302.4A: Quotient Groups

Reviewing the quotient of a group by a normal subgroup. The First and Fourth Isomorphism Theorems.

[余淼-eb]

Hello Professor Salomone, would you mind providing some references for the proof of the Fourth Isomorphism Theorem? It seems that this proof can not be found in Gallian's book. Is there any other text book recommended? Thanks so much.

[MatthewSalomone]

Mont Gum Did you know you can slow down the video playback? Click on the little "Gear" icon and try Speed 0.5. I sound a little funny but the pace is easier to keep up with :)

[xXBRDNXx]

2:48 Cayley Table :) These videos are very helpful, thank you for posting them.

[bonbonpony]

06:09 OK you've lost me here :q Is there a video that I'm missing here that explained what is the "kernel" and "image"?

[discretemacht.]

How can compute the image of (2, 3, 5) in R=Z/3Z xZ/4Z x Z/11Z

[polettix]

These lessons are amazing, thanks. Theorem 40 possibly has a typo, the epimorphism is named psi but the kernel is over phi.

[beback_]

Why would you want to do arithmetic on cosets anyway? What's the intuitive meaning / real-word examples of the concept?

[joannekoratich]

At ~5:54, Z6 is isomorphic to Z2xZ3, so I am confused as to why you list them separately. ? Otherwise I love your clean, clear, colorful approach.

[montgomeryword]

i like your videos a lot, but the speed at which you talk is so fast that i can't keep up. but still you're my favorite so far. just wish i wasn't having a panic attack the whole time.

[alexanderstephens]

In the "Quotient Mapping Theorem", you refer to an epimorphism from G to G/N, but since we aren't necessarily assuming that G/N is a group (since N is not necessarily assumed to be normal), it seems like we should not be using the word "epimorphism".

[kstahmer]

It has to be fast.  This video covers a 50 minute lecture in 11 minutes 36 seconds.  If you find this video ‘fast, ’ please consider...

1. Watch this video multiple times.  Its structure, incomprehensible in the first viewing, becomes comprehensible in subsequent viewings (Gestalt).

2. Do a quotient groups search:


Recommend Millersville:


Also recommend its Abstract Algebra 1 series:

[discretemacht.]

How can find an integer m such that R=Z/mZ? How to computer the ring homomorphism and the inverse

[MatthewSalomone]

" the kernel of any mapping out of G is isomorphic to a normal subgroup of G " Better: the kernel of any homomorphism out of G *is* a normal subgroup of G, not merely up to isomorphism.

[PunmasterSTP]

Quotient? More like "Quite excellent!" 👍

[AbuQuhafahHassan]

how to find the order of quotient group?

[bodnariucdan]

Nice and neat lecture thanks for free help !!!

[AnthonyCasadonte]

So at first I figured that quotient groups are very similar to groups but different in that the elements are not sets but cosets (I also originally wondered why they aren't called cosubgroups haha). Now if my interpretation is correct I see the quotient group to still be a set of elements namely for Z mod 12 { {0, 3, 6, 9}, {1, 4, 7, 10}, {2, 5, 8, 11} }. In set theory these sets (the cosets) are valid elements of a set. So we strip the subgroup of its binary op and use it and the other cosets for the set.

[AnthonyCasadonte]

Okay, I think I get it :) Thanks Professor. And on another note I think you should find I updated the notes pdf in dropbox. I am not used to dropbox, so I hope I did that correctly.

[AnthonyCasadonte]

Actually I think I understand this! So since there is a kernel to the mapping of a homormorphism out of G that is contained in every codomain group and every kernel is a normal subgroup, the kernel of any mapping out of G is isomorphic to a normal subgroup of G okay okay I got it! That is the 1-1 correspondence part Wow so any homomorphism you want and you have associated a normal subgroup in your codomain..quite powerful