GT7. The Commutator Subgroup

[@sebastianskjensvold1830]

lets go Bob, really outdid yourself with this one. Also, neck is looking better than ever.

[@MathDoctorBob]

Thanks, Sebastian! It's been 11 years, but still need size 20 neck for shirts. :)

[@darksecret965]

George from Cannibal Corpse lol

[@notimportant7869]

You do an incredible job explaining these concepts. This is making a somewhat difficult course, for me, bearable.

[@MathDoctorBob]

You're welcome! I say it better than write it. The commutator of y^-1 and x^-1 is in [G, G] by definition; we get our statement showing it is in N.

[@MathDoctorBob]

@cpaniaguam I'll check it out. I'm good with finite group basics due to representation theory, but I haven't studied deep structure. Ronan's Symmetry and the Monster is a great overview of the classification of finite simple groups for non-specialists. - Bob

[@maurizio.giaffredo]

At 11:50, isn't it supposed to be "r to the 2l" instead of "r to the 2l-2k"? The "r to the k" and "r to the -k" terms should cancel out, as the commute with "r to the -l" and their product is the identity. (This - of course - doesn't invalidate the resulting commutator) BTW, thank you for your lectures. I'm totally enjoying them!

[@MathDoctorBob]

Your welcome and good catch! Annotated.

[@stomesukel]

Amazing! I never could've imagined that D_2n /[D_2n, D_2n] voor n being even would turn out to be isomorphic to a fourgroup of Klein. Thank you for the video. :)

[@andrejnj6691]

At 14:05, for n=even, the cycle ends at 2n - 2. Shouldn't it end at n-2? For a 4-gon this would imply that ={e, r^2, r^4, r^6} instead of just {e, r^2}. Thank you for these AWESOME LECTURES!!!! :)

[@MathDoctorBob]

You're welcome, and good catch! I always get confused with dihedral group notation. For symmetries of an n-gon, I grew up with D_n instead of D_2n. I'll annotate.

[@ldb579932]

What do you mean by "want to switch" at 10:50. Do you mean "want to invert"? Seeing cr = (r^-1)c as a relation is confusing. Do you mean (cr)^-1 = (r^-1)c?

[@MathDoctorBob]

Switch as in "change the order". If I have a word in c and r (like crrcrcrrr), I'd like to be able to reduce it to r^k or cr^k. For instance, crc=(r^-1c)c=r^-1.

[@cpaniaguam]

Nice! Another cool subgroup is the Thompson subgroup. A video describing it would be awesome as well.

[@haroldalexandermacedocordo1653]

Thank you very much!! you just saved me from a nervous breakdown, It may sound silly maybe, but I didn't know how to represent the quotient group of S3/[S3 S3]. I have literally searched multiple sources and found only S3/[S3 S3] = Z2

[@MathDoctorBob]

Your welcome! Not silly at all - quotients are weird, you eventually get used to them, and how you think of them will change over time. Another way to think of this example - S_3 is isomorphic to the group of 3 x3 permutation matrices. Because determinant is multiplicative, commutators must have determinant 1, and likewise for any element of the commutators subgroup. That immediately narrows the quotient to {e} or Z/2. But any element of determinant -1 represents the other coset.

[@markshaboabi8513]

MathDoctorBob is the best

[@moimitou]

At 07:42, isn't is supposed to be "belong to [G,G]" instead of "belong to N" under the horizontal bracket? That would finish the proof for two. Thank you by the way, these are fantastic lectures!

[@rahulaggarwal08]

I guess the problem here for most beginner-level peers is that nowhere before have we seen the angle bracket notation nor have we heard about "a subgroup generated by ___ element". I was curious about this throughout the video and could not understand a thing.

[@MathDoctorBob]

GT7 means Part 7 in a series where we do those things. If you think you can learn Abstract Algebra by jumping in on problems like they were Calculus problems, you are going to have a bad time.

[@rahulaggarwal08]

@@MathDoctorBob ok, so I didn't convey my problem clearly in the comment above. I meant that the angle bracket notation and "a subgroup generated by _____ element", these two terms have been used for the very first time in this video lecture series up until this lecture. So there must be an explanation of it before mentioning it and expecting viewers to have enough prior exposure to understand it on their own. Of course, one can google it and find out what it means or watch the video multiple times to understand it, which is what I decided to do. Forgive my bad english and the inability to understand new material in the first go.

[@rahulaggarwal08]

PS- This is not to say that I don't appreciate your work. I was just trying to provide inputs in order to improve it.

[@FreeAsInFreeBeer]

@@rahulaggarwal08 I think you just missed the video "GT2. Definition of Subgroup". It defines what it means for an element to generate a subgroup and the angle-bracket notation. Watching these videos in order has covered all the necessary notation up to here. :)

[@ehsanalizadehkashtiban6839]

Thank you so much sir! 🤩🤩🤩

[@tavrion]

Thank you

[@harrisonbennett7122]

Brilliant

[@snyke1995]

Thanks a bunch! Great lecture.

[@shubhankarpalai8634]

nice one sir

[@atharvchagi8716]

10/10 Neck! Thanks Bob!

[@schirox1]

wow thank you so much

[@chiragchabbra2206]

I can not understand anything