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You explained a confusing topic in the most easiest manner. Thanks a lot.

@zy9662

3 жыл бұрын

I'm still confused as to why she says that every element has an inverse. Is this a consequence of the suppositions or an axiom?

@shreyrao8119

3 жыл бұрын

@@zy9662 Hi, Every element has its own inverse as this is one of the conditions which needs to be met for a set to be classified as a group

@zy9662

3 жыл бұрын

@@shreyrao8119 OK so it's an axiom. Was confusing because the next property she showed (that each element appears exactly once in each column or row) was a consequence and not an axiom

@brianbutler2481

3 жыл бұрын

@@zy9662 In the definition of a group, every element has an inverse under the given operation. That fact is not a consequence of anything, just a property of groups.

@zy9662

3 жыл бұрын

@@brianbutler2481 i think your choosing of words is a bit sloppy, a property can be just a consequence of something, in particular the axioms. For example, the not finiteness of the primes, that's a property, and also a consequence of the definition of a prime number. So properties can be either consequences of axioms or axioms themselves.

The time when you say Cayley table somewhat like to solve a sudoku you win my heart. By the way, you are a good teacher.

I literally went from Struggling in my abstract algebra course to actually loving it !! All love and support from Jordan.

@Socratica

3 жыл бұрын

This is so wonderful to hear - thank you for writing and letting us know! It really inspires us to keep going!! 💜🦉

If anyone else is attempting to find the cayley tables, as assigned at the end: If you take a spreadsheet it makes it really easy. :) Also: she says that 3 of them are really the same. This part is pretty abstract, but what I think this means is that all the symbols are arbitrary, so you can switch 'a' and 'b' and it's really the same table. The only one that's really different (SPOILER ALERT!) is the one where you get the identity element by multiplying an element by itself (a^2 = E, b^2 = E, c^=E).

@dunisanisambo9946

3 жыл бұрын

She says that there are 2 distinct groups because 1 is abelian and the rest of them are normal groups.

@rajeevgodse2896

2 жыл бұрын

@@dunisanisambo9946 Actually, all of the groups are abelian! The smallest non-abelian group is the dihedral group of order 6.

@jonpritzker3314

Жыл бұрын

Your comment helped me without spoiling the fun :)

@fahrenheit2101

Жыл бұрын

@@rajeevgodse2896 Really, I thought I found one of order 5... All elements self inverse, the rest fills itself in. table (only the interior): e a b c d a e c d b b d e a c c b d e a d c a b e What have I missed?

@fahrenheit2101

Жыл бұрын

@@rajeevgodse2896 Nevermind, turns out I needed to check associativity - I'm surprised that isn't a given.

This lesson saved my life omg. Thank you so much for being thorough with this stuff, my professor was so vague!

These videos are really extremely helpful - too good to be true - for learning overall concepts.

love the Gilliam / Python allusions at the end. good work Harrisons, as usual.

These abstract algebra videos are extremely approachable and a lot of fun to watch. I'm really enjoying this series, especially this video! I worked through the exercise at the end and felt great when I got all four tables. Thank you!

Thank you for these videos. I just started exploring abstract algebra and I'm glad I found this series. You make the subject much more approachable than I expected. The groups of order 4 was a fun exercise. Thanks for the tip on the duplicates :) Subscribed and supported. Thank you!

This really helped me because application of caley's table is useful in spectroscopy in chemistry. Symmetric Elements are arranged exactly like this and then we have to find the multiplication. Thanks Socratica for helping once again ^^

I'm from India, your explanation was outstanding.

at 4:10 when she says "e times a" she means "e operating on a" so it could be addition or multiplication ( or even some other operation not discussed so far in this series)

@jeovanny1976andres

3 жыл бұрын

She says actually a times e, but here order it's important. And yes you are allright.

I have loved abstract algebra from the first time I read of it. Google describes it as a difficult topic in math but thanks to Socratica, I'm looking at Abstract algebra from a different view. Thanks Socratica

You teached in such a fantastic way that it is whole conceptualized.... And in the classroom the same topic is out of understanding! Love u for having such scientific approch...! ❤

This reminds me of Sudoku! :-)

Excellent video. Clear, effortless, and instructive.

Your Explaination is great... First time I able to understand abstract algebra.... Thank you much.. Infinite good wishes for you...😊

May God bless you and your channel with good fortune

style of your teaching and delivery of lecture are outstanding Madam Socratica

Thank you. This was an eye opener thought provoking video which cleared many of my doubts which I was searching for.

@sangeethamanickam6002

2 жыл бұрын

U

the background music makes me feel quite intense and wakes me up a lot hahhah. thnak you

Just loved your content , getting easier with each passing minute

I like how these videos are short. Helps it be digestible.

The background music in the first part of video plus the way in which socratica was talking was hypnotizing

Honestly, I watched many videos and read books to really grasp Groups but this presentation is the best hands down. It demystifies Groups and helps to understand it way better. Many thanks!

@randomdude9135

4 жыл бұрын

But how do you know that the associative law holds?

@jonatangarcia8564

4 жыл бұрын

@@randomdude9135 That's the definition of a group, that associative law holds. Now, if you take a concrete set, you have to prove that is a group (Proving that associative law holds).

@randomdude9135

4 жыл бұрын

@@jonatangarcia8564 Yeah how do you prove that the cayley table made by following the rules said by her always follows the associative law?

@jonatangarcia8564

4 жыл бұрын

@@randomdude9135 Cayley Tables are defined using a group, then, associative laws hold, because, since you use a group, and you use the elements of the group and use the same operation of the group, it holds. It's by definition of a Group

I was just thinking "hey we're playing Sudoku!" when Liliana mentioned it at 6:30. As for the challenge. The integers under addition are the obvious first candidate, but the second unique table eluded me. I tried Grey code, but no luck, then I tried the integers with XOR and that seemed to work and produce a unique table.

The explanation was so simple and easy to understand. Thank You !!!

Excellent video. Way better than most college professors. I think, these videos should be named as "demystifying abstract algebra" or rather "de-terrifying abstract algebra"

Thanks, i just had this review on the midterm about it today and now its in my reccomend. Very apt.

This is a great video that demonstrates the road map to the solution of the RSA problem.

I am watching and liking this in 2020!

@markpetersenycong8723

4 жыл бұрын

Guess we are here because of online class due to the Covid-19 😂

@halilibrahimcetin9448

3 жыл бұрын

Been to math village in Turkey?

@sukhavaho

3 жыл бұрын

@@halilibrahimcetin9448 wow - that is cool! will they make you find the prime factors of some random large number before they let you in? (İyi tatiller, BTW!)

@into__the__wild5696

Жыл бұрын

i am in2023

@user-gl7ib3lh3z

9 ай бұрын

2023...

Awesome video, well done as always. One thing that confused me was that group "multiplication" tables actually don't necessarily represent multiplication. Such as when |G|=3 the Cayley table actually represents an addition table rather than a multiplication table. I tend to get confused when terms overlap, luckily that doesn't happen too often.

Thank God Abstract Algebra is back :'''D

Thanks a lot for a clear explanation although the topic is so confusing and hard. God bless you !!!

What a crystal clear explanation. Really enjoyed the explanation here.

Love the vids! I’m binge watching the playlist before the algebra class next semester :D

I kid you not, I used to generate these exact puzzles for myself (well, mine were slightly more broad because I never forced associativity) so it's so good to finally put a name to it: *Group Multiplication Tables.* I used to post questions about this on StackExchange under the name McMath and remember writing algorithms to solve these puzzles in college (before I dropped out lol). I wish I knew abstract algebra existed back then. Liliana de Castro and Team, at Socratica, you're phenomenal!

the weird thing is I have to convince myself that "+" doesn't mean "plus" anymore 😩

@mangai3599

3 жыл бұрын

Yes, that's why you should me more general and abstract and use * instead of + !!!😂

@Abhishek._bombay

2 ай бұрын

Addition modulo 🙌😂

Wished I had you as my teacher when I was at school.

Those "contradiction" sound effects... But on a more serious note, it took me *so* long to piece these things together on my own. I *really* wish I had found Socratica years ago!

It was very very good so never stop.

She should be awarded for the way she explained this concept

Excellent teaching style

I am learning fast with you. Thank you for tutorials,

these videos very well written so far

Loved it. So beautifully explained. 👌

You teaching style is awesome

thank you, no words dear teacher, you gave me the confidence to learn math....

i love the sound fx everytime there's a contradiction

This video was so beautiful that i cannot describe it with words.

Thanks for all vedios you made, they are so exciting and easy to understand ❤❤

The best video on Cayley Table..it got me thinking

this was helpful as a keystone to abstract algebra, thanks for the encouragement.

Nice presentation! Thanks!

Thank you very much.I understood the lesson easily ❤️❤️❤️

Oh dear god, this is the first time I actually engage to a challenge offered in a youtube video!

thanks mam .your lecture is very helpful for me

Thank you very much for explaining this subject. I had a hard time to understand it.

What a beautiful way to teach abstract algebra! Thanks a lot.

I watched some of these for fun before. Now, I'm coming back to supplement the set theory in my discrete mathematics textbook.

very good explanation, love it!

Thank you for this video. It is really helpful

I'm glad that Arthur Cayley was able to speak at the end.

You are amazing, thank you for your work

you are fun to watch, really you are doing a great job, abstract algebra was never fun. Thank you

It's very helpful for everyone interested in mathematics.

Appreciate you for this content.

Great lecture!

We need your classes ❤

Big fan of you... you explained very well❤❤

Thank you soo much 💝💝 I'm not able to express my gratitude.. your videos made me love algebra.. Earlier I didn't like it

So beautiful explanation

This channel is just wayy too good! :)

I've got the 2 groups - spoilers below: Alright, so they're both abelian, and you can quickly work them out by considering inverses. There are 3 non identity elements - call them *a*, *b* and *c*. Note that these names are just for clarity, and interchanging letters still keeps groups the same, so what matters isn't the specific letters, but how they relate. One option is to have all 3 elements be their own inverse i.e. *a^2 = b^2 = c^2 = e* Alternatively, you could have some element *a* be the inverse of *b*, and vice versa, such that *ab = e*. The remaining element *c* must therefore be its own inverse - *a* and *b* are already taken, after all. This means *c^2 = e* That's actually all that can happen, either all elements are self inverse, or one pair of elements are happily married with the other left to his own devices, pardon the depressing analogy. You might be thinking: 'What if *a* was the self inverse element instead?' This brings me back to the earlier point - the specific names aren't that relevant, what matters is the structure i.e. how they relate to one another. Or you could take the point from the video - any 2 groups with the same Cayley table are 'isomorphic', which essentially means they're the 'same', structurally at least. Now, what can these groups represent? Whenever you have groups of some finite order *n*, you can be assured that the integers mod *n* is always a valid group (or Z/nZ if you want the symbols). This is easy to check, and I'll leave it to you to confirm that the group axioms (closure, identity, associativity and inverses) actually hold. In this case, the group where *ab = c^2 = e* is isomorphic to the integers mod 4, with *c* being the number 2, as double 2 is 0, the identity mod 4. (it's also isomorphic to the group of 4 complex units - namely 1, -1, i, -i under multiplication, with -1 being the self inverse element) The best isomorphism I have for the other group is 180 degree rotations in 3D space about 3 orthogonal axes (say *x*,*y* and *z*). Obviously each element here is self-inverse, as 2 180 degree rotations make a 360 degree rotation, which is the identity. It's easy to check that combining any 2 gives you the other, so the group is closed. I wasn't able to come up with any others, though I'm sure there's a nicer one. As for 5 elements? I only found 2, one of which was non-abelian. One had all elements as self-inverse, the other had 2 pairs of elements that were inverses of each other. The latter is isomorphic to Z/5Z but I've got no idea what the other is isomorphic to. Never mind, the other one isn't even a group - you need to check associativity to be safe. It's a valid operation table, but not for a group unfortunately. It does happen to be a *loop*, which essentially means a group, but less strict, in that associativity isn't necessary. There's an entire 'cube' of different algebraic structures with a binary operation, it turns out, going from the simplest being a magma, to the strictest being a group (and I suppose abelian groups are even stricter). By cube I mean that each structure is positioned at a vertex, with arrows indicating what feature is being added e.g. associativity, identity etc. Wow that was a lot.

@stirlingblackwood

9 ай бұрын

Do you know where I can find a picture of this cube?? Sounds both fascinating and like it would give some interesting context to groups.

@fahrenheit2101

9 ай бұрын

@@stirlingblackwood The wiki article for "Abstract Algebra" has the cube if you scroll down to "Basic Concepts" It's been a while since I looked at this stuff though haha - I'm finding myself reading my own comment and being intimidated by it...

@stirlingblackwood

9 ай бұрын

@@fahrenheit2101 Oh boy, now you got me down a rabbit hole about unital magmas, quasigroups, semigroups, loops, monoids...I need to go to bed 😂

@RISHABHSHARMA-oe4xc

2 ай бұрын

@@fahrenheit2101 bro, are you a Math major ?

@fahrenheit2101

2 ай бұрын

@@RISHABHSHARMA-oe4xc haha, I am now, but wasn't at the time. at the time, I think I was just about to start my first term. I know a fair bit more now, for example, any group of prime order must be cyclic. That said, I do need to brush up on Groups, been a while since I looked at it.

Such a great video. It helps me a lot !!!!!!

Nice video. Many thanks!

love this teacher

Hoo it's awesome video if I saw this video before exam I would have attended that question

Great videos !! thank you

No chance of getting an unsubscribed fan !!! 1.Veeeeeeery Clever 2.Ending of the video Booms!!!

Awesome explanation

love these videos!!

Clear explanation.thank you

Thank you for clarification.

Such a good explanation

Very nice explanation

Good job.

Pls include a video on how to find the generators of a cyclic group of multiplicative order

Excellent.i learned very clearly algebra.

Excellent, so easy to understand😘😘

Curiosity has me learning about octionions and above, this video is helpful in that endeavor

So much fun! LOVE!

Thenks so much ...im following you from Algeria 🇩🇿

@Socratica

2 жыл бұрын

Hello to our Socratica Friends in Algeria!! 💜🦉

Woww thank you so much for thr beautiful explanations

Nicely explained

Very useful videos

I highly recommend reading Wikipedia and Proof-Wiki about Cayley tables for how they are used for non associative quasi-groups and other fun stuff.

Thank you so much.

راءع جدا افتهموت اكثر من محاضرات الجامعة لان بالمحاضرة انام من ورة الاستاذ ساعة يلا نفتهم منة معنى الحلقة