Introduction to External Direct Products in Group Theory with Example: ℤ2 ⊕ ℤ3

In Group Theory from Abstract Algebra, given two groups G1 and G2, their external direct product G1 ⊕ G2 is a new group that can be constructed. As a set, it is the same as the Cartesian product G1 × G2 = {(g1,g2) | g1 ∈ G1 and g2 ∈ G2}. The binary operation for G1 ⊕ G2 is defined component-wise. That is, (g1,g2)*(g1',g2')=(g1*g1',g2*g2'). You should realize, however, that for many important examples this binary operation is addition, not multiplication. We then consider the example ℤ2 ⊕ ℤ3 and see that it turns out to be a cyclic group of order 6 (it is isomorphic to ℤ6). Links and resources =============================== 🔴 Subscribe to Bill Kinney Math: https://www.youtube.com/user/billkinneymath?sub_confirmation=1 🔴 Subscribe to my Math Blog, Infinity is Really Big: https://infinityisreallybig.com/ 🔴 Follow me on Twitter: https://twitter.com/billkinneymath 🔴 Follow me on Instagram: https://www.instagram.com/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: https://amzn.to/3eXEmuA. 🔴 Check out my artist son Tyler Kinney's website: https://www.tylertkinney.co/ 🔴 Desiring God website: https://www.desiringgod.org/ AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.