Group Multiplication Tables | Cayley Tables (Abstract Algebra)
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- Опубліковано 27 чер 2024
- When learning about groups, it’s helpful to look at group multiplication tables. Sometimes called Cayley Tables, these tell you everything you need to know to analyze and work with small groups. It’s even possible to use these tables to systematically find all groups of small order!
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Dummit & Foote, Abstract Algebra 3rd Edition
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Milne, Algebra Course Notes (available free online)
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Teaching Assistant: Liliana de Castro
Written & Directed by Michael Harrison
Produced by Kimberly Hatch Harrison
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You explained a confusing topic in the most easiest manner. Thanks a lot.
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?
@@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
@@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
@@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.
@@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.
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).
She says that there are 2 distinct groups because 1 is abelian and the rest of them are normal groups.
@@dunisanisambo9946 Actually, all of the groups are abelian! The smallest non-abelian group is the dihedral group of order 6.
Your comment helped me without spoiling the fun :)
@@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?
@@rajeevgodse2896 Nevermind, turns out I needed to check associativity - I'm surprised that isn't a given.
These videos are really extremely helpful - too good to be true - for learning overall concepts.
This lesson saved my life omg. Thank you so much for being thorough with this stuff, my professor was so vague!
love the Gilliam / Python allusions at the end. good work Harrisons, as usual.
This reminds me of Sudoku! :-)
I'm from India, your explanation was outstanding.
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 ^^
Excellent video. Clear, effortless, and instructive.
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)
She says actually a times e, but here order it's important. And yes you are allright.
I am watching and liking this in 2020!
Guess we are here because of online class due to the Covid-19 😂
Been to math village in Turkey?
@@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!)
i am in2023
2023...
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
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!
But how do you know that the associative law holds?
@@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).
@@jonatangarcia8564 Yeah how do you prove that the cayley table made by following the rules said by her always follows the associative law?
@@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
May God bless you and your channel with good fortune
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the weird thing is I have to convince myself that "+" doesn't mean "plus" anymore 😩
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Addition modulo 🙌😂
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The explanation was so simple and easy to understand.
Thank You !!!
What a crystal clear explanation. Really enjoyed the explanation here.
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.
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She should be awarded for the way she explained this concept
This is a great video that demonstrates the road map to the solution of the RSA problem.
The best video on Cayley Table..it got me thinking
Thanks, i just had this review on the midterm about it today and now its in my reccomend. Very apt.
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Oh dear god, this is the first time I actually engage to a challenge offered in a youtube video!
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.
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i love the sound fx everytime there's a contradiction
Great lecture!
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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!
What a beautiful way to teach abstract algebra! Thanks a lot.
Big fan of you... you explained very well❤❤
I'm glad that Arthur Cayley was able to speak at the end.
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!
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this was helpful as a keystone to abstract algebra, thanks for the encouragement.
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Thank you for clarification.
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I watched some of these for fun before. Now, I'm coming back to supplement the set theory in my discrete mathematics textbook.
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Best explanation in the world
That sound when the contradiction appears after 2:50 is hilariously serious
@ortomy I was looking for someone to comment this hah scared me too!