Normal Subgroups, Quotient groups and Congruence Relations

A discussion of congruence relations on a group and the origins of normal subgroups and quotient groups.

[yuwuxiong]

Thanks for the step by step explanation of the motivation, and the bigger picture in the end!

[sebastiangrijalva]

It's an honor to receive your knowledge through Youtube, I'm a fan of your book on Linear Algebra. Please continue uploading content!

[muzamelyahia]

Good explanation for more intuitive way to describe the notion of the quotient through equivalence relations. Really neat and concise. I hope you continue uploading similar videos.

[callanmcgill]

In the argument at around 27:30 you use that the equivalence relation preserves the product. Instead you want to use that alpha^-1 a ~ 1 and b^-1 beta ~ 1 implies that alpha^-1 a ~ b^-1 beta implies that a b ~ alpha beta. Good pedagogical video.

[geographymathmaster]

Thanks a lot for the video. It was a good review and was nice to see the idea behind the cryptically unmotivated definition of normality. Thank you for making this.
God bless. :)

Notes:
It seems like at 27:30 you might have assumed 2) in the second implication or needed to instead need
a \sim \alpha, b \sim \beta \to a\alpha^{-1} \sim 1, { \beta}b^{-1} \sim 1 so that you could then have the second arrow.
And maybe the third implication assumes 2) as well or just skips the steps of juggling \alpha from one side to the other.

At 41:59 it seems that by accident you wrote a \sim b instead of a\sim \alpha.

[BjarturMortensen]

I just recently started getting really interested in math (had 2 discrete math courses in uni which showed me how different "real" math is), so I am not always able to follow everything. I am probably not the target audience either, as I don't study math. I do feel like I get something out of it, and I very much like the format actually.

It's egotistical of me to say keep going, as I can't be sure I will want to watch future videos or keep my recent interest in math (or pure math) going, but I do like and appreciate the videos.