The symmetric group -- Abstract Algebra 5

[@yoav613]

Best math lectures in youtube!

[@stearin1978]

21:22 - Cycle Notation

[@Nikolas_Davis]

circa 15:25 : isn't this proof of associativity overkill, given that function composition is always associative anyway?

[@iabervon]

Yes, but we kind of skipped discussing properties of function composition so far in this course. Also, it's worth having a few examples of the form of a proof of associativity where the steps are trivial.

[@aashsyed1277]

​@@iabervon yeah it follows from function associativity, but I think it still should be skipped

[@schweinmachtbree1013]

There is no need for the assumption that X is non-empty at 2:25 (the order of S_n is n! even for n=0: 0! = 1 and there is 1 permutation of the empty set, the empty function)

[@pennyfarrar4052]

Really helpful videos. Thanks!

[@Happy_Abe]

Technically a permutation group is a subgroup of a symmetric group. So this videos covers the symmetric group but in the video it’s defined as the permutation group but that will just be any subgroup of this.

[@MGSchmahl]

It seems to me that the group of bijections of infinite sets which leave all but finitely many points fixed is and interesting subgroup of (e.g.) S_C or S_R. Does this have a name, and is it actually interesting?

[@user-en8wj6vb7z]

Are you following Dummit and Foote book?

[@ibn_klingschor]

i feel like the punchline to all your lectures is a proof that in general the roots of a quintic polynomial can not be expressed algebraically because one needs to understand what a symmetric group is first. will you have a video for stuff like alternating groups, galois groups, galois extensions, splitting fields, etc?

[@schweinmachtbree1013]

Alternating groups yes, splitting fields maybe, Galois groups and Galois extensions almost certainly not. (this is introductory abstract algebra - before one gets to Galois theory one typically does first courses in all of group theory, ring theory, and field theory)

[@BenGeorge77]

Would this generally be considered freshman/sophomore level or upper undergraduate material for a math major?

[@schweinmachtbree1013]

freshman/sophomore.

[@homerthompson416]

@@schweinmachtbree1013 At Harvard maybe if you were taking Math 55a.

[@homerthompson416]

When I went to UCLA this material was in Math 110A / 110AH which were considered upper division undergrad courses that math majors typically took junior year, though you could take it earlier if you placed out of some of the required calculus classes through AP tests.

[@schweinmachtbree1013]

​@@homerthompson416 The symmetric group is covered in any first course on group theory, and group theory (sometimes combined with vector spaces; at least it was where I did my undergrad) is usually the first abstract algebra course one takes. So group theory is offered from the second year on all pure math degrees, and from the first year on some math degrees, presumably including Harvard.

[@SilverlightLantern]

[Spoiler alert] cycles of length 5, 2 gives order 10 cycles of length 3,4 gives order 12 cycles of length 3,7 gives order 21