Group Multiplication Tables | Cayley Tables (Abstract Algebra)

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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We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition

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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[sadiqurrahman]

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

[mehulkumar]

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.

[tristanreid]

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).

[MoayyadYaghi]

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

[kirstens]

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

[kingston]

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

[waynelast]

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)

[TheFhdude]

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!

[SaebaRyo]

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 ^^

[sandeepk]

I'm from India, your explanation was outstanding.

[yvanbrunel]

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

[ozzyfromspace]

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!

[mheermance]

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.

[youtwothirtyfive]

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!

[efeuzel]

I am watching and liking this in 2020!

[fg_arnold]

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

[tomasito_]

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

[arrpit]

Just loved your content, getting easier with each passing minute

[JozuaSijsling]

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.

[chrissidiras]

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