Group Multiplication Tables | Cayley Tables (Abstract Algebra) 

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

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.

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).

This reminds me of Sudoku! :-)

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.

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!

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

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!

I am watching and liking this in 2020!

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)

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

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

Excellent video. Clear, effortless, and instructive.

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!

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...! ❤

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

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

May God bless you and your channel with good fortune

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

Just loved your content , getting easier with each passing minute

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.

Thank God Abstract Algebra is back :'''D

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

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!

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

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

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

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"

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

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

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

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

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

You teaching style is awesome

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.

It was very very good so never stop.

Excellent teaching style

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.

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

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

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

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

these videos very well written so far

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!

She should be awarded for the way she explained this concept

Loved it. So beautifully explained. 👌

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

i love the sound fx everytime there's a contradiction

Nice presentation! Thanks!

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

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

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

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

It's very helpful for everyone interested in mathematics.

That sound when the contradiction appears after 2:50 is hilariously serious

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

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

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

thanks mam .your lecture is very helpful for me

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

We need your classes ❤

very good explanation, love it!

Thank you for this video. It is really helpful

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

Appreciate you for this content.

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

So beautiful explanation

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

Great lecture!

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Nice video. Many thanks!

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

Very nice explanation

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

Awesome explanation

Thank you for clarification.

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So much fun! LOVE!

I think the 4 groups are: 1) e a b c 2) e a b c 3) e a b c 4) e a b c a e c b a b c e a c e b a e c b b c e a b c e a b e c a b c a e c b a e c e a b c b a e c b e a However I can't figure out which 3 are identical

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

Hello a good teacher..👋

Thanks you mam, it's very helpful for me.☺️☺️

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

Nicely explained

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Best explanation in the world