External Direct Product of Groups Example: Is ℤ2 ⊕ ℤ2 a Cyclic Group?

The external direct product ℤ2 ⊕ ℤ2 (also written as a Cartesian product ℤ2 x ℤ2) is a non-cyclic group of order 4. We list the elements and make the Cayley table to see that all the non-identity elements have order 2, so none of them is a generator. Therefore, ℤ2 ⊕ ℤ2 is not cyclic. It is in the isomorphism class of the Klein four-group.
🔴 "Contemporary Abstract Algebra", by Joe Gallian: amzn.to/2ZqLc1J
🔴 Abstract Algebra Exam 1 Review Problems and Solutions: • Abstract Algebra Exam ...
🔴 Abstract Algebra Exam 2 Review Problems and Solutions: • Abstract Algebra Exam ...
🔴 Abstract Algebra Exam 3 Review Problems and Solutions: • Abstract Algebra Exam ...
🔴 Abstract Algebra Final Exam Review Problems and Solutions: • Abstract Algebra Final...
🔴 Abstract Algebra Problems with Solutions (including Proofs): • Abstract Algebra Probl...
🔴 Abstract Algebra Lectures Playlist: • Abstract Algebra Cours...
🔴 Check out my blog at: infinityisreal...
#AbstractAlgebra #DirectProduct #GroupTheory
Links and resources
===============================
🔴 Subscribe to Bill Kinney Math: www.youtube.co...
🔴 Subscribe to my Math Blog, Infinity is Really Big: infinityisreal...
🔴 Follow me on Twitter: / billkinneymath
🔴 Follow me on Instagram: / billkinneymath
🔴 You can support me by buying "Infinite Powers, How Calculus Reveals the Secrets of the Universe", by Steven Strogatz, or anything else you want to buy, starting from this link: amzn.to/3eXEmuA.
🔴 Check out my artist son Tyler Kinney's website: www.tylertkinn...
🔴 Desiring God website: www.desiringgo...
AMAZON ASSOCIATE
As an Amazon Associate I earn from qualifying purchases.

[@1337w0n]

For anyone wondering: this is also known as the _Klein 4 Group._

[@billkinneymath]

Thanks for adding that! I also have a video where I talk about that group from a more abstract perspective: th-cam.com/video/FufmDUPxk5E/w-d-xo.html

[@鱼升几何]

Good lecture, But the screen was shaken...

[@billkinneymath]

Thanks. Yes, I try to keep it steady, but it's hard. The table was wobbly. I didn't have any control over that, unfortunately.