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0:00
well I'm mentioning the Super Bowl laughs last time I was thinking along to mention something else that forgot to remember now you if any of you do like watching the Super Bowl you're really in a nice time a nice period of Super Bowl history 2000s and 2010's most of the Super Bowls have been very close very pretty good games when I was your age pretty much all the Super Bowl blowouts you're like well another Super Bowl another blowout so we can be thankful
0:32
I'm really excited today because I have some technology to help me went to the library just five minutes ago and got a clicker and they said I could even borrow it for the whole semester they go through the powerpoints here and we're going to have some we've got some high technology here with PowerPoint went back up we're going to also use some low technology here with an overhead projector today as you might guess we're going to talk about dihedral groups a little bit with the
1:00
symmetries of equilateral triangles and maybe squares if we get to it so I'm pretty excited about that all right so lecture a we'll look at the definition of the formal definition of a group talk about something called caitli tables which you read about maybe do a proof related to that or two related to the definition of a group if we have time or with metic examples and started talking about day 92 a groups that may carry over into lecture four be where we talk a little bit
1:28
more about equivalence relation number theory oops we'll see how we are as far as time goes alright so let's be a little bit more formal now what properties do we require a reserve you think you can name them based on I've been sort of doing this kind of thing before closure again the text states that there is a binary operation this is assumed in our textbook not necessarily in other textbooks that it means it satisfies closure but that does need to be checked for individual
2:02
examples the book mentions are really good example to think about if you think about these irrational numbers which include things like square root of 2 square root of 3 pi e that kind of thing under multiplication and you also include say the number 1 so you make sure you have an identity in it under multiplication the Irrational's are not closed for example square root of 2 times itself is 2 which is not irrational okay it's also not 1 so you do need to check it for some
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specific examples but again as far as the definition goes the book says it's a binary operation that is going to automatically mean it's closed what's the next program
2:48
to be a citizen in property associative property up associativity by the way the binary operation technically speaking as a functions from the Cartesian product of the group with itself to the group itself that takes us an ordered pair a comma B and maps it to a times B say you could put the dot in there may not be four times but its traditional type not to bother like usual in math just right 1880 meaning eight times B so that's what the binary operation does and it's guaranteed
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that to be an element of G we don't usually bother running the ordered pair right we just write the product but that's really what's technically going on in the background the Scioscia tivity means
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it doesn't matter where you put the parentheses a V times C meaning do a times B first and then times C equals this where you do B times C first and then left multiplied by a so that's the social tivity what's the next property sadnesses so they're good you have the existence of an identity
4:19
and in the element there exists a special element called the identity e standing I think they say it's a German abbreviation for something of the word einheit basically means I've come to be a German there existed Eng such that what a times e equals E times a equals a or should I finish it bar I should add something else
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for all Aang
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we use e and the abstract theory some people use the simple one because the identity under multiplication is often one or maybe an identity matrix that people often use the symbol 1 4 as well or capital I say but when we're doing the theory in here we use lottery and existence of what finish it money of inclusive see how the existence of inverses
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for all ang there exists an element
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we will traditionally call it a inverse and write like that I'll go ahead and write it like that right from the start such that when you multiply it on either side of a you get back get in a inverse equals 8 and verse pay equals in anything else that's great here anything else
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creativity well no technically not community is not required why not because there's lots of examples that are not commutative matrix multiplication examples where your elements are matrices square matrices and the group operations multiplication those matrices have to have nonzero determinant as well to be in a general linear group function composition we saw that example where the elements the group of functions and the operation
7:04
was function composition [ __ ] function composition in general is not commutative those kinds of examples are important kinds of examples that we want to be groups and if we want them to be here then we better not require community however some groups are communion like in modular arithmetic Zn under addition is commutative you and we talked about that before talk about it more today under multiplication the group of units modulo n
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is also commutative abelian is what's called more efficient when it is satisfied when commutativity is satisfied that the group is called an abelian group don't say a valium say abelian after all
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okay
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it's nice for nursery communitive they have more properties when they are used but some groups are now let's prove a shoot a few things here based on this so this is pretty broad so you might wonder what you can prove based on this definition you might think it's not much but there's actually quite a bit you can prove basically the entire first month and a half to two months of this course we're doing group theory we are trying to deduce logical
8:30
consequences of this definition so it leads to a lot of structure like you might not have mattered probably the main thing will be how does the structure of the group depend on the number of elements in the group and that's especially interesting for us when we have a finite group well what's an example of a basic kind of theorem that you can derive from this so I'll call it a proposition all right in or is the identity element
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of any group is unique there's only one identity it's not part of the definition we don't say there exists unique II don't put exclamation point there and we we could have because this proposition over there is true but I don't need to assume it because I can prove it you can prove it then you don't need to us
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prove let G be a group and again there's a there's an assumed operation and it won't emphasize that let e and E prime say the identities in G by the rank I'm here is not really anything related to calculus and their derivatives here we actually will talk a little bit about derivatives later in the course or the formal think not in terms of employments so suppose you've got more than one identity essentially if you can show that these two
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things must be the same then you be done that would mean the end of these unique so what do you do we must need to use the fact that their identities that's a real key thing to realize if you are going to prove this you're going to need to use that fact the fact that you are assuming they are in identities just like an induction proofs when you're doing induction proof you I'll always need to use the
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inductive hypothesis in some way otherwise you're not really doing an induction proof
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so we got to use that fact how well you could go back to the definition and start thinking about other elements a for which this equation here would be true for both any crime and you know that you can think about them for a few minutes but it doesn't really get you anywhere until you realized that the a could also be er a prime and that's the key that equation is true if a is also replaced by E I'm just going to write two equalities here and be done essentially then e
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equals E times E prime Y since which I say since E's an identity or you promised an identity since E prime is identity yeah e is like the a a now but now if you turn around and treat the e prime like the a and use the fact that E is the identity and identity and say this equals e Prime done
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I should add a little bit more where the first equality just to be the first equality is because
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the pride is an identity of G and the second
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is because years ago okay and you should have had those kinds of reasons for a proof like this
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so since you've shown yi equals Z Prime for these conceivably different identities at first than the entity must behave it's a little tricky but when it comes down to it's pretty simple all right let's prove another proposition
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by the way again I remind you that a test you are gonna have to write down definitions and theorems oftentimes these definitions and theorems have names or at least can be described the preceding proposition was that the identity of the group is unique so if I say on the test write down the theorem that would be that's probably too easy I probably wouldn't put that right down the theorem that says the enemy if it was unique okay well that basically is that you basically
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stated I stated it for you there this one's got another name it's called the cancellation law
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let G be a group
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and suppose you've got three elements here in the group a B and C
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maybe I'll become a see these are all elements of G with the property that a be equals AC
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what do you think if you take a guess
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yeah that'd be equals like you can cancel the AIDS we're not asserting something community of activities so really it might be better to call this a left cancellation law there is a corresponding right cancellation law as well the a equals CA that equal C as well it's not an axiom it's not something we're assuming the definition so we need to prove if it is indeed true and I can pretty much say the same kind of thing that I started with here let me just
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do this the first sentence can be essentially recopy there in the proposition statement let G be a group and suppose you've got three elements with that being true
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any ideas about how to finish this perfect this is probably at this point is probably most similar to the matrix abstract matrix algebra in linear algebra rather than sector spaces you could pretend these things are matrices yeah hey guys anniversary and what should I do with it yeah multiply was you said yeah multiply both sides by a inverse technically
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maybe I should prove something else ahead of time if I multiply both sides of equation let's say the left by the same thing I do get new equal things I won't bother doing that for today maybe we'll come back to it next time left to multiply and it is good to emphasize left multiply you can also say multiply and left both sides by Anders
17:10
- yet a adversity times in parentheses a B equals a inverse times in parentheses AC again I am really doing that using something that technically should be proved but if you've got two equal things and you multiply both of those things by the same moment say on the left without the new things are also equal that's technically something that should be proved our author doesn't do it
17:53
and now you might think okay I should cancel the a and the ampersand that is true but you really should show one more step first you should use the social tippity by the associative property
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this means we can rearrange the parentheses a inverse and parentheses times B equals a inverse a and parentheses times C so EB equals E C and therefore B equals C by the definition of what an identity is since E is and now I can say the identity Oregon identity if you pervert sometimes I say me and sometimes I say Bob
19:06
it's only in proofs that are in Chapter two that you're going to need to mention the associative property usually you're I'm going to let you use it without mentioning it and again I could do a similar write cancellation law to prove that too so you can always assume for a future reference that the cancellation whoops
19:38
but the group is not commutative and in the again in the general theory it's now assumed to be commutative if you've got an equation like this a equals CA you cannot necessarily hansel the A's B does not necessarily see that the group is commutative abelian then that would be true change the order here in this a that equals AC event ends of the case
20:08
should you ever show this kind of notation a B equals AC / / cancel the A's it's probably not best but I might be forgiving
20:25
rather to think in terms of multiply your love by the members that will help you for certain problems anyway if you focus on thinking about it that way anyway here's something new that you run about sometimes helpful tool called a Cayley table it's a generalization of a multiplication table in the hall entry school let's do something let's make a kale the table for z3
21:00
I think I mentioned this last time but I'll mention it again whenever you see a symbol like this a standard group of Z 3 Z n it is assumed that the group operation here is addition modulo n in this case addition modulo 3 that is assumed how long would you use the symbol when we talk about just one operation actually later once we get into chapter it's 13 one thing I have for so now months maybe I'm going to start talking about Radia theory and
21:35
then we will include also a second operation of multiplication XIII though is actually well Z 3 Z 4 for example is not a group under multiplication on it C 3 happens to be
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and it's not necessarily
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things called zero divisors come into play I won't say what that is right now so z3 has three elements 0 1 & 2 operation is addition modulo 3 make a table just like a multiplication table from elementary school something like this
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and you put the products or in this case the some of these things in these different squares
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you might want to emphasize the operation is plus mod three here in the upper left or maybe just plus sub three for short so that's not typically done so your static 0 plus 0 is 0 0 plus 1 is 1 0 plus 2 is 2 same thing here in here so far we haven't had to mod 1 plus 1 is 2 1 plus 2 here okay now you have to mod is 0 on 3 2 plus 1 is also 0 mod 3 and 2 plus 2 is 1 mod 3 most was born divide 4 by 3 and there is a something interesting to note and is worth noting in
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each column and in each row on the main body of the table right there each group element occurs exactly wants any column looking at what circle here and any column and in any row each group element have occurred exactly once and that will happen gonna kill the table of adding our neck
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Kaleigh cables for infinite rupes I can get some sort of account I get what would happen in that case to at least in certain circumstances and really you really can't make Kayla the cables for example the group of real numbers under edition interpreted are drawn carefully
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let's see for
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now is edition mod for
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and one more element
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see how quickly you can do it if you're doing it with me it's edition mod 4 so 1 plus 2 is no longer 0 and it is 3 plus 3 is 4 which is 0 4
25:30
so again in this main body each element in any column and Eve row each element occurs exactly once you should also note that you can quickly read off the inverse of an element for example the inverse of one would be the thing you add to it to get zero which is the identity three the inverse of one is three you probably should not write it using multiplicative notation here this is an additive group it's better to write the inverse of the one it has the negative ones
26:03
and negative one equals three that's what this is saying in verse the additive inverse of is three and that equation makes sense in both the division algorithm sense and equivalents or congruence modulo four as well if you divide negative 1 by 4 what do you get is a little tricky the quotient is actually negative 1 and the remainder which can't be negative is taken to be through so under think thinking about the division algorithm the remainder is 3 when you
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divide negative 1 by 4 word and therefore that's a reason for this equality if you're thinking in terms of congruence mod 4 but the 3 line there it still works there as well because negative 1 minus 3 is a multiple of 4 it's negative 1 times 4 so you can think in either way either in terms of the original definition of mod in terms of division algorithm and the remainder or in terms of congruence mod 4 take the difference of these two things is it a multiple of that thing
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one more interesting observation in the second example here z4 is the set of group elements just zr1 to form a group unto themselves under addition or meaning you can just consider those two elements and only focus on the parts of the Cayley cable that only involves 0 & 2 this part this part this part and this part it's closed under addition mod for 2 plus 2 is 0 2 Plus 0 is 2 0 plus 0 is 0 my fourth that's interesting that's something
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special it's called that means that that set is what's called a subgroup of the original
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a sober
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of z4 under addition Log form
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our linear algebra just some spaces vector spaces it's a similar concept a subspace of a vector space is a subset that's closed under the same operation vector addition similar kind of thing happens with curves you have cell groups that's something you hear me reading about before right
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are you doing staying awake
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you and Dan grill important groups Zn u-n-d n where N is a finite positive integer are all really important examples of fun anchors remember u n is called the group of units modulo n it's the set of all as a set to set of all positive integers less than n that are relatively prime to n on their multiplication might not addition well they can kill the table for a couple examples in this one too it's called the group of units modulo n again this
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warning the UN can also stand for a different group called the unitary group whose elements are matrices that whose product is matrix multiplication we will I'm not sure if we'll talk about the unitary group at all maybe if we have time someday I'll talk about it
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the unitary major to generalize the idea of an orthogonal interest through a situation of complex number entries the movie or algebra you should have talked about over there orthogonal matrices mace matrices whose inverse was actually their transpose or special matrices a lot of complex number entries you can generalize that idea
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all right let's make Kayla candles for you ain't in you ten
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as a set you wait
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it's all positive integers less than eight that are relatively prime to eight so they better be odd odd positive integers including one one three five seven making Cayley table
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double lines along the edges here
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the operation now though is multiplication mod eight
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what is the identity one times anything is itself quickly fill in those elements again maybe you want to race me help you stay awake see if you can fill this in as fast you can see if it can be needy go
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and you can use the fact that each group element occurs only once in each row in each column
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to help fill in for example these three elements I did those last without even bothering to do a notification
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we shame them even so I was in my cat I was doing the multiplications to fill in these entries hopefully I did not make a mistake and then since I knew each group element only occurs once in each column for example I knew since in this column there was a 3-1 and 7 I needed that one had to be a PI this was five seven one I knew that had to be a three kind of like sit over or something right
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what can we observe about this an interesting thing that's occurring here that did not occur in the previous examples is the identity is occurring along this main diet and if you got the group elements in order green both over here out there that's emphasizing that each element is its own inverse and here you use multiplicative notation one in verses one and in fact in general Ian's verses always e the adverse of three is
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itself the universe of five is itself and the universe of seven is itself again I'm using powers now multiplicative notation because the operation is multiplication
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you can also see the ability an internal group again if you've got the elements in the same order assigning the top there's this inner matrix is like it's it's a symmetric matrix click along the main diagonal look for symmetry seven seven five five five five three three is a symmetric majors it's per transpose many cool itself
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all right let's do you tender
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10 is 2 times 5 so I do not want any even elements I also don't want 5 in here 1 3 7 9 under multiplication a lot of 10
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three seven nine
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go ahead and start filling it in multiplication by 10 see how fast you can
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and triathletes
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so I'm going to say somewhere
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look at the people
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location mod 10 no that's not right I was hitting a confusion through this one this should be it so sorry
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make sure you're careful to think about what group you're thinking about and I don't know why I was doing the multiplications wrong in my head or something - that looks better
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it's easy to lose track of what you're lying by why for the fire this oh I know why as I started filling in this one I did three times three is nine nine eight three hundred seven is twenty-one which is one mountain then I was thinking about the preceding example I didn't bother doing the multiplication three times I was thinking okay I was thinking of you eight examples and saying oh we're missing a five so I just provide
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this one's also a symmetric matrix emphasizing the fact that we group is abelian however we do not have ones on the diagonal all the way only here and here we do not have ones there in there three and seven are now theorem universes but one in nine
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three and verse the universe of three is sevens and vice versa the inverse of 7 is 3 by the way this also true can be proved that the inverse of a inverse it is a just like with matrices it's being Illustrated here with 3 and 7
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to be away for
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so it's different than the end venue a isn't it not just with what the elements are but also in the nature of what's an inverse of what with u8 every element was its own universe and with you 10 that's not true so it's some real fundamental sense ignoring the superficial difference in the numbers themselves and the operation operations themselves these groups you ate and you tend to truly are different there are two different groups non isomorphic
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groups with four elements that's an interesting and useful observation that if your goal is to study the structure of for example contenders to realize there's at least two distinct not isomorphic groups of order four could there be three or four we don't know yet
39:34
okay I guess it's quiz time let's go ahead and take the quiz if nine inverse 100 it's not
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