Chapter 5: Quotient groups | Essence of Group Theory

Quotient groups is a very important concept in group theory, because it has paramount importance in group homomorphisms (connection with the isomorphism theorem(s)). With this video series, abstract algebra needs not be abstract - one can easily develop intuitions for group theory!

In fact, the concept of quotient groups is one way to define modular arithmetic formally, which allows us to prove a lot of number theory theorems once we draw parallels between group theory and number theory. For example, Fermat's little theorem and Euler's totient theorem are just corollaries of the Lagrange's theorem introduced in Chapter 3 of the video series.

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If you want to know more interesting Mathematics, stay tuned for the next video!

If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!

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[mathemaniac]

Thanks for all your support in the previous video! I have never thought it would reach such a wide audience. I really like the comment section on the previous video, which involves some in-depth discussion about the model and the data I used.

This is simply a continuation of the group theory video series. Most of you just subscribed, so I would highly suggest you to watch the entire video series from the start because I referenced quite a few previous videos. If you find this video series useful, consider sharing this video and subscribe if you haven’t already.

[keeleynorman]

So helpful! I wish there were more animated explanations like this on YouTube

[monojitchatterjee]

This is actually pretty awesome! You're a good teacher.

[ThefamousMrcroissant]

Very high quality! Great script, and of course good use of Manim.

[PunmasterSTP]

Quotient groups? More like "Quite useful and dope." These videos are awesome!

[Joel-fszh]

i love you channel. thank you so much for what you do. I think about math all the time.

[carlosraventosprieto]

this was amazing!! thank you so much for these videos

[enderyu]

I think there should be a comment on how normality on H implies the cosets form a group by themselves (some demonstration or intuition). Also, is it a necessary condition as well or simply a sufficient one?

[steveschwartzm.d.]

Your conjugate intuition is brilliant!! And your relative perspective view is brilliant!! Why isn't group theory being used in QCD in physics? It seems they know all about the components, the eight fold way and other factors including shift operators and hyper charge conservation. In algebra we solve for unknowns. Maybe someone can write some software that can tell us using group theory how everything really is working on the atomic and nuclear scale. We have quarks, kaons, pions, mesons and much much more.
I wonder if this perspective capability of group theory to simplify relativistic quantum equations.

[alejrandom]

9:02 So every subgroup of an abelian group is normal?

[tomwellington]

Does this have any use for anything in engineering?

[BleachWizz]

On 4:30 I really think the axiom of choice should state you can choose any rule to pick elements from a group rather than any element from a group

[blacklistnr]

I don't like the example at the end, or maybe I'm not understanding:
The idea is to study the existing structure of e.g. a rubik's cube, but what you do in the example is you introduce some artificial structure by dividing by the arbitrary 12.

I think a better example would be divinding Z into Z- and Z+: you study manipulating just the positive numbers to also understand the negative ones(1 + 2 = 3 => -1 + -2 = -3 ). Similar to how you would study solving a face on the rubik's cube to understand how to solve the others due to symmetry.

There's also the patching of the discontinuities which I didn't hear: e.g. 1-2 takes you from Z+ to Z-. That would make it a lot more clear as how the smaller parts fit into the larger thing we're studying.

[ketv]

Just getting caught up here. I've a pretty decent understanding of algebra, Things like the proof that PSL[n](F[q]) is simple except when n=1 or n=2 and q<=3 give me a headache, but I can plough through them on the 'shut up and calculate' principle. I just don't use the Sylow theorem enough to be facile with it. I'm mostly viewing to see how other people teach the stuff.

8:43 ghg**(-1), given the conventional order of arithmetic operations, would map to (-g)+h+g, wouldn't it? Not important when you're considering only ℝ and ℤ, where the operations are commutative, but if you were to consider matrices under multiplication rather than numbers under addition, it would be a problem! Better to get the notation right at the beginning, because assuming commutativity in non-Abelian groups is a trap for the young players.

[sinkingboat]

What in God's name am I even watching here? xD