it x - One Prime is such that it x * x -1 Prime is equal to e okay so if I have an imposer to the inverse like another element potentially that is equal to the in that that is that is functioning as the invers of X so here it would be x -1 Prime Time X so I I I claim that xus one prime would have to be would have to be equal to xus one why I take for example this part of the equation and I multiply it on the left by x -1 I get x -1 * x * x -1 Prime equals to x -1 * e These Guys these they multiply into e right this multiplies into x - One X - one and this Tays so now I have e times anything is this this other thing so I get x -1 Prime equals to x - one okay so so so inverse of an element is also unique okay so it really means I mean this it kind of follows from the aums I could have I could have required it in the definition but I don't need to it follows automatically from the axom that if you have an inverse if you have a a neutral element there could they could only be one of them you cannot mess around can you please go back to the structure of the z f here show what graphically what that means in the what what is the question exactly the last sentence like the Z the Z minus one Primal to zus one yeah it means that I mean what you know what is the inverse of a of Z2 the inverse of Z2 is is Z3 right so but the there cannot be any other guy apart from Z3 so Z2 minus1 is equal to Z3 yeah and it cannot be equal to any other guy there is it cannot be that Z2 minus one is also equal to Z1 this this would never be because the inverse is unique the inverse is UN that's true and now Prime what is Z3 Prime because we Noe the prime here was a notation just to um I mean suppose I had an element I called it Prime just to kind of distinguish it from x minus one I mean the assumption is that I had an element this is a weird notation but just an element in G that satisfy the the neutral I mean the inverse element equation or the inverse element axing with respect to X okay like suppose that I had an element X and for this element X I had a a a some element that I denote as xus one prime that would satisfy the the inverse axum for X like so right like like is written Then I then the claim here is that this element would have to be the the one that is given in the axom namely X I mean the inverse of x to the minus one yeah okay so this is really just a formal a formal way of phrasing the way understood it is that the prime should be the same for like Z not no this yeah just like a way I Den note things yeah if I if I want to keep the original letter I don't want to change letters too much then I call one thing I call Z and one one I call Z prime or x and x Prime and so on this is like yeah just an notation convention but it's it's important to to get used to this this stuff because this is I mean it's the way we talk about these things all right um questions before we wrap it up Craig we didn't hear from you at all today do you have any any remarks I guess not okay so uh see you next week unless I mean unless I send a message of cancellation thank you with