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0:00
what did we do in abstract algebra for this semester abstract algebra overview well the really big picture overview is we did groups rings and Fields that's sort of thought of as being the main topics of a typical undergraduate abstract algebra class there are other abstract algebra Topics by the way Beyond groups and rings and fields I mean one of them is Vector spaces but there are all sorts of other strange things monoids could look up
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monoids if you want division algebras okay I'm just mentioning a couple names you could look into them if you wanted to these are sort of the core of the subject starting with groups what's the key thing with groups remember math doesn't just come out of nowhere it doesn't appear out of nowhere in people's minds in a vacuum so to speak people historically think about examples first and they only come up with the abstract math when they see commonalities between examples
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the way math is presented in textbooks is the opposite of the way it's discovered you might say in real life people think about examples first they think about oh you know the integers the rational numbers the real numbers we can add integers and get another integer we can add rational numbers and get another rational number we can add two real numbers and get another real number complex numbers of course we can also do multiplication and people knew that but for the
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purposes of unifying simpler ideas under the the idea of having just one operation it's best initially to focus on addition if for no other reason that addition is simpler for example every element having an additive inverse whereas not every element has a multiplicative inverse even in these fields Q, R and C zero does not have a multiplicative inverse in the integers only plus or minus one have multiplicative inverses themselves two yes has a multiplicative inverse 1
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half but it's not an integer so it's better initially to focus on just one operation but then people notice that hey you know there are other kinds of structures and in fact this name of this course used to be different it used to be called algebraic structures there are other kinds of algebraic structures that have an operation that satisfies similar properties Zn as a set consisting of those and numbers under addition mod n is closed when we mod by
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n associative for addition identity is zero it's got additive inverses for example the additive inverse of one would be n minus one right if if I asked you in the abstract what's the additive inverse of one in here it would be n minus one why because 1 + n minus one is n is zero mod n what would the additive inverse of two be it would be n minus two right what particular examples we can think about that perhaps a little bit more easily but
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you could even think about it in the abstract it's also happens to be commutative excuse me as all these were are there any non-c commutative groups way to attach the name Abelian or non Abelian to these things of people notice hey there are certain sets of functions that are closed under function composition there's Dn the dihedral group of order 2 N right this has got order 2 N Dn itself if you wanted to write it in the abstract would include n rotations which if we're writing
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the rotation amount in degrees in the abstract we could write that as r0 R uh for example of D if n were four the minimal rotation would be a 90° rotation 360 divid 4 in general you divide 360 by n then multiply that by two 2 * 360 Over N Etc the last one would be R subn minus1 * 360 Over N talking about the degree of the rotation and then you'd have n flips as well n Reflections F1 F2 through FN about various axes of symmetry I'm not going to try to
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even draw a diagram but here's how you could represent Dn in the abstract not specifying how to make the full Cayley table I would take a lot of work but it's got order 2N you've got uh SN we saw symmetric group on N objects
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all the permutations of n objects the elements are permutations the operation is function composition do you remember how many elements this had what's its order n factorial yeah for example S3 has three factorial elements six elements what are they there's the identity Epsilon which you could also write like that while you notation you got two cycles one two 1 3 and two three right that's up to four elements remember the cycle representation is not unique one two is
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the same as 2 one 1 three would be the same as 31 two three would be the same as 32 and you've got three Cycles this one and and this one and there are no others though there are other ways of writing these one two three could be written as 231 or 312 132 could be written as 321 or 213 and you also can write these as products of two cycles with a little trick um this trick here in both two
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cycles start with a one put the last number in the three cycle in the first two cycle with the second to last and the next one Etc if you had a longer cycle this does work one does get mapped to work from right to left here this Maps one to two two gets mapped to itself so ultimately one gets mapped to two here this Maps two to one but then one gets mapped to three so ultimately does get
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mapped to three and this Maps three to itself but then three gets mapped to one just like it does there and this one is one two 13 and those are different if the Cycles are not disjoint and these are not disjoint there's a common one in them they do not necessarily commute only disjoint Cycles commute a an has as n factorial over two elements that's the set of all even permutations of n objects what's an even permutation it's one where the cycle the elements the permutations
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can be written as an even number of two cycles a product of an even number of two cycles like these two yes they are three Cycles but they're technically even permutations because they can be written as an even number of two cycles these three are not even permutations even though they are two cycles and two is even because they're writing them as one two cycle not an even number the identity permutation is even you can for example write it like
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that it can be written as an even number of true Cycles so for example A3 consists of the identity permutation and then these two three Cycles which are even permutations 3 factorial over two is three elements it is a subgroup a an is a subgroup of SN and in fact a an is a normal subgroup of SN as well in fact here's a more general fact that's worth knowing if H is a subgroup of G with index two then H is actually a normal subgroup of G that's a that's a general fact worth knowing I
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haven't really emphasized it in class I think I did emphasize it in the lectures in writing all this I'm focusing on examples I've ignored a lot of stuff in the first six five chapters or so including if in talking about normal subgroups here I guess I'm in index I'm talking about chapters seven and 8 seven and n as well what other kinds of things did we discussed that was important in group Theory we mostly focused on finite group Theory so yes while these
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are groups they were not main our main examples that we focused on WE focused on finite examples mostly because that's sort of for beginners the more interesting thing to focus on and you might say the simpler thing to focus on infinite groups are harder to understand in the abstract
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and you got things like you think about the order of a group as being the number of elements in the group you got the order of an element a if it's n that means and is the smallest positive power where this is true it doesn't mean if this is true that the order is n and's got to be the smallest positive power of a bringing you back to the identity for the order rate to be in these are not equivalent to each
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other that's the smallest positive power where this equation is true and we did lots of examples just try to understand these Concepts where you were you know Computing groups Computing K tables to help you compute orders of elements to understand these definitions and we had the idea of what is a subgroup
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real quickly a subgroup is a subset first of all that is itself a group under the exact same operation and so that would also mean that even though say for example Z4 is a subset of Z8 in terms of its elements 0 through three is a subset of the bigger set 0 through 7 it's not a subgroup but Z4 is not a subgroup of Z8 because the operation's different that's a fundamental thing to know to realize there the operations addition mod four there it's addition mod 8 it's a different
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operation therefore they're completely different groups even though they are related as being one being a subset of the other similar kind of thing happens with the groups one could be a subset of the other but the operation is different so it's not a subgroup relation on the other hand if you make an appropriate identification an appropriate isomorphism S3 for example could be thought of as a subgroup of S4 under an appropriate identification of if you got a permutation
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that permutes three elements and you're wondering can I think about it as being a permutation that permutes four elements yes just keep that fourth element fixed I that's kind of implicit in cycle notation when I write a cycle like this I'm sort of implicitly saying it's an element of S3 but there's nothing stopping me from thinking of it as an element of S4 so in a sense you could think of S3 as being a subgroup of S4 as long as you make the
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right identification of realizing okay we're taking the number four and fixing it likewise DN for example could be a subgroup of d2n because the the it's not only a subset but the operation is still the same it's still function composition so whether something's a subgroup or not it's a little bit more subtle than maybe we typically think about but we also did have the the subgroup test subgroup test that was very important we use that a
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lot not because it's the most important thing in group Theory but because it just makes for nice exercises getting practice in in doing basic kinds of exercises in group Theory so it's it's nice for homework problems and it's nice for exam problems there were there were three subgroup tests really we mostly use the fir one step subgroup test and the two step subgroup test test there was also a finite subgroup test in all cases you had to First verify that H the
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proposed subgroup was nonempty that was kind of the the zero step for the onestep subgroup test you were given One Step you were given A and B in h and you want to show a b inverses also an H for the twostep you're given A and B and H and you want to show two things show that a * B is an h and a inverse is an H or B inverse AG why bother with the two-step when you can do the one step because sometimes the two-step just feels a little easier feels a little
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easier to think about being closed under multiplication and closed under inverses individually one step at a time you might say rather than sort of doing them both at once for the finite subgroup test if G is finite you're trying to show a subset h a nonempty subset H is a subgroup it's good enough actually just verify closure turns out under multiplication but we wanted to get practice with these so we mostly use
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these cyclic groups are really important right we always were talking about cyclic groups by definition that's this that's a cyclic subgroup generated by an element it's it itself is always a cyclic every group has plenty of cyclic subgroups that doesn't mean the overall bigger group is cyclic though because it doesn't mean any of those cyclic subgroups equal the entire group when they do that's nice we're happy when when it's not when that makes things
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harder G is Click if and only if g equals a cyclic subgroup generated by some element a or some a not necessarily all a in in the group but for some a in the group then we got in the later chapters in the group Theory chapters 6 7 8 9 10 11 that dealt with more abstract ideas isomorphisms initially
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but then we also had automorphisms and ultimately homomorphisms and the author does emphasize gallan emphasizes that though these are similar ideas their roles in group Theory are different the role of the idea of an isomorphism not only being operation preserving but also one to one and onto is to identify when two groups are the same or not
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are they the same up to isomorphism or not if you can find an isomorphism from one to the other then they are if you can't and if you can prove you can't then they're not and there are various properties of isomorphisms that more easily allow you to say when two groups are not isomorphic simplest one is that they have a different number of elements they have a different number of elements you can't even come up with
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a one to one and onto function between them whether it's operation preserving or not is irrelevant if they do happen to have the same number of elements the next thing to do probably is to focus on what are the orders of the elements if they don't all match up if they don't have this for example the same number of elements of order four then they will not be isomorphic because isomorphisms preserve all group theoretical
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properties between the groups automorphisms are isomorphisms from a group to itself and for us that provided a way to construct new groups that might tell us something about the old group and inner automorphisms played a role later in also thinking about classification of groups find those kinds of ideas in chapter nine for example and 10 later on homomorphisms chapter 10 there only want operation preserving you're not necessarily one to one or onto and
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the author emphasized that the role of homomorphisms is really more to take a group and find another group that's simpler than the original that is a homomorphic image got the domain group over here you map it homomorphically onto perhaps some smaller group that's simpler probably it's it's on to but not one to one being operation preserving though is enough to preserve some of the properties of the original group but the the new group the
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smaller group The simpler group is easier to understand is sort of the big picture idea of homomorphism in between there we had other stuff we got lr's theorem of course the most important theorem in finite group group Theory quickly remind you what it says you got a finite Group G and a subgroup of g h the order of H has to divide the order of G the number of left coets of H and G is the order of G divided by the order of
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H H the order of H divides the order of G so you can actually do that fraction and it's an integer and then there are various cories
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including the orbit stabilizer theorem and we also considered examples of um external direct products and all this stuff its purpose was to lead to classification classification facts for example any group of order of prime order is cyclic
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if you got a group G of order P where p is prime it's isomorphic to zp in fact that's one classification fact another classification fact just off the top of my head is any group of order P Squared where p is prime must be a billion in fact if G has order p^ s it's isomorphic to either a cyclic group of order p^ s or X external direct product of zp with itself there's a classification fact for or groups of order 2p they either cyclic of order 2p or isomorphic to DP at least when p
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is bigger than two DP being the dihedral group of order 2p not cyclic not even AB billion and ultimately in chapter 11 there 's the fundamental theorem of finite ailion groups that really completely classifies finite ailion groups that was our main purpose with group Theory right classification kind of an abstract goal we're trying to play the role of group theorists and understand groups classifying groups is how you understand them because you're
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completely studying them you completely understand certain groups just like a a botanist would completely understand well as much as possible certain animals then after that it was rings and Fields fields are rings so really when I say rings and Fields I'm really mostly just talking Rings more broadly Fields is a specialized subject within rings but it's interesting enough in its own right to say that when we're
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studying Fields we're we're doing field Theory but ring Theory includes field Theory rings and Fields now we're allowing two operations two binary operations an addition and a multiplication are they always addition in multiplication actually they're not always addition in multiplication there are some rings where the ring operations are not literal addition and not literal multiplication we never really studied any
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of those but they do exist though sometimes you think about rings of matrices and we certainly plenty of times thought about rings of pols that was our main emphasis in fact that's our main goal with ring and field theory is understanding pols their factorizations their Roots as much as we can but we did it start abstract a ring is always a group and a billion group in fact under addition if you ignore the multiplication but then there is a multiplication which minimally satisfies
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closure it's a binary operation associativity and distributive property a new property over addition though because you're not assuming that the multiplication is communative you do need to write down two distributive properties essentially a left distributed property and a right distributed property those are two separate properties in the abstract if your ring does happen to be communative then they're the same property really and yes communative means communative with respect
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to multiplication and we say communative we don't say abil different historical developments historically that's what happened ring thir and group theorists they didn't even talk to each other really really that much is my impression so when group theorists were talking about a billion groups ring through they're like huh what's that okay I might be exaggerating but that's sort of the picture we use the word communative for multip Rings don't have to have a one they
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don't have to have a Unity though they often do tu Z for example all the integers the the even integers is a ring it's closed under addition and multiplication and it satisfies all the ring properties it's got an additive identity zero but it doesn't have a multiplicative identity there's no one although we saw rings can be weird that doesn't have a Unity but
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this set does have a Unity this has a Unity Unity under multiplication not mod four not Mod Five but mod six so you can think of it as a subing of Z6 and it is a subing of Z6 and it turns out to have a unity and the unity is um four not anything else it's not zero it's not two and there's there's no one in there but four acts like the one amazingly it's weird but true if you think about Z6 containing 0 1 2 3 four and five under multiplica addition and
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multiplication mod six that's a ring where the unity is one and not four forx is the identity with these numbers but not with uh for example four time one is not one four * 3 is not three 4 * 3 mod 6 is in fact zero which brings up another weird thing that can happen zero divisors those things happen in the abstract but we kind of like avoiding those things when we get down to it so then we start talking about integral domains no zero divisors commut
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rings with unity when there's no zero divisors Fields communitive ring with unity in which every nonzero element has a multiplicative inverse meaning it's a unit and then we started getting into stuff with ideals being the analog of normal subgroups Factor Rings instead of factor groups sometimes those Factor Rings were Fields when the idea was a maximal ideal for example well that's when it would be a field we were interested in those ones that were Fields because we were
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interested in maximal ideals because we were interested in irreducible polynomials principal ideals generated by irreducible polynomials were maximal so we were always thinking about okay is that's let's create a field for example of some finite order let's find a polynomial that's irreducible over the field create the principal ideal generated by that and consider the corresponding factoring and
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we know it's a field we did that a lot so that very fast yeah but you did that a lot and there were irreducibility tests in chapter 17 then since the last exam we talked about Vector spaces field extensions splitting Fields examples various kinds of theorems related to these how to write this field extension like we just even talked about today finite fields and ultimately Galois Theory
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okay you can go into much more depth on any of these topics group Theory ring Theory field Theory individually and there are other topics that we didn't get to it's an active field of study plenty of people still do new things in group theory that nobody's ever done before day by day okay that that's happening now in pretty much every area of math including abstract algebra
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